Trajectory of the Universe - Mathematics and Physics Notebook of Markus MauteNothing in life is to be feared, it is only to be understood. Now is the time to understand more, so that we may fear less.

''16 Vectors'' are a generalization of Lorentz 4 vectors. A 16 vector is understood as a $2^4$-dimensional vector in [[P-space|Polyvector Space]]. A description by means of 16 vectors therefore implies that space-time is 4-dimensional.

Here are $25$, intentionally provocative statements (I beg you to keep that in mind when reading the following), as to why superstring theory is the wrong theory when it comes to a correct description of nature - at least this is the way I see it:
# It is an "ugly" theory (that was my first impression of string theory in the 80s). Strings are just as "ugly" (and an idealisation) as are point particles. I am happy that meanwhile we have at least branes which make this problem appear a bit milder to me.
# It is dogmatic, why should one assume quantum mechanics and the Lorentz symmetry as fundamental ? People have been trying to unify physics based on these two principles for around a hundred years now and have failed.
# String theory predictions are in a realm that is not accessible to experiments. Therefore any prediction will do.
# It cannot predict the standard model (the emphasis is on "predict"). This I regard as the key failure of this ambitious theory.
# The issue of background dependence: So what is the background anyway ?
# It is lacking a fundamental principle. (A statements by Ed. Witten was, that a principle like the equivalence principle in GR is missing).
# \M-Theory is still "wishful thinking".
# Why should anomalies be bad and have to be eliminated ? They appear for example naturally if one assumes that spacetime geometry is non-associative. Hence they could well be physical ! A hint thereof can even be found in a book co-authored by the "superstring-guru" himself [???]. (Isn't this the way one should interpret an anomalous Jacobian ?). Why aren't string theoreticians consequent and in the same spirit as they try to eliminate anomalies are trying to eliminate non-trivial ("anamolous") commutators ?
# The mathematical tools used (by physicists) in string theory are quite limited and conservative (mainly group theory). Mathematicians have way more at stake. So why restricting to string theory in trying out to describe nature in a unified way.
# Where are all the supersymmtric particles ?
# Supersymmetry is a mathematical stigma.
# String theory is not fully renormalizable.
# There are (obviously) no extra dimensions, they are science fiction. (Many people like science fiction. Is this the reason why superstring theory is so popular ?)
# It predicts the landscape, i.e. it predicts "nothing". (This was the point where I gave up with this theory, being deeply disappointed and seeing all my high hopes and expectations destroyed).
# It cannot predict why spacetime (we see) is $4$-dimensional. Every theory that claims to be a TOE should do that. The nearly arbitrarily compactifiable dimensions of Calabi and Yau have yet only been good for producing knots in the brains of physicists.
# Who says that gravity has to be quantized at all and why ?
# It lacks creativity in that it very much sticks to tradition when it comes to the tools of trade used. This way one avoids "painful thinking".
# It has been cast in a form, well suited for teaching in academia - i.e. it exhibits expedience, irrespective of it describing the true nature of nature. Put it in other words, it is a theory formed such that it can be "understood" by teachers and pupils as well. (Yet I claim that there is no true understanding of the theory).
# It has lead to a dogmatic science industry. This claim has been put forward and described by others repeatedly and thus I'm not going to dwell on it any further.
# In many respects, noncommutative geometry has demonstrated to be superior to superstring theory (although not being the last word, I guess). So there are evidently alternatives.
# String theory has already failed when it comes to the description of the strong interaction (at least in parts, as QCD has turned out to be the better framework). So maybe history repeats and the description of gravity by means of the theory will flop as well.
# The theory can be seen as idiosyncratic and an outgrowth of "Western" sciences and thinking. (One could even go as far as to claim that it is a result of cold war). Superstring theory is ignoring much of excellent physics and mathematics done elsewhere in the world (in particular in Russia and other "Eastern" countries, where due to the political separation in the second half of the last century a "second school" has emerged, quite apparently contrasting the "Western" one. Since the resolution of this political separation, superstring theory seems to penetrate the Eastern countries as well.
# Superstring theory appears pretty much more as a "dumping place" of all kinds of ideas than as a fundamental theory.
# The input of string theory is a weird algebra (supersymmetry) and the output not surprisingly a weird $26$-dimensional spacetime. Yet, this is one of the hailed properties of the theory as it means that it constrains the dimensionality of "a" world and therefore must be a candidate for a "theory of everything" (TOE). Wouldn't it be more satisfactory to input another algebra instead and get as an output "our" $4$-dimensional world ? - Something that is well possible, giving up the paradigm that everything must be "Lie".
# It has been conjectured that superstrings are objects having sizes considerably larger than the Planck scale. Therefore it is conceivable that superstring theory is not a fundamental theory at all (and therefore not a TOE), rather it would still be an effective theory.

Writing down such theses, one probably cannot avoid doing injustice to some people. I'm sorry for that.

Even if only 50% of these statements are correct, superstring theory still has a big problem and cannot be a viable theory.

Honestly, these theses rest a bit on shaky grounds because I am nowhere near to having an overview over the vast field of superstring theory. (I even heard that string theoreticians themselves are not able any more to keep an overview, which is yet another argument that this theory cannot be the final answer.)

Links:
* [[Warren Siegel - Physics Parodies|http://insti.physics.sunysb.edu/~siegel/plan.html]]

Videos:
* [[Is Our Universe Unique, and How Can We Find Out? [Part 4]|http://www.youtube.com/watch?v=ms_VSVR7vhc&feature=related]]

...I'm shure it urges you to comment
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Videos:
* [[Edward Witten Lecture - Dimensional Gravity Revisited|http://www.youtube.com/view_play_list?p=36BF70A0707857EF]]

''Albert'' is a computer algebra system for doing calculus with [[nonassociative algebras|Nonassociative Algebra]].

Links:
* [[Albert website|http://www.cs.clemson.edu/~dpj/albertstuff/albert.html]]

Google books: 
* [[Genius: In Their Own Words - The Intellectual Journeys of Seven Great 20th-century Thinkers - D. R. Steele, K. Mommer|http://books.google.de/books?hl=de&lr=&id=0mc4BKpAyr0C&oi=fnd&pg=PR7&ots=JuDZNqInRc&sig=u0vDtCSmEmyD9TNOV5p9GoGtIls]] [[local|google_books/GeniusInTheirOwnWord.pdf]] bct. 0

!!!!An Essay ...
{{center{[img(306px+, )[images/Sierpinsky.gif]]}}}

Let's start with the real numbers or any set that is bijectively equivalent. We'll call this meta-reality. We can speak of it in a non-constructive way, but we cannot "write down" all real numbers explicitly.

We define physical reality as all those numbers that can explicitly be named. These will we called "algorithmically reachable" (or AR numbers)${}^*$. E.g. $1$, $0.5$, $0.25$, $0.125$ are easily "reachable" such numbers. (However each one in the sequence is harder to reach and requires more time and algorithmic brain power = energy.) Roughly speaking, algorithmic reachability is closely related to the number of after comma digits. Notice that saying $1.12$ with $12$ periodic implies that this is not a AR number, as one really had to write it down digit by digit.

The limit of algorithmical reachability is where it would require more than all the possible resources of the universe to construct a machine (a computer) that generates an explicit output of an \AR-number. All non-AR numbers will be called non-constructive numbers (\NC-numbers). An example is $\pi$ with $10^{100000000}$ digits. We can name this number but we cannot explicitly write it down. Therefore non-constructive mathematical truths can be regarded as metaphysics, as they are not part of our physical reality.

In a different universe different mathematics might prevail and the AR numbers might be different. Due to the speciality of our universe and its evolution, the set of AR numbers must be quite special with a high degree of symmetry which finds its counterparts in the "beautiful" mathematical structures that govern our mathematics.
This could also help to resolve Wigner's paradox of the "unreasonable effectiveness of mathematics in the natural sciences".
As the universe ages, it might well be that the number of AR numbers changes.
Notice, that the section of AR numbers in the continuum is completely disconnected. I.e. we cannot write down two adjacent AR numbers, or put it differently, every AR number is surrounded by infinitesimally many NC numbers. (E.g. the number $0.99999\ldots$ is a NC number and is therefore not adjacent to $1$). The measure of the AC numbers is $0$.
Maybe this explains why building reality starting with $1$ is so successful. (Decomposition of $1$ or equivalently decomposition of probabilities).

In respect to the continuum hypothesis, I claim that it will always remain a hypothesis, as we can only talk about the AR number world with $\aleph_0$ in terms of logic defined by means of NC numbers (which we can code in a computer). The real numbers are the union of the AR and the NC numbers and there is therefore a natural split of meta-reality into those two number systems with different cardinality.

Interestingly, the number of AR numbers might grow. This could be related to a growth of the forward light cone which makes causally connected spacetime grow and hence the available resources to build a computing machine. However there is no algorithmic means within a subset of AR numbers to determine how they will grow, the only way to find out is to let the whole set of AR numbers grow, i.e. let our universe grow. Therefore only the algorithm representing the whole of the universe "knows what comes next". Every subset (i.e. any human brain) is limited in its ability to predict the exact growth of any subset of AR numbers. Furthermore the change of any such subset depends on the change of the whole algorithm, i.e. in a way is determined (I hesitate to use the word deterministic) by the whole algorithm representing "our" reality.

Certain very big symmetries within the continuum just don't exist in mathematics because they are not algorithmically reachable. I.e. there is a limit on the size of mathematical structures. But within the AR numbers we can hope for a classification of structures to a certain degree. And those are the ones that matter anyway, because they are part of reality and might practically serve to "describe it". Therefore the claim is, that no NC number will ever be useful in a physical theory. Or put it differently, the theory can equally well be defined without referring to them. They are part of the realm of metaphysics.

AR numbers are physically realizable numbers, be it in the brain or in a machine/computer. So they really exist, can be autentically represented by a physical system. But how could a physical system represent a line or a circle, which are mathematical idealisations. Any physical system can at best "approximate" them (and give us a pseudo smooth representation of these abstract objects).

To sum up: Reality is a non-dense subset of the continuum.

${}^*$ After having written this essay I found the notion of "feasible number" in literature, which I regard as equivalent.

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Quantization can spoil classical symmetries. As a consequence, symmetry currents, whose classical conservation is assured by [[Noether's theorem|Noether Theorem]], cease to be conserved. Such currents are called ''anamolous''. They possess an anomalous divergence, and the coupling of gauge fields to this current becomes problematical.
The problem afflicts:
* Continuous chiral symmetries in any even-dimensional space-time
* Gravitational symmetries of massless (Weyl) fermions  in space-times with dimensionality $4k + 2, k = 0,1,2,...$
* Discrete symmetries (P,T) in odd dimensions
* Scale/conformal symmetries in any dimension: ''Weyl anomaly'' (also called ''Trace Anomaly'' or ''Conformal Anomaly''), that is, the breakdown of conformal invariance upon quantization. Classically, this invariance leads to the vanishing of the energy–momentum tensor, while its breakdown in the quantum theory leads to a nonvanishing value.
The mathematical connection has come to a sharper focus in the characterization of an anomalous gauge theory by the fact that commutators of gauge transformation generators are anomalous and do not follow the Lie algebra of the gauge group.
For the gauge fields one has
\[
D_\mu D_\nu G^{\mu\nu} = D_\mu J^\mu \propto  G_{\mu\nu} \tilde{G}^{\mu\nu} \ne 0 = \text{ anomaly}
\]
with
\[
G_{\mu\nu}\tilde{G}_{\mu\nu} = \partial_\mu K_\mu
\]
and
\[
K_\mu = 2\epsilon_{\mu\nu\alpha\beta} \left( A_\nu \partial_\alpha A_\beta + \frac{2}{3} i g A_\nu A_\alpha A_\beta \right)
\]

Papers:
* [[Twenty Years of the Weyl Anomaly - M. J. Duff|http://scholar.google.de/scholar?hl=de&lr=&cites=2182335914955876668]] {{t100Cite{[[pct. 183|http://scholar.google.de/scholar?hl=de&lr=&cites=2182335914955876668]]}}}
* [[Topological Invariants, Instantons and Chiral Anomaly on Spaces with Torsion - O. Chandia, J. Zanellia|http://arxiv.org/PS_cache/hep-th/pdf/9702/9702025v1.pdf]] [[pct. 66|http://scholar.google.de/scholar?hl=de&lr=&cites=13079915837245870961]]
* [[On the Chiral Anomaly in Non-Riemannian Spacetimes - Y. N. Obukhov, E. W. Mielke, J. Budczies, F. W. Hehl|http://www.citebase.org/fulltext?format=application/pdf&identifier=oai:arXiv.org:gr-qc/9702011]] [[pct. 28|http://scholar.google.de/scholar?hl=de&lr=&cites=215116949989095649]]
* [[Non-abelian Chiral Anomalies and Wess-Zumino Effective Actions - J. L. Petersen|http://th-www.if.uj.edu.pl/acta/vol16/pdf/v16p0271.pdf]] [[pct. 21|http://scholar.google.de/scholar?hl=de&lr=&cites=16546363659818067930]]
* [[The Axial Anomaly Revisited - P. Federbush|http://www.ma.utexas.edu/mp_arc/c/96/96-316.ps.gz]] [[pct.9|http://scholar.google.de/scholar?hl=de&lr=&cites=5532133868412867919]]
* [[What’s Wrong with Anomalous Chiral Gauge Theory? - T. D. Kieu|http://psroc.phys.ntu.edu.tw/cjp/download.php?d=1&pid=834]] [[pct. 6|http://scholar.google.de/scholar?hl=de&lr=&cites=10696231753488158395]]
* [[Effective Action, Conformal Anomaly and the Issue of Quadratic Divergences - K. A. Meissner, H. Nicolai|http://arxiv.org/PS_cache/arxiv/pdf/0710/0710.2840v2.pdf]] [[pct. 6|http://scholar.google.de/scholar?hl=de&lr=&cites=13813838645528182828]]
* [[Algebraic versus Topologic Anomalies - V. Aldaya, M. Calixto, J. Guerrero|http://repositorio.bib.upct.es/dspace/bitstream/10317/519/6/avt.pdf]] pct. 0
Links:
* [[Schoolarpedia - Axial anomaly|http://www.scholarpedia.org/article/Axial_anomaly]]

According to the ''Antropic Principle'' the universe is the way it is because we exist. 

The term "anthropic principle" was introduced by B. Carter in 1974 and defined in a non-controversial form: "What we can expect to observe must be restricted by the conditions necessary for our presence as observers". He calls this the ''Weak Anthropic Principle'' and defines the controversial ''Strong Anthropic Principle'' in the form: "The universe necessarily has the properties requisite for the existence of life at some stage in its history".

The ''Antibracket Formalism'' is used to [[quantize|Quantization]] a [[gauge theory|Gauge Transformation]] and appears to be the most powerful method to do this.
Contrary to the Faddeev\-Popov  functional integration procedure it does not fail for:
* So called "open gauge algebras" which only close on-shell. Such algebras occur in gravity and [[supergravity|Supergravity]].
* "Reducible theories" where the gauge generators are all not independent,
* [[Yang-Mills theories|Yang-Mills Theory]], if one considers exotic gauge-fixing procedures for which "extraghosts" appear.

For each field and ghost one introduces an antifield, thereby doubling the total number of original fields. The antibracket is an odd non-degenerate symplectic form on the space of fields and antifields and plays the role of the Poisson bracket. As a consequence, Hamiltonian concepts, such as canonical transformations, can be formulated and used. The original classical [[action|Action Principle]] is extended to a new action, in an essentially unique way, to arrive at a theory with manifest [[BRST symmetry|BRST Quantization]]. The antibracket formalism proceeds via the functional integral, hence the powerful techniques of functional integration are available.

Papers:
* [[Antibracket, Antifields and Gauge-Theory Quantization - J. Gomis, J. Paris, S. Samuel|http://arxiv.org/PS_cache/hep-th/pdf/9412/9412228v1.pdf]] {{t100Cite{[[pct. 212|http://scholar.google.de/scholar?cites=8635365789647031556&hl=de&as_sdt=2000]]}}}

Links:
* [[WIKIPEDIA - Apophis|http://en.wikipedia.org/wiki/99942_Apophis]]

The ''Ashtekar Formalism'' is a canonically equivalent description of [[general relativity|General Relativity]].

In the ''Ashtekar Formalism'' the [[SL(2,C)]]-invariance of the [[Einstein equations|Einstein Field Equations]] is written in such a way as to have a manifest invariance under the complex group [[SU(2)]] so that the gauge fields can be separated into self-dual and anti self-dual parts. 

The main advantage of the Ashtekar formulation is that it allows to take the [[spin-connection|Spin Connection]] as the canonical variables while the inverse [[tetrads|Tetrad]] are taken as the conjugate variables, which allows to formulate a theory for loop quantum gravity.

Papers: 
* [[New Variables for Classical and Quantum Gravity - A. Ashtekar|http://www.montgomerycollege.edu/Departments/planet/planet/Numerical_Relativity/QLG/PRL57p2244_1986.pdf]] {{t500Cite{[[pct. 804|http://www.montgomerycollege.edu/Departments/planet/planet/Numerical_Relativity/QLG/PRL57p2244_1986.pdf]]}}}

A curve $x^\sigma = x^\sigma(\tau)$ is said to be ''autoparallel'' (with respect to the [[connection|Connection]] $\Gamma^\sigma_{\mu\nu}$) if the covariant derivative $ \frac{D}{D\tau} \equiv \frac{d}{d\tau} + \Gamma^\sigma_{\mu\nu}$  along the curve of the tangent vector $u^\sigma(\tau) = \frac{dx^\sigma(\tau)}{d\tau}$ is zero at every point:
\begin{eqnarray}
&&\boxed{\frac{Du^\sigma }{D\tau}  = \frac{d u^\sigma}{d\tau} + \Gamma^\sigma_{\mu\nu} u^\mu u^\nu \equiv  0}
\end{eqnarray}
Using $\frac{d}{d\tau} = \frac{d x^\mu(\tau)}{d\tau} \frac{\partial}{\partial x^\mu} = u^\mu(\tau) \frac{\partial}{\partial x^\mu} $ this can also be expressed as
\begin{eqnarray}
\frac{D u^\sigma}{D\tau}  & = & u^\mu \frac{\partial u^\sigma}{\partial x^\mu} + \Gamma^\sigma_{\mu\nu} u^\mu u^\nu  \\
& = & u^\mu \left (\frac{\partial u^\sigma}{\partial x^\mu}+ \Gamma^\sigma_{\mu\nu}  u^\nu \right ) \\
\end{eqnarray}
Hence the autoparallelity condition reads
\begin{eqnarray}
&&\boxed{u^\mu D_\mu u^\sigma = u^\mu \frac{Du^\sigma}{D x^\mu} = 0}
\end{eqnarray}
with $D^\mu$ the [[covariant derivative|Covariant Derivative]] in respect to the coordinates $x^\mu$.

Only if [[torsion|Torsion]] is totally antisymmetric ($T_{ijk} = -T_{jik} = -T_{ikj}$) are autoparallels identical to [[geodesics|Geodesic Equation]] (i.e. the connection reduces to a [[Levi-Civita connection|Levi-Civita Connection]]). That is to say autoparallels are a ''generalization of geodesics''.

!!!!Physical interpretation
An autoparallel describes a trajectory of an intertial system, that is one, in which no forces act. Therefore the momenta in this system are conserved. Locally this is a system with an orthonormal basis $\{\mb e_a\}$.
Therefore
\begin{eqnarray}
0 & = &\mb a = \frac{d}{d\tau} \mb p = \frac{d  p^a}{d\tau}\mb e_a = \frac{d}{d\tau} ( p^a \mb e_a) \\
   & \equiv & \frac{d}{d\tau} (p^\mu \mb e_\mu) \\
   & = & \frac{D  p^\mu}{D\tau} \mb e_\mu \\
\end{eqnarray}
Hence
\[
\frac{D  p^\mu}{D\tau} = 0
\]
!!!!Generalizations
See [[Polyvector Autoparallelity]].

Papers:
* [[New Action Principle for Classical Particle Trajectories in Spaces with Torsion - P. Fiziev, H. Kleinert|http://arxiv.org/PS_cache/hep-th/pdf/9503/9503074v1.pdf]] [[pct. 36|http://scholar.google.de/scholar?cites=1374319893785598759&hl=de]]
* [[Autoparallels From a New Action Principle - H. Kleinert and A. Pelster|http://arxiv.org/PS_cache/gr-qc/pdf/9605/9605028v2.pdf]] [[pct. 13|http://scholar.google.de/scholar?cites=15539175244601932971&hl=de]]

See [[Kalb-Ramond field|Kalb-Ramond Field]].

''BRST Quantization'' (or the ''BRST Formalism'') is a differential geometric approach to performing consistent, [[anomaly|Anomaly]]-free perturbative calculations in a non-abelian gauge theory. It is due to C. M. Becchi, A. Rouet, R. Stora and I. V. Tyutin.
In the BRST approach, one selects a perturbation-friendly __gauge fixing procedure__ for the action principle of a gauge theory using the differential geometry of the gauge bundle on which the field theory lives. One then quantizes the theory to obtain a Hamiltonian system in the interaction picture in such a way that the "unphysical" fields introduced by the gauge fixing procedure resolve gauge anomalies without appearing in the asymptotic states of the theory. The result is a set of Feynman rules for use in a Dyson series perturbative expansion of the S-matrix which guarantee that it is unitary and renormalizable at each loop order—in short, a coherent approximation technique for making physical predictions about the results of scattering experiments.

After quantization there remains a nilpotent, odd, global symmetry involving transformations of both fields and [[ghosts|Ghost Field]] which is called ''Becchi\-Rouet\-Stora\-Tyutin (BRST) Symmetry''.

Links:
* [[Bel and Bel-Robinson Tensors|http://www.phy.olemiss.edu/~luca/Topics/b/bel.html]] [[local|html/bel.html]]

''Berry’s Phase'' is an example of [[holonomy|Holonomy]], the extend to which some variables change when other variables or parameters characterizing a system return to their initial values.

Papers:
* [[Berry's Phase in the Relativistic Theory of Spinning Particles (1987) - I. Bialynicki-Birula, Z. Bialynicka-Birula|http://www.cft.edu.pl/~birula/publ/BerryPhase.pdf]] [[pct. 19|http://scholar.google.de/scholar?cites=16175902696327481478&as_sdt=2005&sciodt=2000&hl=de]] TRD
* [[The Berry Phase of D0-Branes - C. Pedder, J. Sonner, D. Tong|http://aps.arxiv.org/PS_cache/arxiv/pdf/0801/0801.1813v3.pdf]] [[pct. 6|http://scholar.google.de/scholar?cites=14313039222530863931&as_sdt=2005&sciodt=2000&hl=de]]

The ''"Big Desert" Hypothesis'' asserts that, apart from the [[Higgs boson|Higgs Mechanism]], all particles relevant at the [[grand unification|GUT]] scale are already discovered.

* Is there a principle - supposedly mathematical in nature - which allows one to derive all of physics and if so, what is it ?
* What is [[consciousness|Consciousness]] ?
* Can we be [[immortal|Immortality]] ? If we are not by nature, can we engineer immortality or at least (considerably) increase human lifespan ?

A ''Bilinear Covariant'' is a product of the form $\mb {\bar \Psi \Gamma \Psi}$, where $\mb {\Gamma}$ is a $4 \times 4$ complex matrix and $\bs \Psi$ and $\mb {\bar \Psi}$ are a [[Dirac spinor|Dirac Spinor]] and its adjoint respectively.
$16$ linearly independent $\mb {\bar \Psi \Gamma \Psi}$ are maximally possible (which actually span the underlying Clifford algebra [[Cl(1,3)]]). They are given by:
\begin{eqnarray}
S'(\mb x') &= &S(\mb x) = \mb {\bar \Psi}(\mb x) \bs {\Psi} (\mb x)\quad \text{ (1 scalar)} \\
&=& \vert \bs \psi_a \vert^2 +  \vert \bs \psi_b \vert^2 -  \vert \bs \psi_c \vert^2 - \vert \bs \psi_d \vert^2\\
&=& \psi_1^2 +  \psi_2^2 +  \psi_3^2 +  \psi_4^2 - \psi_5^2 -  \psi_6^2 -  \psi_7^2 -  \psi_8^2 \\
\mb V'^{\mu}(\mb x')& =& \mb{V^\mu}(\mb x) =  \Lambda^\mu_\nu \bar {\mb \Psi}(\mb x) \bs \gamma^\nu \bs {\Psi} (\mb x) \quad \text{ (4 vectors)} \\
\mb B'^{\mu\nu}(\mb x')& =& \mb B^{\mu\nu}(\mb x) =  \frac{i}{2} \Lambda^\mu_\rho \Lambda^\mu_\sigma \mb {\bar \Psi} (\mb x) [\bs \gamma^\rho,\bs \gamma^\sigma] \bs {\Psi} (\mb x) \quad \text{ (6 bivectors/antisymmetric tensors)} \\
\mb T'(\mb x') &= &\mb T(\mb x) = \det(\bs \Lambda) \Lambda^\mu_\nu \mb {\bar \Psi(\mb x)} \bs \gamma^\nu \bs \gamma^5 \bs {\Psi}(\mb x) \quad \text{ (4 pseudovectors)} \\
P'(\mb x') &= &P(\mb x) = \det(\bs \Lambda) \mb {\bar \Psi}(\mb x) \bs \gamma^5 \bs {\Psi} (\mb x) \quad \text{ (1 pseudoscalar)} \\
\end{eqnarray}
where the Dirac representation of the [[gamma matrices|Gamma Matrices]] was used.

These bilinear covariants play an important role in determining the possible Lorentz invariant couplings of the Dirac spinor field to other fields.

!!!!Examples
Coupling between the Dirac field and the
* electromagnetic field: $\mathcal L \propto \bar { \mb \Psi}(\mb x) \bs \gamma^\mu \bs {\Psi}(\mb x)  A_\mu(\mb x)   $
* pseudoscalar meson field: $  \mathcal L \propto \mb {\bar \Psi (\mb x)} \bs \gamma^5 \bs {\Psi} (\mb x) \Phi (\mb x)$

<html><center><iframe name="content" src="http://www.markus-maute.de/trajectory/advert_60.html" width=51% height=86></iframe></center></html>Papers:
* [[Observables, Operators, and Complex Numbers in the Dirac Theory - D. Hestenes|http://www.intalek.com/Index/Projects/Research/Observ-opers.pdf]] {{t100Cite{[[pct. 100|http://scholar.google.de/scholar?cites=1346678541492404014&as_sdt=2005&sciodt=2000&hl=de]]}}}
* [[The Electromagnetic Form of the Dirac Electron Theory - A. G. Kyriakos|http://redshift.vif.com/JournalFiles/V11NO2PDF/V11N2KYR.pdf]] [[pct. 3|http://scholar.google.de/scholar?cites=5255194155955097275&hl=de&as_sdt=2000]] 

Lectures:
* [[Spatial Reflection, Bilinear Covariants, Charge Conjugation, and Time Reversal|http://www.physics.buffalo.edu/phy511/Chapter%2012%20RQM.pdf]]

''Birefringence'' or ''Double Refraction'' is the decomposition of a ray of light into two rays (the ordinary ray and the extraordinary ray) when it passes through an anisotropic material.
For a medium having no birefringence means that it has a single lightcone.

Papers:
* [[On the Physical Interpretation and the Mathematical Structure of the Combinatorial Hierarchy - T. Bastin, H. P. Noyes, J Amson, CW Kilmister|http://www.slac.stanford.edu/pubs/slacpubs/2250/slac-pub-2304.pdf]]  [[pct. 32|http://scholar.google.de/scholar?cites=9803140199832021391&hl=de]]
* [[A Short Introduction to BIT-STRING PHYSICS - H. P. Noyes|http://arxiv.org/PS_cache/hep-th/pdf/9707/9707020v1.pdf]] [[pct. 17|http://scholar.google.de/scholar?cites=15737130025158251635&hl=de]]
* [[A Finite Particle Number Approach to Quantum Physics - H. P. Noyes|http://www.slac.stanford.edu/pubs/slacpubs/2750/slac-pub-2906.pdf]] [[pct. 7|http://scholar.google.de/scholar?cites=346310480967340547&hl=de]]
* [[Fractal Strings as the Basis of Cantorian-fractal Spacetime and the Fine Structure Constant - C. Castro|http://www.scribd.com/doc/13049520/Fractal-Strings-and-Cantorian-Spacetimes-]] pct. 0
* [[From Bit-Strings (part way) to Quaternions - H. P. Noyes|http://www.slac.stanford.edu/pubs/slacpubs/5250/slac-pub-5431.pdf]] pct. 0

Google books:
* [[The Theory of Indistinguishables - A. F. Parker-Rhodes|http://books.google.com/books?id=hHG0IuGm2V8C&dq=The+Theory+of+Indistinguishables&printsec=frontcover&source=bl&ots=LcSyuRupqm&sig=WyIUI6i63Y1Mxcx9o3Sjc7BZ9iQ&hl=de&ei=RZn0SbCJJoKO_Qal56jsCQ&sa=X&oi=book_result&ct=result&resnum=7#PPP1,M1]] [[local|google_books/TheTheoryOfIndistinguishables.pdf]] [[bct. 38|http://scholar.google.de/scholar?cites=5917646935173349348&hl=de]]

>The objective world simply is, it does not happen.
> - Hermann Weyl -

The ''Block Universe'' view of the universe affords equal (ontological) status to all points in space-time, thus regarding temporality as an illusory human construct with no reference to reality.

This view may have come about as a consequence of the usual way of modelling the mathematics of general relativity as a theory about the curvature of an eternally existing arena of space-time.

Papers:
* [[How Time Passes - G. Franck|http://www.iemar.tuwien.ac.at/publications/GF_2003c.pdf]] [[pct. 2|http://scholar.google.com/scholar?hl=de&lr=&cites=9298222414182727390&um=1&ie=UTF-8&ei=fRPCSo39GZXsmwPO-qCxBg&sa=X&oi=science_links&resnum=1&ct=sl-citedby]]

Links:
* [[WIKIPEDIA - Boltzmann Brain|http://en.wikipedia.org/wiki/Boltzmann_brain]]

Bore Hole experiments allow for testing possible violations of Newton's inverse-square law. Such violations have been reported and are referred to as ''Bore Hole Anomaly''.

Papers:
* [[Test of Newton's Inverse-Square Law in the Greenland Ice Cap - M. E. Ander, M. A. Zumberge, T. Lautzenhiser, R. L. Parker, C. L. V. Aiken, M. R. Gorman, M. M. Nieto, A. P. R. Cooper, J. F. Ferguson, E. Fisher, G. A. McMechan, G. Sasagawa, J. M. Stevenson, G. Backus, A. D. Chave, J. Greer, P. Hammer, B. L. Hansen, J. A. Hildebrand, J. R. Kelty, C. Sidles, J. Wirtz|http://www.whoi.edu/science/AOPE/people/achave/Site/Next_files/28.pdf]] [[pct. 40|http://scholar.google.com/scholar?hl=de&lr=&cites=10079090251362993710&um=1&ie=UTF-8&ei=57jBSqmvMZOe4QbnxMyLCA&sa=X&oi=science_links&resnum=1&ct=sl-citedby]]

Papers:
*[[Quantum Physics, Semester 1 2008 - J. D. Cresser|http://www.physics.mq.edu.au/~jcresser/Phys301/LectureSlides/Phys301Lectures.pdf]] - (download very slow)
* [[Quantum Physics Notes, Chapter 9: Operations on States - J. D.  Cresser|http://www.physics.mq.edu.au/~jcresser/Phys301/Chapters/OperationsOnStates.pdf]]

Unlike bosonic [[p-branes|P-Brane]] which can be formulated in arbitrary spacetime dimensions $D$, [[supersymmetric|Supersymmetry]] $p$-branes only can be formulated for certain combinations of $d = p + 1$ and $D$. This restriction, enforced by supersymmetry, gives rise to the so called ''Brane Scan''.

<html><center><img src="images/brane_scan.jpg" style="width: 350px; "/></center></html>
The brane scan only tells us which branes are not forbidden by supersymmetry. If these branes actually exist as solutions to any supersymmetric field theory is another question.

In 11 space-time dimensions one only has 2 possible $p$-branes, therefore in case of [[M-theory|M-Theory]] supersymmetry is quite restrictive in respect to possible p-branes.

In so called ''Brane World Scenarios'' which are cosmological models with extra dimensions it is assumed that ordinary matter is confined to a surface, called a brane, embedded in a higher dimensional spacetime.

These models are in contrast with [[Kaluza-Klein models|Kaluza-Klein Theory]] where matter fields also extend to the extra compact dimensions.

Example: [[Randall-Sundrum model|Randall-Sundrum Model]].

Papers:
* [[Einstein-Cartan Gravity Excludes Extra Dimensions - N. J. Poplawski|http://arxiv.org/PS_cache/arxiv/pdf/1001/1001.4324v1.pdf]] pct. 0
* [[Gravity, Higher Dimensions, Nanotechnology and Particle Physics - M. Ito|http://www.iop.org/EJ/article/1742-6596/89/1/012019/jpconf7_89_012019.pdf?request-id=b3f418bd-8e76-40c4-b483-15282196284f]] pct. 0

<<player id=1 windows audio/HeimTheory.mp3 220 45>>
''Burkhard Heim'' has developed a theory (a.k.a. ''Heim Theory'') which is an attempt to quantize gravity. It reproduces remarkably well many of the known fundamental parameters of nature. Still the theory has found little recognition as an established physical theory in the physicists community, especially when compared to other theories of quantum gravity. One reason might be that Heim's work has not been subject to peer reviewing.

A basic idea of the theory is that the gravitational connection is composed of the classical symmetric [[Christoffel part|Christoffel Symbols]] and an antisymmetric part. Consequently the [[stress energy tensor|Stress Energy Tensor]] also contains an antisymmetric part. The latter represents matter and is quantized.

The fundamental building block of physical space is taken to be an area element, called a ''Metronom'' $\tau$. It is a constant of $\mathbb R^3$, i.e. independent of it's curvature and is therefore regarded as a natural constant, given by
\[
\tau = \frac{3\pi\hbar G}{4 c^3} =  6.15 \cdot 10^{-70} m^2
\]

Compare this with the [[Planck area|Planck Units]] given by
\[
l_P^2 = \frac {\hbar G} {c^3} = 2.61223 \cdot 10^{-70} m^2
\]

!!!!A bit of the mathematics of Heim's theory
!!!!!Discrete functions
Smooth $C^\infty$-functions are replaced by discrete function $\varphi(n)$ with $\varphi, n \in \mathbb N$, called ''Metronomic Functions''.

!!!!!Discrete differential calculus
The argument of a metronomic function $\varphi(n)$, $n \in \mathbb N$ can only be varied in discrete steps $\pm 1$. This requires a revision of differential calculus for such functions.

Differential operators $\partial$ are replaced by so called ''Metronomic Operators'' which are a discrete realisation of the former. Given a smooth function $f \in C^\infty$ and letting $\partial$ act on it, it "picks" another function $g$ of the $C^\infty$-space. In a special case if one requires that $f$ and $g$ are linearly dependent, i.e. $f = \lambda g$, one gets an [[eigenvalue equation|Eigenvalue Theory]] $\partial f = \lambda f$.

In case of Heim's theory $f$ is assumed to be discrete a priori, i.e. $f_i \in \mathbb N$. Therefore the analogue of the partial differential operator, the metronomic operator, denoted $\partial^{\!\!/}_i$ "picks" a discrete subset (a number series) instead. It is therefore also called ''Selector''. The action of selecting a subset of $g_i \subset f_i$ by  $\partial^{\!\!/}_i$ is indicated by a semicolon, i.e. $g_i \equiv \partial^{\!\!/}_i; f_i$.
A characteristic equation therefore reads:
\[
\partial^{\!\!/}_j; \phi_i = C_{ij} \phi_i
\]
with the eigenvalues $C_{ij} \in \mathbb N$.


Algebraic properties of selectors:
* Existence of a zero-selector $0$ with $0;n = 0$.
* Existence of a unity selector $E$ with $1;n = 1$.
* Distributivity and associativity in respect to addition and multiplication.
* Commutativity in respect to addition.
* Non-commutativity in respect to multiplication.

!!!!!Pros
* As it is a discrete theory based on natural numbers, spacetime singularities cannot appear. It furthermore circumvents the philosophical and mathematical problems associated with the [[continuum|Continuum Hypothesis]].
* The theory avoids the dichotomy between smooth eigenfunctions and discrete eigenvalues, only the latter being accessible to experiments. It can be dispensed with a philosophical digress into the ontological meaning of real-valued wavefunctions.
* The space-time manifold is spectral in its nature as is the case in [[non-commutative geometry|Noncommutative Geometry]]. The arguments made there in favour of such a structure therefore also apply here.
!!!!!Cons
* Predicts a neutral electron and two more neutrinos, the latter being excluded by LEP experiments.
* The theory is quite hard to understand, in parts due to the fact that B. Heim uses his own mathematical language. (I have bought two of his books and they are extremely densely packed with formula. I believe, that it is indispensable to include the original literature in the reading to understand the theory better. My impression is, that the Heim theory and the mathematics used are quite serious. By the way, B. Heim was a student under supervision of no one less than Carl Friedrich von Weizäcker).

Papers:
* [[Heim's Theory of Elementary Particle Structures (1992) - T. Aauerbach, I. Ludwiger|http://www.scientificexploration.org/journal/jse_06_3_auerbach.pdf]] [[pct. 2|http://scholar.google.de/scholar?cites=11972683543239715477&hl=de&as_sdt=2000]]

Links:
* [[WIKIPEDIA - Heim Theory|http://en.wikipedia.org/wiki/Heim_theory]]

Lectures:
* [[Grundgedanken einer einheitlichen Feldtheorie der Materie und Gravitation|http://www.engon.de/protosimplex/downloads/02%20heim%20-%20mbb%201.2.pdf]]
@@display:block;text-align:right;font-size:12pt;font-family:Scripts;{{stretch{[img[My comments ...|images/Heim.jpeg][Comments]]}}}&nbsp;@@

The ''CPT Theorem'' says that physics should be invariant under a combination of a charge-, parity- and time [[transformation|CPT-Transformations]].
It follows that any [[Lorentz invariant|Lorentz Transformation]] local [[quantum field theory|Quantum Field Theory]] with a Hermitian Hamiltonian must have a CPT symmetry.

Usually a CPT violation is considered to be due to [[breaking of Lorentz invariance|Lorentz Violation]]. However, it is possible, that locality is the less fundamental requirement than is Lorentz invariance, therefore, experiments searching for CPT violations of both types are of interest.

So far no signals of CPT violations have been observed despite numerous experimental tests.

Papers:
* [[Testing of CP, CPT and Causality Violation with the Light Propagation in Vacuum in Presence of the Uniform Electric and Magnetic Fields. (2002) - |http://arxiv.org/PS_cache/hep-ph/pdf/0211/0211217v1.pdf]] pct. 0

Links:
* [[WIKIPEDIA - CPT Symmetry|http://en.wikipedia.org/wiki/CPT_symmetry]]

<html><center><img src="images/cpt.jpg" style="width: 230px; "/></center></html>

The ''Cartan Tensor'' $C^\rho_{\mu\nu}$ (a.k.a. ''Modified Torsion Tensor'') is defined by
\begin{equation}
C^\rho_{\mu\nu} = T^\rho_{\mu\nu} + T_\mu \delta_\nu^\rho + T_\nu \delta_\mu^\rho
\end{equation}
with $T^\rho_{\mu\nu}$ the uncontracted and $T_\mu = T^\nu_{\mu\nu}$ the contracted [[Cartan torsion tensor|Torsion]] (a.k.a ''Torsion Vector''). For the latter different normalisations are found in literature.

Papers:
* [[On a Completely Antisymmetric Cartan Torsion Tensor - L. Fabbri|http://arxiv.org/PS_cache/gr-qc/pdf/0608/0608090v2.pdf]] [[pct. 2|http://scholar.google.de/scholar?hl=de&lr=&cites=1653535039168438446]] - There exists yet another different version.

A ''Catalan Number'' $C_n$ is defined as:
\[
C_n = \frac{1}{n+1}{2n\choose n} = \frac{(2n)!}{(n+1)!\,n!} , \quad n\ge 0
\]
!!!!Examples
| !$n$ | !$C_n$|
| $0$ | $1$ |
| $1$ | $1$ |
| $2$ | $2$ |
| $3$ | $5$ |
| $4$ | [[42|http://en.wikipedia.org/wiki/Phrases_from_The_Hitchhiker%27s_Guide_to_the_Galaxy]] |
| $5$ | $132$ |
| $6$ | $429$ |
See also: Sloane's [[A000108|http://www.research.att.com/~njas/sequences/A000108]]. 


The following (important) recursion formula for Catalan numbers holds: 
\[
C_{n+1} = \sum_{k=0}^n C_n C_{n-k} 
\]
Hence one has the sequence
\begin{eqnarray}
C_0 &=& 1 \\
C_1 &=& C_0C_0 \\
C_2 &=& C_1 C_0 + C_0 C_1\\
C_3 &=& C_2C_0 + C_1C_1 + C_0C_2 \\
C_4 &=& C_3C_0 + C_2C_1 + C_1C_2 + C_0C_3\\
\ldots
\end{eqnarray} 
which can be represented in terms of a triangle, depicted in the following with values inserted

{{centeredTable{
| | | |  |  | $1$ |  | | | | |
| | | |  |  | $1$ |  | | | | |
| | | |  | $1$ | | $1$ | | | | |
| | | |  $2$ |   | $1$ | | $2$   | | |
| | | $5$  |  | $2$  |     | $2$ |   | $5$  | | |
| | $14$ |  | $5$   |  |  $4$ |  |   $5$  | | $14$ | |
| $42$ | | $14$  |  | $10$  |     | $10$ |   | $14$ |  | $42$ |
}}}

Catalan numbers are ubiquitous in combinatorics. E.g. Richard Stanley has listed 173 combinatorial interpretations for them. 

The following example describes their occurrence in the [[Taylor series expansion|Taylor Series]] of a [[nonassociative algebra|Nonassociative Algebra]]. This is due to the fact that they count the number of ways one can put parentheses in products part of the expansion of a given order, i.e. how many different association types there exist for that order.  
| !$Order$ | !$Fundamental\;\, tensor {}^*$ |! | ! | ! |!  |!  |!  |! |! |!  |
| $1$ | | | |  |  | $ \color{orange}  \partial $ |  | | | |
| $2$ |$T$ | | |  |  | $ \color{orange} {\partial \partial} \equiv T $ |  | | | |
| $3$ |$A \equiv \partial T + R$  | | |  | $ \color{blue} {\partial} ( \color{orange} {\partial \partial }) = \partial T$ | |  $( \color{orange} {\partial \partial }) \color{blue} {\partial} = T \partial \equiv R $| | | |
| $4$ |$Q \equiv \partial A + TT + A\partial$ | | |  $ \color{violet} \partial A \; [2]  $ |   | $(\color{orange} {\partial \partial } ) (\color{orange} {\partial \partial }) = TT  \; [1]  $ | |$ A \color{violet} \partial \; [2]  $   | |
| $5$ |$P \equiv \partial Q + TA + AT + Q\partial$  | |$ \color{\lightgreen} \partial Q  \; [5]$  |  | $(\color{orange} {\partial \partial } ) A = TA \; [2]$  |     | $  A (\color{orange} {\partial \partial } ) = AT \; [2] $ |   | $ Q  \color{\lightgreen} \partial  \; [5]$  | |
| $6$ |$H \equiv  \partial P + TQ + AA + QT + Q\partial$ |$ \color{cyan} {\partial} P \; [14]$ |  | $ TQ \; [5] $ | |  $AA \; [4] $ |  | $ QT \; [5] $ |  | $ P  \color{cyan} \partial  \; [14]$ |
${}^*$ Note, that for the [[torsion tensor|Torsion]] $T$ and the [[nonassociativity tensor|Nonassociativity Tensor]] $A$ it is known that they are fundamental tensors of order $2$ and $3$ respectively. For the higher order objects this is merely a conjecture yet, based on analogy.

For associative algebras one recovers known facts. In this case all terms of orders higher than $2$ must vanish. Roughly speaking the nonassociativity tensor coincides with the Riemann tensor $R$ and the cyclic sum over it must vanish, which is the [[first Bianchi identity|First Bianchi Identity]]. 
In fourth order one is left with terms of the kind $\partial A = \partial R$ and in essence they represent the [[second Bianchi identity|Second Bianchi Identity]], when doing a cyclic summation.

This shows that [[Riemannian geometry|Riemann Space]] is neatly embedded in [[nonassociative geometry|Quasigroup Manifold]] and is recovered in a certain limit.


In terms of [[association types|Association Type]] the decomposition looks as follows:
{{centeredTable{
~~
| |  |  | $\partial = \mb e$ |  | | |
| |  |  | $\partial \partial = \partial \mb e $ |  | | |
| |  | $ \partial (\partial \partial) = \partial ( \mb {ee}) $ | |  $(\partial \partial) \partial = (\partial \partial) \mb e$ | | |
| |  $ \{\partial((\partial\partial) \partial),  \partial(\partial (\partial \partial)) \} =$ <br> $ \partial((\mb {ee}) \mb e), \partial(\mb e (\mb {ee})) \} $ |   | $ (\partial \partial) (\partial \partial) =$<br>$(\partial \partial) (\mb {ee}) $ | |$ \{ ((\partial\partial) \partial)\partial,  (\partial (\partial \partial))\partial \} =$<br>$\{ ((\partial\partial) \partial) \mb e,  (\partial (\partial \partial))\mb e \}$ |
| $ \{\partial(((\partial\partial)\partial)\partial), \partial(\partial((\partial\partial)\partial)),$<br>$\partial((\partial\partial)(\partial\partial)), \partial(\partial(\partial\partial)\partial),$<br>$\partial(\partial(\partial(\partial\partial)))\} =$<br>$\{\partial(((\mb {ee})\mb e)\mb e), \partial(\mb e((\mb{ee})\mb e)),$<br>$\partial((\mb{ee})(\mb{ee})), \partial(\mb e(\mb{ee})\mb e),$<br>$\partial(\mb e(\mb e(\mb{ee}))) \}$ | | $\{ (\partial \partial) ((\partial \partial)\partial), (\partial \partial) (\partial (\partial\partial)) \} = $ <br> $\{ (\partial \partial) ((\mb {ee})\mb e), (\partial \partial) (\mb e (\mb {ee})) \}$ |     | $ \{((\partial \partial)\partial)(\partial\partial), (\partial (\partial\partial))(\partial\partial) \} = $ <br> $ \{((\partial \partial)\partial)(\mb{ee}), (\partial (\partial\partial))(\mb {ee}) \} $ |   | $\{(((\partial\partial)\partial)\partial)\partial, (\partial((\partial\partial)\partial))\partial,$<br>$((\partial\partial)(\partial\partial))\partial, (\partial(\partial\partial)\partial)\partial,$<br>$(\partial(\partial(\partial\partial)))\partial\} =$<br>$\{(((\partial \partial)\partial)\partial)\mb e, (\partial((\partial \partial)\partial ))\mb e,$<br>$((\partial \partial)(\partial \partial))\mb e, (\partial (\partial \partial)\partial)\mb e,$<br>$(\partial(\partial(\partial\partial)))\mb e \}$ |
~~}}}
The (left- right-) symmetry of the triangle reflects the "left- right-mirror-symmetry" of the association types. 

!!!!Historical
Although the Catalan numbers carry the name of Eugène Charles Catalan, they have already been described at least as early as 1751 by the great Leonhard Euler.


Links:
* [[Catalan Numbers - R. M. Dickau|http://mathforum.org/advanced/robertd/catalan.html]]
Papers:
* [[Catalan Numbers - T. Davis|http://www.geometer.org/mathcircles/catalan.pdf]]
Books:
* [[Catalan Numbers with Applications - T. Koshy|books/ThomasKoshy_CatalanNumbersWithApplications.pdf]] [[bct. 3|http://scholar.google.de/scholar?cites=15579281873459284328&hl=de]]

Papers: 
* [[Space-Time Structure of Weak and Electromagnetic Interactions - D. Hestenes|http://geocalc.clas.asu.edu/pdf-preAdobe8/ST&EW.pdf]] [[pct. 34|http://scholar.google.de/scholar?cites=14346070686986909264&hl=de&as_sdt=2000]]
* [[Gauge Gravity and Electroweak Theory - D. Hestenes|http://arxiv.org/PS_cache/arxiv/pdf/0807/0807.0060v1.pdf]] [[pct. 3|http://scholar.google.de/scholar?cites=18168197432238270851&hl=de&as_sdt=2000]]
* [[The Glashow-Salam-Weinberg Electroweak Theory in the Real Algebra of Spacetime - R. Boudet|http://clifford-algebras.org/v7/v7%28S%29/BOUDET95.pdf]] [[pct. 2|http://scholar.google.de/scholar?cites=7818588189175864172&hl=de&as_sdt=2000]]
* [[Electroweak Theory - A. Lewis|http://cosmologist.info/notes/old/electroweak.ps]] pct. 0
* [[Electroweak Fields and Duality in the STA-Dirac Equation - F.M.C. Witte|http://www.phys.uu.nl/~witte/dualitySTA.pdf]] pct. 0

The ''Coleman\-Mandula Theorem'' (or ''CM Theorem'') states that under certain natural hypotheses the symmetry group of the S-matrix must be a direct product of the [[Poincaré group|Poincaré Transformation]] and an internal symmetry. It is usually interpreted as a no-go theorem$\,^{}^{*)}$, forbidding a nontrivial mixing of spacetime and internal symmetries.

[[Supersymmetry]] ([[Haag-Łopuszański-Sohnius theorem|Haag-Łopuszański-Sohnius Theorem]]) and certain [[quantum groups|Quantum Group]] (i.e. [[noncommutative geometries|Noncommutative Geometry]]) famously manage to avoid the theorem: in these cases the symmetry is not an ordinary [[Lie group|Lie Group]], as assumed by the theorem. 
[[Dual numbers|Dual Number]] offer another possibility to avoid the obstruction imposed by the Coleman\-Mandula theorem (shown by Wald).

Furthermore the CM theorem does not hold in [[de Sitter space|De Sitter Space]] (see [1]).

\*)
$\quad\;$<html><img src="images/Coleman.jpg" style="width: 138px "/></html>
$\quad$ [[Sidney Coleman|http://en.wikipedia.org/wiki/Sidney_Coleman]]

Papers:
* [[All Possible Symmetries of the S Matrix - S. Coleman, J. Mandula|http://hep.phy.tu-dresden.de/Lehre/SS2009/SUSY/literatur/coleman_madula_p1251_1.pdf]]  {{t500Cite{[[pct. 566|http://scholar.google.de/scholar?cites=2608624804188340434&hl=de&as_sdt=2000]]}}} - The original paper on the topic.
* [[Universal Enveloping Algebras and Some Applications in Physics - X. Bekaert|http://www.ulb.ac.be/sciences/ptm/pmif/Rencontres/ModaveI/Xavier.pdf]] [[pct. 4|http://scholar.google.de/scholar?cites=3148727214875758285&hl=de]]
* [[[1] Mixing Internal and Spacetime Transformations: Some Examples and Counterexamples - R. Percacci|http://arxiv.org/PS_cache/arxiv/pdf/0803/0803.0303v1.pdf]] [[pct. 1|http://scholar.google.de/scholar?cites=17660056454374112215&hl=de&as_sdt=2000]]

Your comments are very welcome. Please refer to a [[tiddler|What is a Tiddler]] or topic if possible. Thanx a lot !

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The ''Conformal Weyl Group'' is the $10$-parameter [[Poincaré group|Poincaré Transformation]] supplemented with a $1$-parameter group of scale transformations.
\[
x'_{\mu}  = e^{\theta} x_\mu\text{,} \quad \bs \psi' (\mb x') = e^{?k\theta} \bs \psi (\mb x)\text{;} \quad k,\theta= const.
\]

Links:
* [[Quantum Consciousness - Stuart Hameroff|http://www.quantumconsciousness.org/]]

Papers:
* [[Quantum Computation in Brain Microtubules|http://www.cs.indiana.edu/classes/b629-sabr/QuantumComputationInBrainMicrotubules.pdf]] {{t100Cite{[[pct. 119|http://scholar.google.de/scholar?cites=17233805516325330389&hl=de]]}}}
* [[Theory of Brain Function, Quantum Mechanics and Superstrings - D. Nanopoulos|http://arxiv.org/PS_cache/hep-ph/pdf/9505/9505374v1.pdf]] [[pct. 27|http://scholar.google.de/scholar?cites=15335628828834998581&hl=de]]
* [[Non-Computability of Consciousness - D. Song|http://arxiv.org/PS_cache/arxiv/pdf/0705/0705.1617v1.pdf]] [[pct. 2|http://scholar.google.de/scholar?cites=4241235277389295173&hl=de]]

Videos:
* [[Supercomputing the Brain's Secrets - H. Markram|http://www.youtube.com/user/TEDtalksDirector#p/u/91/LS3wMC2BpxU]]

The ''Copernican Principle'' states that the universe is homogenous - when viewed on a very large scale, different parts of the universe look essentially the same.

The Copernican principle is a built-in assumption of the current favoured solutions to [[Einstein's equations|Einstein Field Equations]], called the Friedmann\-Robertson\-Walker space-times.

The ''Cosmological Constant'' $\Lambda$ can be introduced via the [[Einstein equations|Einstein Field Equations]] as a global curvature term that adds to the [[Ricci curvature|Ricci Tensor]]. Modern field theory associates this term with the energy density of the vacuum. The measured value is negative which means that in the (near) absence of gravity spacetime is curved negatively.

''Cosmological Constant Problem:''
If the universe is described by an effective local quantum field theory down to the Planck scale, one would expect a cosmological constant of the order of $m_{\rm pl}^4$. In fact the measured value is smaller than that by a factor of $10^{120}$.

''Experimental status:''
Current experimental findings are consistent with the idea of [[dark energy|Dark Energy]] behaving like Einstein's cosmological constant,  i.e. it describes a density and pressure associated with "empty" space.
The latest Hubble data contradict theories that postulate that dark energy behaved differently billions of years ago to how it does today. The observations also confirmed that the expansion rate of the cosmos began speeding up about five to six billion years ago. This is when astronomers believe that dark energy's repulsive force starts dominating over the gravitational force.

Papers:
*[[New Hubble Space Telescope Discoveries of Type Ia Supernovae at z ? 1: Narrowing Constraints on the Early Behavior of Dark Energy - A. G. Riess, L.-G. Strolger, S. Casertano, H. C. Ferguson, B. Mobasher, B. Gold, P. J. Challis, A. V. Filippenko, S. Jha, W. Li, J. Tonry, R. Foley, R. P. Kirshner, M. Dickinson, E. MacDonald, D. Eisenstein, M. Livio, J. Younger, C. Xu, T. Dahlen, D. Stern|http://xxx.lanl.gov/PS_cache/astro-ph/pdf/0611/0611572v2.pdf]]  {{t100Cite{[[pct. 221|http://scholar.google.de/scholar?cites=14144653772503374306&hl=de]]}}}

> You cannot compare two tensors with two different base points ... just by comparing their components. ... They lie in different tensor spaces and have absolutely nothing to do with each other.
> - Bjørn Felsager - Geometry, Particles, and Fields

Roughly speaking a ''Covariant Derivative'' $D$ describes the change of a tensor on a manifold. 
As generally the coordinate basis changes from point to point, one has to add an extra term $\Gamma$, called the [[connection|Connection]], to compensate for this change and to "restore the Cartesian situation". This way one gets the "true change" of the tensor, i.e. one can compare the infinitesimally changed tensor with the original one. 
If the tensor is a vector, the compensatory term amounts to the fact that the vector is what is called parallel transported.
The covariant derivative therefore is defined as:
\[
D = \partial + \Gamma
\]
The general form for a tensor $T^{\mu_1 \ldots \mu_r}{}_{\nu_1 \ldots \nu_s}$ is given by:
\begin{eqnarray} \large
(D_\lambda T)^{\mu_1 \ldots \mu_r}{}_{\nu_1 \ldots \nu_s}& \equiv& T^{\mu_1 \ldots \mu_r}{}_{\nu_1 \ldots \nu_s;\lambda} \\
&=& \frac{\partial}{\partial x^\lambda}T^{\mu_1 \ldots \mu_r}{}_{\nu_1 \ldots \nu_s} + \\
& & \Gamma ^{\mu_1}{}_{\sigma\lambda} T ^{\sigma \ldots \mu_r}{}_{\nu_1 \ldots \nu_s} + \ldots + \Gamma ^{\mu_r}{}_{\sigma\lambda} T ^{\mu_1 \ldots \mu_{r-1}\sigma}{}_{\nu_1 \ldots \nu_s} - \\
& &  \Gamma ^\sigma {}_{\nu_1 \lambda} T ^{\mu_1 \ldots \mu_r}{}_{\sigma \ldots \nu_s} - \ldots - \Gamma ^\sigma {}_{\nu_s \lambda} T ^{\mu_1 \ldots a_r}{}_{\nu_1 \ldots \nu_{s-1} \sigma}
\end{eqnarray}
where the $ \Gamma ^\mu {}_{\nu \sigma}$ are the connection coefficients.

For a contravariant vector $V^\nu$ for example on gets:
\[
D_\mu V^\nu  = V^\nu{}_{;\mu} = \frac{\partial V^\nu}{\partial x^\mu} + \Gamma^\nu{}_{\sigma\mu}V^\sigma
\]

Is it conceivable that the universe we life in has been created by some kind of intelligence in a parent (predecessor) universe ?

Some questions in this respect:
* What does it take to create a universe ?{{floatright { <html><span style="padding-left:8px"><iframe name="content" src="http://www.markus-maute.de/trajectory/advert_sky.html" width=180 height=620 scrolling="no"></iframe> </html> }}} (If some preceding intelligence has done it, it seems likely that we are pretty much like them and we could do it as well one day). Hence to understand how to build a universe might allow us to better understand the origin of our own universe.
* Why would some kind of intelligence in a universe be interested in sprouting a child universe ? 

Some thoughts, ideas and speculations: 
* A complex system doesn't appear "out of nothing" or out of randomness, rather it should have a long evolutionary history. As our universe, in particular as it harbours intelligent life, is a pretty complex system, it may well be that the few billion years since the big bang are not enough to explain __all__ the necessary steps evolution had to take to end up with a universe the way we see it today. Therefore, introducing an ancestry of universes gives evolution more time, as now the time available to produce our universe as an output is a few billion years times the number of predecessor universes. 
* The big bang singularity has a ridiculously low entropy whereas the entropy of a black hole is high. (An oddity [[Roger Penrose|Roger Penrose]] over and over again has pointed to). If one assumes that a black hole naturally gives rise to a baby universe, it seems quite mysterious what effect it would take to convert the initial high entropy state to a low entropy one. An alternative is "intelligence". I.e. only such universes are low entropy universes that have been created due to the intervention of intelligence. The natural creation may well exist alongside, but these offspring universes most probably are low in complexity (as high in entropy) and therefore will not give rise to intelligence.
* The explanation of the emergence of conciousness and intelligence could be that it guarantees the replication of the universe, a necessary step in an evolutionary process. I.e. consciousness is one possible trait of a universe guaranteeing survival in the evolutionary process. (Expressed in a more colloquial way: Consciousness is the "sexuality of the universe", destined to reproduce it and to contribute to the evolution of its kind). If this it true, evolution also is taking place on a higher level than what we are accustomed to.
* If our universe is "programmed" for life, this means that the [[strong anthropic principle|Anthropic Principle]] holds and the scenario alluded to here is a concrete realisation of it. This implies that the universe may not have completely unfolded yet and it offers an explanation as to why we see an ongoing progress, characterized by a directionality towards more complexity and organisation which is contrary to what one would naively expect due to the second law of thermodynamics. (For an extreme conclusion from this fact, see [[omega point theory|Omega Point]]).
* In this respect I came up with a totally strange Gedankenexperiment: Suppose we create matter and anti-matter by means of pair production. From the anti-matter we form a black hole whereas the matter we assemble in a low entropy state. (Maybe one could even create a conscious brain out of the matter part. Curiously enough, experts say that to create a universe in the laboratory it only takes a few pounds of matter, incidentally, just about the mass of a brain of a highly developed intelligent species). If the anti-matter black hole gives rise to a daughter universe then it seems that the whole of this universe is EPR\-correlated with this complex structure in our universe. (And if this complex structure is conscious, this consciousness is EPR\-correlated with the whole of a universe). We can turn this argument upside down and wonder if the whole of our universe is EPR\-correlated with some preceding intelligence. So then, could our consciousness be correlated with some maybe further developed intelligence in another universe ? Could this explain how ideas come into the world ? I.e. could that mean that seemingly new things we come up with are just inherited from some foregoing intelligence and not really new ?
* Furthermore this model could serve to explain the [[fine tuning|Fine Tuning]] of our universe.
* If this scenario is true, we could speak of [["intelligent design"|Intelligent Design]] of our universe. Yet this does not necessarily imply the existence of a monotheistic God, rather the existence of beings that are not more of a God than what we would be one day if we were to create a universe in the laboratory having the potential to bring about intelligent/conscious life suffices as an explanation. Besides God and the multiverse (or landscape) this offers a "third way" (one "in between") for making plausible why the world around us is so special and unlikely.

See also: [["Organic universe"|Organic Universe]].

Papers: 
* [[The Natural Selection of Universes Containing Intelligent Life (1995) - E. R. Harrison|http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1995QJRAS..36..193H&amp;data_type=PDF_HIGH&amp;whole_paper=YES&amp;type=PRINTER&amp;filetype=.pdf]] [[local|papers/natural_selection.pdf]] [[pct. 29|http://scholar.google.de/scholar?cites=17506788885958317714&hl=de]] 
* [[Message in the Sky (2005) - S. Hsu, A. Zee|http://scholar.google.de/scholar?cites=92794270126152221&as_sdt=2005&sciodt=2000&hl=de]] [[pct. 10|http://scholar.google.de/scholar?cites=92794270126152221&as_sdt=2005&sciodt=2000&hl=de]]
* [[Child Universes in the Laboratory (2006) - S. Ansoldi, E. I. Guendelman|http://arxiv.org/PS_cache/gr-qc/pdf/0611/0611034v1.pdf]] [[pct. 5|http://scholar.google.de/scholar?cites=8964676878467787935&as_sdt=2005&sciodt=2000&hl=de]]
* [[The Universe out of a Monopole in the Laboratory? (2006) - N. Sakai, K. Nakao, H. Ishihara, M. Kobayashi|http://arxiv.org/PS_cache/gr-qc/pdf/0602/0602084v3.pdf]] [[pct. 3|http://scholar.google.de/scholar?cites=820591780757734784&as_sdt=2005&sciodt=2000&hl=de]]
* [[How to Create a Universe (2007) - G. McCabe|http://philsci-archive.pitt.edu/archive/00003196/01/Spec.pdf]] pct. 0
* Is it Possible to Create a Universe in the Laboratory by Quantum Tunneling? (1990) - E. Farhi, A. H. Guth, J. Guven {{t100Cite{[[jct. 170|http://scholar.google.de/scholar?cites=10837268565401913655&as_sdt=2005&sciodt=2000&hl=de]]}}}
* An Obstacle to Creating a Universe in the Laboratory - E. Farhi, A. H. Guth [[jct. 97|http://scholar.google.de/scholar?cites=17855786670394600441&as_sdt=2005&sciodt=2000&hl=de]]

Links: 
* [[Scholarpedia - Time's Arrow and Boltzmann's Entropy|http://www.scholarpedia.org/article/Time%27s_arrow_and_Boltzmann%27s_entropy]]

Audio:
* [[Build Your Own Universe|http://www.npr.org/templates/story/story.php?storyId=6545246]]

The ''D'Alembert Equation'' is given by
\[
\partial_\mu\partial^\mu \Phi (\mb x) \equiv  \square  \Phi (\mb x) = 0
\]
with $ \Phi (\mb x)$ a scalar field.

<html><center><img src="images/dbrane.jpg" style="width: 191px;"/></center></html>In [[Superstring Theory]] a string can either be closed or open. For a closed string one has periodic, for an open string Dirichlet- or Von Neumann-boundary conditions. The latter were considered unphysical in the beginning, as they break Lorentz invariance. However when [[p-branes|P-Brane]] were discovered it was realized, that open strings can end on them and satisfying the Dirichlet boundary condition. Such special $p$-branes are called $Dp$-branes, with the "$D$" standing for "Dirichlet" or short ''$D$-branes''. 

In $D$-branes, coordinates of space-time become [[noncommuting|Noncommutative Quantum Field Theory]] and $U(N)$ matrix-valued. A [[metric|Metric Tensor]] on such spaces will also become matrix-valued.
Open string theories as well as $D$-branes in the presence of a background antisymmetric [[B-field|Kalb-Ramond Field]] give rise to noncommutative effective field theories. This is equivalent to field theories deformed with the [[star product|Star Product]].

Papers:
* [[Dark Eneregy and 3-Manifold Topology - T. Asselmeyer-Maluga, H. Rosé|http://th-www.if.uj.edu.pl/acta/vol38/pdf/v38p3633.pdf]]

Links:
* [[WIKIPEDIA - Dark Flow|http://en.wikipedia.org/wiki/Dark_flow]]

<html><center><a href="http://apod.nasa.gov/apod/ap070516.html"><img src="images/DarkMatter.jpg" style="width: 482px; "/></a></center></html>
''Dark Matter'' was introduces to "explain", based on [[Einsteins field equations|Einstein Field Equations]],
* the gravitational field needed for the galactic rotation curves,
* gravitational lensing of galaxies,
* the formation of structures in the universe.
It also appears in the spectral decomposition of the cosmic microwave background radiation.  However, there is no single observational hint at particles which could make up this dark matter. As a consequence, there are attempts to describe the same effects by a modification of the gravitational field equations, e.g. of Yukawa form, or by a modification of the dynamics of particles, like the [[MOND]] ansatz. Due to the lack of direct detection of dark matter particles, all those attempts are on the same footing.

Videos: 
* [[Hubblecast EPISODE 05: Hubble Finds Ring of Dark Matter|http://www.space.com/php/video/player.php?video_id=150407Dark_matter]]

Links: 
* [[Galaxy Cluster Cl 0024+17 (ZwCl 0024+1652)|http://imgsrc.hubblesite.org/hu/db/images/hs-2007-17-b-print.jpg]]

!!!!De Sitter Space-time
De Sitter Spacetime is the most symmetric spacetime, namely a homogeneous constant-curvature spacetime with cosmological constant $\Lambda$.
For a $n$-dimensional space-time the relevant symmetry groups are:
\begin{equation}
    G = \left\{
      \begin{array}{cll}
          SO(n,1) & \Lambda > 0  &  \text { (de Sitter)}\\
          ISO(n-1,1)&  \Lambda = 0 & \text { (Minkowski)} \\
          SO(n-1,2) &  \Lambda < 0 &\text { (anti de Sitter)}
      \end{array}
      \right.
\end{equation}
The metric is given by
\begin{equation}
d\tau^2 = dt^2 - e^{-2\Lambda t} dr^2
\end{equation}
Papers:
* [[The de Sitter and Anti-de Sitter Sightseeing Tour - U. Moschella|http://www.bourbaphy.fr/moschella.ps]] [[pct. 3|http://scholar.google.de/scholar?hl=de&lr=&cites=8046863754320787269]] [[local|papers/Moschella.ps]]

[[Welcome]]

> You insist that there is something a machine cannot do. If you will tell me precisely what it is that a machine cannot do, then I can always make a machine which will do just that!
> - John von Neumann -

There are many variants of ''Digital Physics'' (also referred to as ''Digital Philosophy''), but most of them have in common that physical reality and mental activity is viewed as digitized information processing.

Digital philosophy can be regarded as a modern reinterpretation of Gottfried Leibniz's monist metaphysics, one that replaces Leibniz's monads with aspects of the theory of cellular automata, assuming that the universe is a gigantic Turing-complete cellular automaton.
So far there is no unambiguous physical evidence against the possibility that "everything is just a computation".

Some people that are regarded as adherers to the concept of digital philosophy are: Gottfried Wilhelm Leibniz, Konrad Zuse, Edward Fredkin, Stephen Wolfram, [[Gregory Chaitin]], Jürgen Schmidhuber and Seth Lloyd.

Jürgen Schmidhuber pointed out that the simplest explanation of the universe would be a very simple Turing machine programmed to systematically execute all possible programs computing all possible histories for all types of computable physical laws. Furthermore there is an optimally efficient way of computing all computable universes based on Leonid Levin's universal search algorithm. He expanded this work by combining Ray Solomonoff's theory of inductive inference with the assumption that quickly computable universes are more likely than others.

The idea of a fundamental discrete entity being the building block of physical reality has appeared over and over again in history in many different guises, as for example:
* [[Planck units|Planck Units]]
* Monads (Leibnitz)
* Urs (Weizäcker)
* Bits (Wheeler)
* Metrons ([[Heim|Burkhard Heim]])
* Ons (Goertzel)

See also:
* [[Cellular automaton|Cellular Automaton]]
* [[Process physics|Process Physics]]
* [[Discrete spacetime|Discrete Spacetime]]
* [[Spin networks|Spin Network]]
* [[World crystal|World Crystal]]
* [[Ultrafinitism]]

Links:
* [[WIKIPEDIA - Digital Physics|http://en.wikipedia.org/wiki/Digital_physics]]
* [[Zuse's Thesis: The Universe is a Computer - Jürgen Schmidhuber|http://www.idsia.ch/~juergen/digitalphysics.html]]
* [[Digital Philosophy.org|http://www.digitalphilosophy.org/]]

Papers:
* [[Algorithmic Theories of Everything - J. Schmidhuber|http://arxiv.org/PS_cache/quant-ph/pdf/0011/0011122v2.pdf]] [[pct. 46|http://scholar.google.de/scholar?cites=7282820845356865291&hl=de]]

The ''Dilaton field'' $\Phi$ is a scalar field which modifies gravity. 

Its appearance in physical models is related to the question of mass and scales. Its mass scale is much larger than the weak scale and could be as large as the GUT scale or Planck scale.

!!!![[Noncommutative geometry|Noncommutative Geometry]]
The dilaton field does not appear in the classical [[spectral action|Spectral Action]], which is to be contrasted with the [[Connes-Lott formulation|Connes-Lott Model]] of the noncommutative action where the dilaton field is part of the gravitational interactions. However a modified spectral action can do justice to the dilation, where only the bosonic part of the action is affected by the field (i.e. "fermions don't feel the dilaton").

!!!![[String theory|Superstring Theory]]
The dilaton is always part of the low energy spectrum in string theory.  

Furthermore it shows up in so called ''String Inspired Models'', where it is also considered for spacetimes with dimensionalities different from those typical in string theory. 

!!!![[Inflation|Inflation]]
The dilaton plays a fundamental role in models of inflation.


<html><center><iframe name="content" src="http://www.markus-maute.de/trajectory/advert_60.html" width=51% height=86></iframe></center></html>Papers:
* [[Scale Invariance in the Spectral Action - A. H. Chamseddine, A. Connes|http://arxiv.org/PS_cache/hep-th/pdf/0512/0512169v3.pdf]] [[pct. 22|http://scholar.google.de/scholar?cites=320964133535099019&as_sdt=2005&sciodt=2000&hl=de]]

The [[Dirac equation|Dirac Equation]] in Minkowski space-time has global [[SL(2,C)|SL(2,C)]] invariance, and it is natural to require that this invariance be promoted to become local by introducing the [[spin-connection|Spin Connection]] as a gauge field. There is, however, a need to introduce the [[vierbein|Tetrad]] $h_a^\mu$ as an external field. 

In curved spacetime the Dirac equation then takes the form:
\[
\mb e^a h_a^\mu \mb D_\mu \bs \Psi =  - i m\bs \Psi
\]
i.e. one has to consider a local [[(tetrad-)basis|Tetrad]] which amounts to replacing the differential operator $ \partial_{\mu}$ by a [[covariant differential operator|Spinor Derivative]]
\[
\mb D_\mu = \partial_\mu - \frac{i}{4} \omega_{\mu cd} \bs \sigma^{cd}
\]
with
\[
\bs \sigma^{cd} = \frac{i}{2}[\mb e^c,\mb e^d]
\]
the generators of the [[Lorentz group|Lorentz Transformation]] and $\omega_{\mu cd}$ the spin connection coefficients.

Therefore:
\[
\mb e^a h_a^\mu (\partial_\mu + \frac 18 \eta_{cb}\omega^b_{\mu d} [\mb e^c,\mb e^d]) \bs \Psi = - i m \bs \Psi
\]

\begin{eqnarray}
\omega^c_{\mu d} (\mb x) = - h^\nu_d (\mb x) \partial_\mu h^c_\nu (\mb x) + h^\nu_d (\mb x)  h^c_\rho (\mb x)  \Gamma^\rho_{\mu\nu}  (\mb x)
\end{eqnarray}


Inserting the explicit form of the spin connection
\[
\omega^b_{\mu d} = - h^\nu_d \partial_\mu h^b_\nu + h^\nu_d h^b_\rho \Gamma^\rho_{\mu\nu} 
\]
one gets:
\[
\mb e^a h_a^\mu (\partial_\mu + \frac 18 \eta_{cb} (- h^\nu_d (\partial_\mu h^b_\nu) + h^\nu_d h^b_\rho \Gamma^\rho_{\mu\nu} ) [\mb e^c,\mb e^d]) \bs \Psi = - i m \bs \Psi
\]
or 
\[
(\partial \! \! \! \mb/ + \frac 18 \eta_{cb} (- h^\nu_d (\partial_\mu h^b_\nu) + h^\nu_d h^b_\rho \Gamma^\rho_{\mu\nu} ) h_a^\mu \mb e^a [\mb e^c,\mb e^d]) \bs \Psi = - i m \bs \Psi
\]

The Dirac equation in curved spacetime can be obtained from the following action integral 
\[
S = \int d^4x \sqrt{-g}\bar {\mb \Psi} (i \mb e^a h_a^\mu (\partial_\mu + \bs \Omega + m) \bs \Psi 
\]
!!!!Torsion
Torsion has been considered e.g. by Gürsey (1957), Finkelstein (1960),  Hehl, Datta (1971) and Hehl et al. (1976). Even when starting from slightly different assumptions, torsion induces nonlinear terms in the Dirac equation. 

!!!![[Poincaré Incariance|Poincaré Transformation]]
The idea of extending the homogeneous Lorentz invariance to the inhomogeneous Lorentz invariance was has also been exploited. In this case the group has translation generators in addition to the rotation generators.
The field strengths associated with the translation generators are constrained to be zero, allowing the identification of the translation gauge parameters with the [[diffeomorphisms|Diffeomorphism]] parameters. This is where a [[gauge theory of gravity|Gauge Theory of Gravity]] differs from the usual Yang\-Mills type gauge theory, as the constraint of vanishing translational field strength allows to solve for the spin connection as function of the __vierbein, which in this case is the gauge field associated with translations__. This makes the field strength associated with rotations depend on second derivatives of the vierbein, and becomes identified with the curvature of the metric formed from the vierbein. Yet, these constraints render the theory non-renormalizable and formulating gravity as a gauge theory of the inhomogeneous Lorentz group does not lead to improvements in the renormalizability of the theory.

!!!!!An observation and a vague idea 
The term $\mb e^a[\mb e^c,\mb e^d]$ corresponds to half of a term of the [[Jacobi identity|Jacobian]]. The other part, namely $[\mb e^c,\mb e^d] \mb e^a $ corresponds to the [[Riemann curvature tensor|Riemann Tensor]] and mass. If one assumes an "anomalous" Jacobian, i.e. a non-vanishing associator, the second term could be interpreted as the mass-generating term of the Dirac-equation. (This statement however requires further scrutinization). 

Papers: 
* [[Spin and Anholonomy in General Relativity - R. Aldrovandi, P. B. Barros, J. G. Pereira|http://arxiv4.library.cornell.edu/PS_cache/gr-qc/pdf/0402/0402022v2.pdf]] [[pct. 1|http://scholar.google.de/scholar?cites=730740112177101284&hl=de&as_sdt=2000]]

Theses: 
* [[Higher-Order Theories of Gravitation - L. Fabbri|http://amsdottorato.cib.unibo.it/333/1/libro.pdf]] [[tct. 2|http://scholar.google.de/scholar?cites=15206276095350270271&hl=de&as_sdt=2000]]

Journals:
* [[Dirac Equation in Space-Time with Torsion - A. Zecca|journals/DiracAndTorsion.pdf]] [[jct. 7|http://scholar.google.de/scholar?cites=12061691060650090411&hl=de&as_sdt=2000]]

Links:
* [[Gennadi Sardanashvily, Publications - Selected Articles|http://gnsardan.appfarm.ru/lp_sa2.html]] - Contains several publications together with Ivanenko. 

Google books: 
* [[Clifford (Geometric) Algebras with Applications to Physics, Mathematics, and Engineering - W. E. Baylis|http://books.google.com/books?id=0Nji78YQKfQC&pg=PA335&lpg=PA335&dq=bivector+%22spin+connection%22+%22dirac+equation%22&source=bl&ots=C4Q0HMBWhX&sig=iQhHqVdwL_jApSASgFko6A_9jjQ&hl=de&ei=J4ioS5X1E4mh_AaMyPDnAw&sa=X&oi=book_result&ct=result&resnum=8&ved=0CDoQ6AEwBw#v=onepage&q=bivector%20%22spin%20connection%22%20%22dirac%20equation%22&f=false]] [[bct. 96|http://scholar.google.de/scholar?cites=8553518639729290440&hl=de&as_sdt=2000]]

Papers:
* [[Noncommutativity and Discrete Physics - L. H. Kauffman|http://www2.math.uic.edu/~kauffman/NCDP.pdf]] [[pct. 14|http://scholar.google.de/scholar?cites=3342187083863786167&hl=de]]

Presentations:
* [[Numerical Simulations of Causal Dynamical Triangulations - J. Ambjørn, A. Görlich, J. Jurkiewicz, R. Loll|http://www.pact.cpes.sussex.ac.uk/~dl79/CLAQG/Jurkiewicz.pdf]]

Abstracts:
* [[Quantum Computation and Combinatorial Spacetime - D. Madina|http://www.qci.jst.go.jp/eqis02/program/abstract/poster36.pdf]]

See also:
* [[World crystal|World Crystal]]
* [[Spin networks|Spin Network]]

Links:
* [[NDT Ressource Center - Linear Defects - Dislocations|http://www.ndt-ed.org/EducationResources/CommunityCollege/Materials/Structure/linear_defects.htm]]

The ''Dixon\-Souriau Equations'' are a generalization of the [[Mathisson-Papapetrou equations|Mathisson-Papapetrou Equations]] in that an additional electromagnetic field is assumed.
In the absence of [[torsion|Torsion]] the equations are given by:
\begin{eqnarray}
\frac{D\tilde p^\mu}{D\tau} & = &  -\frac{1}{2} {R^\mu}_{\nu\lambda\sigma} S^{\nu\lambda} u^\sigma + eF^\mu{}_\nu u^\nu  -\frac\lambda2 S^{\nu\sigma}
\partial^\mu F_{\nu\sigma} \\
 \frac{DS^{\mu\nu}}{D\tau}& = &\tilde p^\mu u^\nu- \tilde p^\nu
 u^\mu +\lambda [S^{\mu\sigma}F_\sigma^\nu - S^{\nu \sigma}F_\sigma^\mu]
\end{eqnarray}
with
\[
\tilde{p}^{\mu} \equiv p^\mu - \frac{DS^{\mu\nu}}{D\tau}u_\nu
\]
In  addition to the Mathisson\-Papapetrou equations the equations contain the [[electromagnetic field strength tensor|Field Strength Tensor]] $F^{\mu\nu}$ and $\lambda$, which is an electromagnetic coupling scalar.

!!!!Special Cases
The Dixon\-Souriau equations reduce to the Van Holten equations whenever the particle’s four-momentum and four-velocity become co-linear. It has also been shown that the equations reduce to the well known Bargmann\-Michel\-Telegdi equations in the limit of the weak and homogeneous external field.

Papers:
* [[On the Electrodynamics of Spinning Particles - J. W. Van Holten|http://www.nikhef.nl/pub/services/biblio/preprints/h90-22.pdf]] [[local|papers/h90-22.pdf]] [[pct. 36|http://scholar.google.de/scholar?cites=5311923282338670619&hl=de&as_sdt=2000]]
* [[Modèle de Particule à Spin Dans le Champ Electromagnétique et Gravitationnel - J. M. Souriau|http://www.jmsouriau.com/Publications/JMSouriau-ModPartSpin1974.pdf]] [[local|papers/JMSouriau-ModPartSpin1974.pdf]] [[pct. 20|http://scholar.google.de/scholar?cites=4757212981966671457&hl=de&as_sdt=2000]] - One of the original papers.
* [[Charged Particles with Spin in a Gravitational Wave and a Uniform Magnetic Field - M. Mohseni|http://arxiv.org/PS_cache/gr-qc/pdf/0510/0510094v2.pdf]] [[pct. 3|http://scholar.google.de/scholar?cites=445315149535384639&hl=de&as_sdt=2000]] - With excellent literature review on the topic.
* [[Spin-Rotation Couplings: Spinning Test Particles and Dirac Field - D. Bini, Luca Lusanna|http://arxiv.org/PS_cache/arxiv/pdf/0710/0710.0791v1.pdf]] [[pct. 1|http://scholar.google.de/scholar?cites=7777017156572952812&hl=de]]
* [[Spinning Particles in General Relativity - F. Cianfrani, G. Montani|http://arxiv.org/PS_cache/gr-qc/pdf/0701/0701080v1.pdf]] pct. 0

Links:
* [[Site Officiel de Jean-Marie Souriau|http://www.jmsouriau.com/]]

Journals:
* [[Spinning Particles in Schwarzschild Spacetime - R. H. Rietdijk, J. W. Van Holten|journals/SpinningParticleSchwarzschildMetric.djvu]] [[jct. 36|http://scholar.google.de/scholar?cites=15970824269076798034&hl=de&as_sdt=2000]]

[[Tiddlers|TiddlyWiki]] that are indicated to be a ''Draft'' are preliminary and not fully worked out. This concerns mostly tiddlers that contain own ideas.      

Therefore, content earmarked this way should be taken with a grain of salt. I cannot exclude the possibility that it is fundamentally flawed.  

Your comments may be very helpful to improve such content. (Please refer to the tiddler name when commenting): 
<html><center> <br><iframe name="content" src="http://www.markus-maute.de/trajectory/comments/index.php?do=add_form&page=1" width=65% height=450></iframe></center>
</html>
~~The credit belongs to the man who is actually in the arena; whose face is marred by sweat and blood; who strives valiantly; who errs and comes short again and again because there is no effort without error and shortcoming; who knows the great enthusiasms, the great devotion, spends himself in a worthy cause; who at best knows in the end the triumph of high achievement; and who at worst, if he fails, at least fails while daring greatly, so that his place shall never be with those cold and timid souls who have never tasted victory or defeat.
- Theodore Roosevelt -

The [[energy-momentum tensor|Stress Energy Tensor]] does not change if an [[electromagnetic field|Electrodynamics]] is transformed by a so called ''Duality Rotation'':
\[
F'_{\mu\nu} = F_{\mu\nu} cos (\delta) + \tilde F_{\mu\nu} sin (\delta)
\]
Consequently, although a given electromagnetic tensor uniquely defines the electromagnetic energy-momentum tensor $T_{\mu\nu}$, the converse is not true. Given $T_{\mu\nu}$, $F_{\mu\nu}$ is defined only up to duality rotations.
Furthermore the currents transform according to
\[
j'_{\mu} = j_{\mu} cos (\delta) + \tilde j_{\mu} sin (\delta)
\]

More explicitely one has
\begin{eqnarray}
\vec E' & =& \vec E \cos (\alpha) + \vec B \sin (\alpha)  \\
\vec B'& = &\vec B\cos (\alpha) - \vec E \sin (\alpha)
\end{eqnarray}
and Gauß's law becomes
\[
\vec \nabla \times \vec E'  + \frac{\partial \vec B'}{\partial t} = \rho'
\]

If one assumes $\alpha = \pi/2$, one gets
\begin{eqnarray}
\vec E' & =&\vec B \\
\vec B'& = &-\vec E
\end{eqnarray}
which defines a [[duality involution|Duality Involution]] ${}^\sim$.
The associated transition $F \rightarrow \tilde F$ corresponds to the [[duality|Duality]] of electric and magnetic fields, i.e. the map:
\[
\vec E \rightarrow -\vec B, \quad  \vec B \rightarrow \vec E
\]
Lectures:
* [[Problems and Solutions - G. Mammadov|http://gmammado.mysite.syr.edu/notes/Electromagnetic_Field_Strength_Tensor.pdf]]

Google books:
* [[Modern Nonlinear Optics, Part 2 - M. W. Evans|http://books.google.com/books?id=9p0kK6IG94gC&pg=PA333&lpg=PA333&dq=%22Larmor%22+%22Rainich+group%22&source=bl&ots=tR3pyIOp_a&sig=e79EOZT3diri9gmkjGtNZhh9s5A&hl=de&sa=X&oi=book_result&resnum=1&ct=result#PPA332,M1]] [[bct. 89|http://scholar.google.de/scholar?cites=16148624411202458834&hl=de&as_sdt=2000]]

A ''Dyon'' is a particle that carries electric and magnetic charges.

Latest Fermilab experiments with the Tevatron indicate that the precise prediction of the [[Higgs|Higgs Mechanism]]-mass of $161.8033989$ \GeV by E-infinity theory can be excluded. 

Papers:
* [[A Review of E Infinity Theory and the Mass Spectrum of High Energy Particle Physics - M.S. El Naschie|http://www.complexity.ru/papers/science25.pdf]]{{t100Cite{ [[pct. 331|http://scholar.google.de/scholar?cites=14121921044845368187&hl=de]]}}}
* [[The VAK of Vacuum Fluctuation, Spontaneous Self-Organization and Complexity Theory Interpretation of High Energy Particle Physics and the Mass Spectrum - M.S. El Naschie|http://www.el-naschie.net/bilder/file/7.%20The%20VAK%20of%20vacuum%20fluctuation,%20spontaneous.pdf]] [[pct. 41|http://scholar.google.de/scholar?cites=16744730194865314043&hl=de]]
* [[From Arthur Cayley via Felix Klein, Sophus Lie, Wilhelm Killing, Elie Cartan, Emmy Noether and Superstrings to Cantorian Space–Time - L. Marek-Crnjac|http://www.el-naschie.net/bilder/file/Crnjac_From_Arthur_Cayley.pdf]] [[pct. 5|http://scholar.google.de/scholar?cites=10159834154703874053&hl=de]]
* [[Exceptional Lie Groups, E-infinity Theory and Higgs Boson - A. A. El-Okaby|http://arxiv.org/ftp/arxiv/papers/0709/0709.2394.pdf]] [[pct. 1|http://scholar.google.de/scholar?cites=16105425813544884648&hl=de]]

The ''ECE (Einstein\-Cartan\-Evans) Theory'' was developed by Myron Evans. 

One of the paradigms of the theory is that the unification of quantum mechanics and general relativity occurs by accepting objectivity and causality and rejecting indeterminacy.

Papers:
* [[The Bianchi Identity of Differential Geometry M. W. Evans, H. Eckardt|http://aias.us/documents/uft/paper88.pdf]] pct.0

Links: 
* [[Evans on Torsion|http://www.americanantigravity.com/documents/Myron-Evans-Interview.pdf]] 
@@display:block;text-align:center;[img[My comments ...|images/comment.gif][Comments]]&nbsp;@@

>I believe there are
>$15.747.724.136.275.002.577.605.653.961.181.555.468.044.717.914.527.116.709.366.231.425.076.185.631.031.296 $
>protons in the universe and the same number of electrons.
> - Arthur Eddington, Mathematical Theory of Relativity (1923) -
Eddington arrived at the outrageous conclusion of the citation above after a series of convoluted (and wrong!) calculations in which he first "proved" that the value of the so-called fine-structure constant was exactly $1/136$. This value appears as a factor in his prescription for the number of particles (protons + electrons; neutrons were not discovered until 1930) in the universe: $2 \cdot 136 \cdot 2^{256} = 17 × 2^{260} = 3.149544\ldots \cdot 10^{79}$ (double the number written out in full in the quote above). This is the Eddington number, notable for being the largest specific integer (as opposed to an estimate or approximation) ever thought to have a unique and tangible relationship to the physical world. However, experimental data gave a slightly lower value for the fine-structure constant, closer to $1/137$. Unfazed, Eddington simply amended his "proof" to show that the value had to be exactly $1/137$, prompting the satirical magazine Punch to dub him "Sir Arthur Adding\-One."

Papers:
* [[Special-Relativistic Resolution of Ehrenfest's Paradox: Comments on Some Recent Statements by T. E. Phipps, Jr. -O. Gron|http://128.112.100.2/~mcdonald/examples/mechanics/gron_fp_11_623_81.pdf]]

>Newton successfully wrote apple = moon, but you cannot write apple=neutron.
>- J. L. Synge -

>There does not exist any known property of particles showing that spin is less important than mass. 
> - F. Lurçat -

''Einstein\-Cartan Theory'' (abbreviated ''EC Theory'') extends [[general relativity|Einstein Field Equations]] to allow for the description of spin. It presents the simplest model of a [[Poincaré gauge theory of gravity|Gauge Theory of Gravity]].
It is of fundamental importance to have both, mass and spin, as underpinning concepts of the space-time structure, as these quantities can be identified as classification labels of the irreducible representations of the [[Poincaré group|Poincaré Transformation]]. 

EC theory differs only very slightly from the classical Einstein theory. The effects of spin and torsion are significant only at very high densities of matter, however still much smaller than the Planck density at which quantum gravitational effects are believed to dominate.
The underlying space of EC theory is a [[Riemann-Cartan space|Riemann-Cartan Space]]. Space-time therefore is described by Non\-Riemannian geometry. A $4$-dimensional Cartan space-time is commonly denoted $\mathbb U_4$.

The EC theory implies that gravity cannot be described by the [[metric|Metric Tensor]] only, but that torsion has independent dynamical importance.
Thus the matter content in each spacetime point of the universe is presented in addition to the stress-energy tensor $T_{\mu\nu}$ by the the [[spin angular momentum tensor|Spin Angular Momentum Tensor]] $M^\rho_{\mu \nu}$. While $T_{\mu\nu}$ describes the mass density of spacetime, $M^\rho_{\mu \nu}$ describes its spin density.

The fundamental ''postulates of EC theory of gravitation'' are that the [[Cartan tensor|Cartan Tensor]] $C^\rho_{\mu \nu}$ is proportional to the spin angular momentum tensor and that the [[Einstein tensor|Einstein Tensor]] $G_{\mu \nu}$ is proportional to the stress-energy tensor (as is also the case in Einstein's theory of gravity), i.e.
\begin{eqnarray}
C^\rho_{\mu \nu} &=& T^\rho_{\mu\nu} + 2T_{[\mu} \delta_{\nu]}^\rho \equiv \kappa M^\rho_{\mu \nu} \\
G_{\mu \nu} &=&  R_{\mu\nu} - {1\over2} g_{\mu\nu}R \equiv \kappa T_{\mu \nu} 
\end{eqnarray}
with $\kappa = {8\pi G\over c^4}$.
The two sets of equations are called ''Einstein\-Cartan Equations''.
Although the latter have the same form as the [[Einstein equations|Einstein Field Equations]], they differ in that neither the [[Ricci tensor|Ricci Tensor]] nor the [[energy-momentum tensor|Stress Energy Tensor]] need to be symmetrical.

The dynamics of the tensor fields is governed by the following two equations 
\begin{eqnarray}
D_\rho C^\rho_{\mu\nu} &=& 2 R_{[\mu\nu]} + T_\rho T^\rho{}_{\mu\nu} \\ 
D_\rho G_\mu^\rho &=& T^\rho{}_{\sigma\tau} \left (\frac 12 R^{\tau\sigma}{}_{\rho\mu} + g_\mu^\tau R_\rho{}^\tau \right )
\end{eqnarray}
!!!!Coupling of the Electromagnetic Field
The (classical) electromagnetic field cannot be minimally coupled to torsion without breaking gauge invariance of the Lagrangian. (I .e. one cannot replace the flat with the curved metric and the derivative with the covariant derivative). This result finds a natural justification in the framework of the [[Poincaré gauge field theory of gravitation|Gauge Theory of Gravity]]. Several attempts have been made to solve this problem. They are successful, however, only at the cost of imposing arbitrary geometrical constraints upon torsion, or introducing a modified definition of a gauge transformation. Therefore these attempts lie beyond the limits of the EC theory. The most simple hypothesis within the framework of  Einstein\-Cartan theory seems to be to assume that photons neither produce nor feel torsion. This however can be assumed to be strictly valid only as long as the electromagnetic field is treated classically. If quantized, this condition only holds up to the first order of the perturbative expansion.

!!!!Properties
EC gravity has the potential to solve fundamental cosmological problems:
# The [[cosmological constant problem|Cosmological Constant]]: $\Omega_\Lambda \approx 1$.
# Predict the present mass density of the universe: $\Omega_{matter} $.
# Allow for singularity free solutions, contrary to the singular cosmological solutions in Einstein's gravity (due to the Penrose\-Hawking\-Geroch singularity theorems).
# Has sufficient freedom to exclude non-causal solutions (closed timelike curves).
# Can explain the source of density inhomogeneities in terms of quantum fluctuations of spin. (However for cosmological models based on chaotic inflation it is claimed that spin-torsion interactions are effective only at a very early stage of inflation and are therefore not affecting the standard inflationary scenario nor the density perturbations [1]).
# Torsion could be an essential factor for quantizing spacetime, due to its interpretation as a topological defect of the spacetime manifold.
Yet the problem of [[anomalies|Anomaly]] persists.

!!!!Historical
After Einstein's formulation of general relativity (1915), Cartan (1922) realized the possible importance of torsion for the geometrical theory of gravity. However, only forty years later Kibble and Sciama proposed a dynamical definition of spin connected with torsion, and this theory was further elaborated by Hehl and Trautman.

!!!! A personal remark
\Einstein-Cartan theory can be regarded as paradigmatic in respect to a unified geometrical description of spacetime and matter, as contrary to general relativity where the energy stress tensor is "put in by hand", here it is given by the geometry (namely the connection) of the space-time manifold.

Papers:
* [[On Primordial Cosmological Density Fluctuations in the Einstein-Cartan Gravity and COBE Data - D. Palle|http://arxiv.org/PS_cache/astro-ph/pdf/9811/9811408v1.pdf]] [[pct. 27|http://scholar.google.de/scholar?hl=de&lr=&cites=17371051949502840858]]  
* [[The Einstein-Cartan Theory - A. Trautman|http://www.fuw.edu.pl/~amt/ect.pdf]] [[pct. 18|http://scholar.google.com/scholar?hl=de&lr=&cites=5528988117515193029&um=1&ie=UTF-8&ei=uF0zSqSWH86O_AaO3pTIDQ&sa=X&oi=science_links&resnum=1&ct=sl-citedby]]
* [[Spin and the Structure of Space-Time - F. W. Hehl, P. von der Heyde|http://archive.numdam.org/ARCHIVE/AIHPA/AIHPA_1973__19_2/AIHPA_1973__19_2_179_0/AIHPA_1973__19_2_179_0.pdf]] [[local|papers/AIHPA_1973__19_2_179_0.pdf]] [[pct. 10|http://scholar.google.com/scholar?hl=de&lr=&cites=16863964518475846626&um=1&ie=UTF-8&ei=YFAzSunOL4mC_AaUi_WnCg&sa=X&oi=science_links&resnum=1&ct=sl-citedby]]
* [[Constitutive Theory in General Relativity: Spin-Material in Spaces with Torsion - H. J. Herrmann, G. Rückner, W. Muschik|http://www.emis.de/journals/RSMT/58-2/141.pdf]] [[pct. 5|http://scholar.google.de/scholar?hl=de&lr=&cites=15827073801215214613]]
* [[On a Completely Antisymmetric Cartan Tensor - L. Fabbri|http://arxiv.org/PS_cache/gr-qc/pdf/0608/0608090v2.pdf]] [[pct. 2|http://scholar.google.de/scholar?hl=de&lr=&cites=1653535039168438446]]
* [[Totally Asymmetric Torsion on Riemann-Cartan Manifold Y. Lam|http://arxiv.org/PS_cache/gr-qc/pdf/0211/0211009v1.pdf]] [[pct. 2|http://scholar.google.de/scholar?cites=13805618791711710878&hl=de]]
* [[Gravimetry, Relativity and the Global Navigation Satellite Systems - Second Lesson: Introduction to Differential Geometry - A. Tarantola|http://www.ipgp.fr/~tarantola/Files/Professional/Teaching/Seminar/Lessons/DifferentialGeometry/Text/DifferentialGeometry.pdf]] [[local|papers/DifferentialGeometry.pdf]]  prl. 0
* [[[1] Spin-Torsion in Chaotic Inflation - L. C. Garcia de Andrade, R. O. Ramos|http://arxiv.org/PS_cache/gr-qc/pdf/9910/9910053v1.pdf]] pct. 0

Theses:
* [[Exact String-Type Solutions in the Einstein-Cartan Theory of Gravity - R. A. Puntigam|http://www.puntigam.org/roland/pdf/thesis.pdf]]

Presentations:
* [[Gravimetry, Relativity, and the Global Navigation Satellite Systems - A. Tarantola|http://www.ipgp.fr/~tarantola/Files/Professional/Teaching/Seminar/Lessons/DifferentialGeometry/Slides/DifferentialGeometrySlides.pdf]] [[local|papers/DifferentialGeometrySlides.pdf]] prl. 10

Links:
* [[WIKIPEDIA - Einstein-Cartan Theory|http://en.wikipedia.org/wiki/Einstein%E2%80%93Cartan_theory]]

''Einstein–Cartan Theory'' extends general relativity to allow for the description of spin.
The description of spacetime includes [[torsion|Torsion]] and the energy-momentum tensor is generally nonsymmetric.
The Einstein–Cartan theory differs only very slightly from the Einstein theory. The effects of spin and torsion are significant only at very high densities of matter, however still   much smaller than the Planck density at which quantum gravitational effects are believed to dominate.

Papers:
* [[The Einstein–Cartan Theory - A. Trautman|http://www.fuw.edu.pl/~amt/ect.pdf]]
*[[On a Completely Antisymmetric Cartan Tensor - L. Fabbri|http://arxiv.org/PS_cache/gr-qc/pdf/0608/0608090v2.pdf]]

The ''Electroweak Gauge Potential'' is given by:
\[
\mb{W}_{\mu} = (W^0_\mu, W^1_\mu, W^2_\mu, W^3_\mu) \equiv (B_\mu, W^1_\mu, W^2_\mu, W^3_\mu)
\]
One therefore has $16$ field components for the electroweak field.

The ''Electro\-Weak [[Field Strength Tensor|Field Strength Tensor]]'' derived from it is
\[
\mb{W}_{\mu\nu}  = \partial_{\mu} \mb{W}_{\nu} - \partial_{\nu} \mb{W}_{\mu} + g \mb{W}_{\mu} \times \mb{W}_{\nu}
\]
which is antisymmetric.

In component notation this reads
\[
W_{\mu\nu}^{a}  = \partial_{\mu} W^{a}_{\nu} - \partial_{\nu} W^{a}_{\mu} + g \epsilon_{abc}W^{b}_{\mu}W^{c}_{\nu}
\]

''Electroweak Currents'' are in general decomposed into a vector current $V^\mu_a$ and an axial-vector current $A^\mu_a$.
The vector parts of the charge changing current and the isovector piece of the electromagnetic current are three components of a vector in isospace. All $3$ components are conserved. 
The axial current is not conserved, even in the chiral limit.

Books:
* [[Electroweak Theory - E. A. Paschos|books/electroweak_theory_emmanuel_paschos.pdf]] [[bct. 6|http://scholar.google.de/scholar?cites=2474110183633045376&hl=de&as_sdt=2000]]

The ''Weak Equivalence Principle'' states that all particles follow the same path in a gravitational field independent of their mass.

The ''Strong Equivalence Principle'' states that an accelerated reference frame is equivalent to gravitation, or that mass curves space, and accelerated motion is due to the curvature. Technically speaking this means that all physical laws that hold in flat Minkowski space (i.e. “special relativity”) continue to hold in every reference frame provided one replaces derivatives by covariant derivatives.

Videos:
* [[The Quest for a Living World|http://discovermagazine.com/video/science-videos/quest-for-a-living-world]]

Letting the symmetric group $S_7$ act on the [[Fano plane|Fano Plane]] one gets $\operatorname{ord}(S_7) = 7!$ different labellings.
Equivalently one can describe the Fano plane as a [[Steiner triple system|Steiner Triple System]] on seven points, i.e. a $STS(7)$, given for example by:
\begin{eqnarray}
STS(7) &= &\{(\mb e_1,\mb e_2,\mb e_3),(\mb e_1,\mb e_4,\mb e_5),(\mb e_1,\mb  e_6,\mb  e_7),(\mb  e_2,\mb  e_4,\mb  e_6),\\
&&\;\;(\mb  e_2,\mb  e_5,\mb  e_7),(\mb e_3,\mb e_4,\mb e_7),(\mb e_3,\mb  e_5,\mb  e_6) \}
\end{eqnarray}
The action of a permutation $\sigma \in S_7$ on it is given by
\begin{eqnarray}
\sigma(STS(7))& =&\{(\mb e_{\sigma(1)},\mb e_{\sigma(2)},\mb e_{\sigma(3)}),(\mb e_{\sigma(1)},\mb e_{\sigma(4)},\mb e_{\sigma(5)}),(\mb e_{\sigma(1)},\mb  e_{\sigma(6)},\mb e_{\sigma(7)}),(\mb  e_{\sigma(2)},\mb  e_{\sigma(4)},\mb  e_{\sigma(6)}), \\
&& \;\;(\mb  e_{\sigma(2)},\mb  e_{\sigma(5)},\mb  e_{\sigma(7)}),(\mb e_{\sigma(3)},\mb e_{\sigma(4)},\mb e_{\sigma(7)}),(\mb e_{\sigma(3)},\mb  e_{\sigma(5)},\mb  e_{\sigma(6)}) \}
\end{eqnarray}
[[Automorphisms|Automorphism]] are those permutations the preserve the groupings of the triples (blocks). These are the elements of the automorphism group [[PSL(2,7)]].

Example: $\sigma(1) = 1,\, \sigma(2) = 4,\, \sigma(3) = 5, \, \sigma(4) = 2, \, \sigma(5) = 3,\, \sigma(6) = 6,\, \sigma(7) = 7$
\begin{eqnarray}
\Rightarrow \, \sigma(STS(7)) &=&\{(\mb e_1,\mb e_4,\mb e_5),(\mb e_1,\mb e_2,\mb e_3),(\mb e_1,\mb  e_6,\mb e_7),(\mb  e_4,\mb  e_2,\mb  e_6),(\mb  e_4,\mb  e_3,\mb  e_7),(\mb e_5,\mb e_2,\mb e_7),(\mb e_5,\mb  e_3,\mb  e_6) \} \\
&=& STS(7)
\end{eqnarray}

$PSL(2,7)$ divides the order of $S_7$ by $168$ such that one is left with $30$ [[cosets|Coset]] (which are still isomorphic to one another). These $30$ cosets are considered in the following, picking one representant of each, resulting in a set of $30$ Fano planes, which will be refered to as "different Fano planes".

The $30$ differentFano planes form one [[orbit|Orbit]] under the symmetric group $S_7$ and two orbits of length $15$ each under the [[alternating group|Alternating Group]] $A_7$.

Alternatively they can be partitioned into $6$ [[orbits|Orbit]] of orders $7,7,7,7,7,1,1$ respectively under the action of a cyclic shift. (See example below).

$2$ among the the $30$ labellings have either $0$, $1$ or $3$ lines (=triples of labels) in common.
There is a unique partition of the $30$ different Fano planes into $2$ sets of $15$ planes each, such that any $2$ Fano planes in one of the sets have exactly $1$ line in common. Both sets allow for the construction of the Fano tetrahedron of the projective geometry [[PG(3,2)]] or the [[Hoffman-Singleton graph|Hoffman-Singleton Graph]].

!!!!Class 1:
The following $2$ equivalent pictures show the $15$ Fano planes of the first class mentioned above:

<html><center><img src="images/FanoPlanes.gif" style="width: 750px; "/></center></html>
<html><center><img src="images/15planes1.jpg" style="width: 420px; "/></center></html>
The associated $15$ [[Steiner triple systems|Steiner Triple System]] (STS(7)) ordered lexicographically are given by:
\begin{eqnarray}
O_1: \{1,2,3\}, \{1,4,5\}, \{1,6,7\}, \{2,4,7\}, \{2,5,6\}, \{3,4,6\}, \{3,5,7\}\\
\{1,2,3\}, \{1,4,6\}, \{1,5,7\}, \{2,4,5\}, \{2,6,7\}, \{3,4,7\}, \{3,5,6\}\\
\{1,2,3\}, \{1,4,7\}, \{1,5,6\}, \{2,4,6\}, \{2,5,7\}, \{3,4,5\}, \{3,6,7\}\\
\{1,2,4\}, \{1,3,5\}, \{1,6,7\}, \{2,3,6\}, \{2,5,7\}, \{3,4,7\}, \{4,5,6\}\\
\{1,2,4\}, \{1,3,6\}, \{1,5,7\}, \{2,3,7\}, \{2,5,6\}, \{3,4,5\}, \{4,6,7\}\\
\{1,2,4\}, \{1,3,7\}, \{1,5,6\}, \{2,3,5\}, \{2,6,7\}, \{3,4,6\}, \{4,5,7\}\\
\{1,2,5\}, \{1,3,4\}, \{1,6,7\}, \{2,3,7\}, \{2,4,6\}, \{3,5,6\}, \{4,5,7\}\\
\{1,2,5\}, \{1,3,6\}, \{1,4,7\}, \{2,3,4\}, \{2,6,7\}, \{3,5,7\}, \{4,5,6\}\\
\{1,2,5\}, \{1,3,7\}, \{1,4,6\}, \{2,3,6\}, \{2,4,7\}, \{3,4,5\}, \{5,6,7\}\\
\{1,2,6\}, \{1,3,4\}, \{1,5,7\}, \{2,3,5\}, \{2,4,7\}, \{3,6,7\}, \{4,5,6\}\\
\{1,2,6\}, \{1,3,5\}, \{1,4,7\}, \{2,3,7\}, \{2,4,5\}, \{3,4,6\}, \{5,6,7\}\\
\{1,2,6\}, \{1,3,7\}, \{1,4,5\}, \{2,3,4\}, \{2,5,7\}, \{3,5,6\}, \{4,6,7\}\\
\{1,2,7\}, \{1,3,4\}, \{1,5,6\}, \{2,3,6\}, \{2,4,5\}, \{3,5,7\}, \{4,6,7\}\\
O_1: \{1,2,7\}, \{1,3,5\}, \{1,4,6\}, \{2,3,4\}, \{2,5,6\}, \{3,6,7\}, \{4,5,7\}\\
\{1,2,7\}, \{1,3,6\}, \{1,4,5\}, \{2,3,5\}, \{2,4,6\}, \{3,4,7\}, \{5,6,7\}
\end{eqnarray}
An example of a cyclic shift is the following: We take the first STS and get:
\begin{eqnarray}
\{1,2,3\}, \{1,4,5\}, \{1,6,7\}, \{2,4,7\}, \{2,5,6\}, \{3,4,6\}, \{3,5,7\} \rightarrow\\
\{2,3,4\}, \{2,5,6\}, \{2,7,1\}, \{3,5,1\}, \{3,6,7\}, \{4,5,7\}, \{4,6,1\} = \\
\{2,3,4\}, \{2,5,6\}, \{1,2,7\}, \{1,3,5\}, \{3,6,7\}, \{4,5,7\}, \{1,4,6\} = \\
\{1,2,7\}, \{1,3,5\}, \{1,4,6\}, \{2,3,4\}, \{2,5,6\}, \{3,6,7\}, \{4,5,7\}\\
\end{eqnarray}
which is the second last of the STS listed above, denoted $O_1$.

!!!!Class 2:
The class set of $15$ Fano planes is depicted in the following:

<html><center><img src="images/15planes2.jpg" style="width: 420px; "/></center></html>
The associated $15$ Steiner triple systems ordered lexicographically are:
\begin{eqnarray}
\{1,2,3\},\{1,4,5\},\{1,6,7\},\{2,4,6\},\{2,5,7\},\{3,4,7\},\{3,5,6\}\\
\{1,2,3\},\{1,4,6\},\{1,5,7\},\{2,4,7\},\{2,5,6\},\{3,4,5\},\{3,6,7\}\\
\{1,2,3\},\{1,4,7\},\{1,5,6\},\{2,4,5\},\{2,6,7\},\{3,4,6\},\{3,5,7\}\\
\{1,2,4\},\{1,3,5\},\{1,6,7\},\{2,3,7\},\{2,5,6\},\{3,4,6\},\{4,5,7\}\\
\{1,2,4\},\{1,3,6\},\{1,5,7\},\{2,3,5\},\{2,6,7\},\{3,4,7\},\{4,5,6\}\\
\{1,2,4\},\{1,3,7\},\{1,5,6\},\{2,3,6\},\{2,5,7\},\{3,4,5\},\{4,6,7\}\\
\{1,2,5\},\{1,3,4\},\{1,6,7\},\{2,3,6\},\{2,4,7\},\{3,5,7\},\{4,5,6\}\\
\{1,2,5\},\{1,3,6\},\{1,4,7\},\{2,3,7\},\{2,4,6\},\{3,4,5\},\{5,6,7\}\\
\{1,2,5\},\{1,3,7\},\{1,4,6\},\{2,3,4\},\{2,6,7\},\{3,5,6\},\{4,5,7\}\\
\{1,2,6\},\{1,3,4\},\{1,5,7\},\{2,3,7\},\{2,4,5\},\{3,5,6\},\{4,6,7\}\\
\{1,2,6\},\{1,3,5\},\{1,4,7\},\{2,3,4\},\{2,5,7\},\{3,6,7\},\{4,5,6\}\\
\{1,2,6\},\{1,3,7\},\{1,4,5\},\{2,3,5\},\{2,4,7\},\{3,4,6\},\{5,6,7\}\\
\{1,2,7\},\{1,3,4\},\{1,5,6\},\{2,3,5\},\{2,4,6\},\{3,6,7\},\{4,5,7\}\\
\{1,2,7\},\{1,3,5\},\{1,4,6\},\{2,3,6\},\{2,4,5\},\{3,4,7\},\{5,6,7\}\\
\{1,2,7\},\{1,3,6\},\{1,4,5\},\{2,3,4\},\{2,5,6\},\{3,5,7\},\{4,6,7\}
\end{eqnarray}
!!!![[MAGMA|http://magma.maths.usyd.edu.au/calc/]]^^[[Help|MAGMA]]^^ examples
* [[Code File|code/MAGMAFanoPlanes.txt]]

Papers:
* [[A Note on the Covering of all Triples on 7 Points with Steiner Triple Systems - A. E. Brouwer|http://www.win.tue.nl/~aeb/preprints/zn63.pdf]] [[pct. 4|http://scholar.google.de/scholar?cites=15007749364926377386&hl=de&as_sdt=2000]]
* [[YEA WHY TRY HER RAW WET HAT A Tour of the Smallest Projective Space - B. Polster| http://www.qedcat.com/articles/yea.pdf]] [[local|papers/yea.pdf]] [[pct. 3|http://scholar.google.com/scholar?cites=11018116709271941363&hl=de&as_sdt=2000]]

Presentations:
* [[Some New and Old Results Regarding Room Squares and Related Designs - J. Dinitz|http://www.emba.uvm.edu/~dinitz/mcccc.09.ppt]] [[local|presentations/mcccc.09.ppt]]

The ''Fermi Paradox'' is the apparent contradiction between high estimates of the probability of the existence of extraterrestrial civilizations and the lack of evidence for such civilizations and contact with them.

Links:
* [[WIKIPEDIA - Fermi Paradox|http://en.wikipedia.org/wiki/Fermi_paradox]]

!!!!Properties
* For flat spaces the path integral formalism is completely equivalent to the [[Schwinger action principle|Action Principle]]. This equivalence holds equally well for quantum field theory, which is usually regarded as involving a flat configuration space.

Papers:
*[[The Schwinger Action Principle and the Feynman Path Integral for Quantum Mechanics in Curved Space - D. J. Toms|http://arxiv.org/PS_cache/hep-th/pdf/0411/0411233v1.pdf]] [[pct. 1|http://scholar.google.de/scholar?cites=12337870966204431579&hl=de]]
* [[Challenges to Path Integral Formulations of Quantum Theories - R. Jackiw|http://arxiv.org/PS_cache/arxiv/pdf/0711/0711.1514v1.pdf]] [[pct. 1|http://scholar.google.de/scholar?cites=7114550071588992508&hl=de]]

!!!!Electrodynamics
The field strength tensor $F^{\mu\nu} $ is defined by:
\[
F_{\mu\nu}  = \partial_{\mu}A_{\nu} - \partial_{\nu} A_{\mu}
\]
Written out explicitely one has
\[
F_{\mu\nu} \equiv
\left(\begin{matrix}
0  &  E_x &  E_y & E_z \\
-E_x &   0  &  -B_z & B_y \\
-E_y & B_z &   0  &  -B_x \\
-E_z &  -B_y & B_x &   0  \\
\end{matrix}\right)
\]
Its [[dual|Duality Rotation]] $\tilde F^{\mu\nu} $ is defined by:
\[
\tilde{F}^{\mu\nu} \equiv \frac{1}{2}\, \varepsilon^{\mu\nu\alpha\beta}\,F_{\alpha\beta}  =
\begin{pmatrix}
0  & -B_x & -B_y & -B_z \\
B_x &   0  &  E_z & -E_y \\
B_y & -E_z &   0  &  E_x\\
B_z &  E_y & -E_x &   0 \\
\end{pmatrix}
\]
!!!!!Properties
* Antisymmetry: $F_{\mu\nu} = ? F_{\nu\mu}$
* Tracelesness: $F_{\mu\mu} = 0$
* $6$ independent components

A long range ''Fifth Force'' is predicted by some extensions of the [[standard model|Standard Model]].
It has been hypothesized to impact large-scale structure formation. If it was attractive and very long range, it would effectively increase the gravitational field strength and thus accelerate structure formation. Previous studies have shown that such a force could reduce discrepancies between observations and predictions, e.g. by increasing the number of galaxy clusters and superclusters and reducing voids, which would agree better with observations.

Papers:
* [[Fifth Force from Fifth Dimension: A Comparison between two Different Approaches - F. Dahia, E. M. Monte, C. Romero|http://arxiv.org/PS_cache/gr-qc/pdf/0303/0303044v1.pdf]] [[pct. 1|http://scholar.google.de/scholar?cites=17847174983934044315&hl=de]]

See also: 
* [[Organic universe|Organic Universe]]
* [[Anthropic principle|Anthropic Principle]]
* [[Intelligent design|Intelligent Design]]

Papers:
* [[Why the Universe is just so (2000) - C. J. Hogan|http://llacolen.ciencias.uchile.cl/~vmunoz/download/papers/h00.pdf]] [[local|papers/h00.pdf]] {{t100Cite{[[pct. 111|http://scholar.google.de/scholar?cites=2011791294939170025&as_sdt=2005&sciodt=2000&hl=de]]}}}

Links:
* [[WIKIPEDIA - Feinabstimmung der Naturkonstanten|http://de.wikipedia.org/wiki/Feinabstimmung_der_Naturkonstanten]]

Videos: 
* [[Alpha-Centauri: Was ist die Beryllium-Barriere?|http://www.br-online.de/br-alpha/alpha-centauri/alpha-centauri-beryllium-barriere-2005-ID1207917317644.xml]]

<html><center><img src="images/p_mannheim.jpg" style="width: 680px; "/></center></html>$\quad\quad\quad\quad$ - Philip Mannheim -

Papers:
* [[Living with Ghosts - S. W. Hawking, T. Hertog|http://arxiv.org/PS_cache/hep-th/pdf/0107/0107088v2.pdf]] [[pct. 68|http://scholar.google.de/scholar?hl=de&lr=&cites=5775590995509111619]]
* [[On the History of Fourth Order Metric Theories of Gravitation - R. Schimming, H. J. Schmidt|http://arxiv.org/PS_cache/gr-qc/pdf/0412/0412038v1.pdf]] [[pct. 11|http://scholar.google.de/scholar?hl=de&lr=&cites=11897331899145799901]]

The exceptional [[Lie group|Lie Group]] ''$G_2$'' ist the [[automorphism group|Automorphism]] of the [[octonions|Octonion]], i.e.
\[
G_2 = Aut (\mathbb O) = \{g \in GL(\mathbb O), \mb O_1, \mb O_2 \in \mathbb O: g(\mb O_1 \mb O_2) = g(\mb O_1) g(\mb O_2) \}
\]
The octonions themselves cannot form an automorphism group since they are not associative. However a special (associative) subset of them, consisting of the elements of $G_2$, can meet the requirements.
The group $G_2 \in SO(7)$ contains a subset of seven-dimensional rotations. Essentially the group elements of $G_2$ are simultaneous rotations in two planes of $\mathbb R^7$.

The $G_2$ automorphisms preserve the multiplication table of the octonions and hence leave invariant the relation
\[
\mb E_i \mb E_j = C_{ijk} \mb E_k .
\]

For the split form of $G_2$ which is related to the [[split octonions|Split Octonion]], see [[Split G2]].

Papers:
* [[Euler Angles for G2 - Sergio L. Cacciatori, B. L. Cerchiai, A. D. Vedova, G. Ortenzi, A. Scotti|http://arxiv.org/PS_cache/hep-th/pdf/0503/0503106v2.pdf]] [[pct. 15|http://scholar.google.com/scholar?hl=de&lr=&cites=13587689812387963729&um=1&ie=UTF-8&ei=3zjsS_2LGMXD-QanxoG0BA&sa=X&oi=science_links&resnum=2&ct=sl-citedby&ved=0CCQQzgIwAQ]]
* [[A Construction of G2 Holonomy Spaces with Torus Symmetry - O.P. Santillan|http://arxiv.org/PS_cache/hep-th/pdf/0208/0208190v3.pdf]] [[pct. 8|http://scholar.google.de/scholar?cites=13483596094794087435&hl=de&as_sdt=2000]]
*  [[Associative Cones and Integrable Systems (2006) - S. Kong, C.-L. Terng, E. Wang|http://math.uci.edu/~cterng/Assoc_cone_11_14_05.pdf]] [[pct. 2|http://scholar.google.de/scholar?cites=17860433930639169933&hl=de&as_sdt=2000]]
* [[Manifolds with G2 Holonomy (2003) - J. D. Olson|http://people.maths.ox.ac.uk/~hausel/m392cr/olson.pdf]] pct. 0

* [[GAP Online Manuals|http://www.gap-system.org/Doc/manuals.html]]
** [[GAP Release 4.4.12 Reference Manual|http://www.gap-system.org/Manuals/doc/ref/manual.pdf]] [[local|documents/GAPReferenceManual.pdf]]
** [[GUAVA - A GAP4 Package for Computing with Error-correcting Codes|http://www.gap-system.org/Manuals/pkg/guava3.10/doc/manual.pdf]] [[Html-version|http://www-history.mcs.st-and.ac.uk/~gap/Manuals/pkg/guava3.10/htm/chap0.html]] [[local|documents/GUAVAManual.pdf]]
** [[Loops Package|GAP Loops Package]]
** [[AtlasRep|http://www.math.rwth-aachen.de/~Thomas.Breuer/atlasrep/index.html]] - An interface between GAP and the Atlas of Group Representations, a database that comprises representations of many almost simple groups and information about their maximal subgroups.

Examples:
* [[Applied Abstract Algebra - D. Joyner, R. Kreminski, J. Turisco|http://www.usna.edu/Users/math/wdj/book/book.html]]

''GTG'' is an acronym for ''Gauge Theory of Gravity'' which is a theory that describes gauge fields in a unified way by means of the [[spacetime algebra|Spacetime Algebra]] and [[Clifford geometric calculus|Clifford Geometric Algebra]]. It is based on a different approach than are [[gauge theories of gravity|Gauge Theory of Gravity]] of the Poincaré group. 

Papers:
* [[Gravity, Gauge Theories and Geometric Algebra - A. Lasenby, C. Doran, S. Gull|http://www.mrao.cam.ac.uk/~clifford/publications/ps/gravity.pdf]] [[local|papers/gravity.pdf]] {{t100Cite{[[pct. 106|http://scholar.google.de/scholar?cites=15459370966736119609&hl=de&as_sdt=2000]]}}}
* [[Spacetime Geometry with Geometric Calculus  - D. Hestenes|http://geocalc.clas.asu.edu/pdf/SpacetimeGeometry.w.GC.proc.pdf]] [[pct. 1|http://scholar.google.com/scholar?hl=de&lr=&cites=9273923840321077641&um=1&ie=UTF-8&ei=MfOsS9L3IJagsQb-tpGcAw&sa=X&oi=science_links&resnum=2&ct=sl-citedby&ved=0CBUQzgIwAQ]] prl. 9 - To be able to better compare the notation in the paper with the notation used in this WIKI, the following correspondences may be helpful: $L_{\mu\nu}^\lambda \sim\Gamma_{\mu\nu}^\lambda$, $g_\mu  \sim \mb e_\mu (\mb x)$,  $\gamma_a  \sim \mb e_a$. 

Links:
* [[Cambridge University Geometric Algebra Research Group Home Page|http://www.mrao.cam.ac.uk/~clifford/index.html]] - Web site of the "inventors" of the theory.

>A final word about Clifford algebras in general: they are, like Lie groups, profligate. There are too many of them, an infinite number of both Lie groups and Clifford algebras that are physically irrelevant, not a part of the design of reality. This is and always was the problem with GUT theories based on a unifying large Lie group, and it is and always will be the problem with unification theories based on large Clifford algebras. In both cases it is the principal of the educated guess that leads to the choice of unifying algebraic object. This is unsatisfactory. Nature can not be so arbitrarily ugly.
>- Geoffrey M. Dixon - Division algebras: octonions, quaternions, complex numbers, and the Algebraic Design of Physics -

Group embeddings of the standard model:
[[E8]] $\supset$ [[E7]] $\supset$ [[E6]] $\supset SO(10) \supset SU(5) \supset SU(3)_c \times SU(2)_L  \times U(1)_Y$
However the irreducible representations of a GUT should be [[chiral|Chirality]]. The inclusions [[E8]] $\supset$ [[E7]] $\supset$ [[E6]] must therefore be ruled out. Instead it is conjectured that the gauge group is $E_8 \times E_8$ or [[SO(32)]] which one has to somehow break down to e.g. $E_6$.

!!!! Unification with Clifford Algebras:
<html><center><img src="images/clifford_unification.jpg" style="width: 420px; "/></center></html>
Papers:
* [[Aspects of Grand Unification in Higher Dimensions - A. A. Wingerter|http://deposit.ddb.de/cgi-bin/dokserv?idn=976474522&dok_var=d1&dok_ext=pdf&filename=976474522.pdf]] [[local|papers/dokserv.pdf]] [[pct. 2|http://scholar.google.de/scholar?cites=6613607189809313311&hl=de&as_sdt=2000]]
* [[The Algebra of Grand Unified Theories - J. Baez, J. Huerta|http://arxiv.org/PS_cache/arxiv/pdf/0904/0904.1556v1.pdf]] [[pct. 1|http://scholar.google.de/scholar?cites=1321160442974930747&hl=de&as_sdt=2000]]

The ''(Dirac) Gamma Matrices'' $\bs \gamma_\mu$ form a [[Clifford algebra|Clifford Algebra]], i.e. they satisfy
\begin{equation}
\{\bs{\gamma}_\mu,\bs{\gamma}_\nu\} = 2 \eta_{\mu\nu} \bs{1}
\end{equation}
''Dirac representation''
\begin{equation}
\bs \gamma_{0}=\left(\begin{matrix}\bs 1 & \bs 0\\
\bs 0 & -\bs 1\end{matrix}\right),\quad \bs \gamma_{m}=\left(\begin{matrix}\bs{0} & \bs \sigma_{m}\\
-\bs \sigma_{m} & \bs 0 \end{matrix}\right) \quad m= 1,2,3
\end{equation}
with $\bs\sigma_{m}$ the [[Pauli matrices|Pauli Matrices]].

Explicitely this reads
\begin{equation}
\bs \gamma_{0} = \begin{pmatrix}\mb 1 &\mb  0 & \mb  0 & \mb 0\\
\mb 0 & \mb 1 &\mb  0 &\mb  0 \\
\mb 0 & \mb 0 &- \mb 1 &\mb  0\\
\mb 0 & \mb 0 & \mb 0 &-\mb 1\end{pmatrix}
\end{equation}
\begin{eqnarray}
\bs \gamma_{1} =  \begin{pmatrix}\mb  0 & \mb  0 &\mb  0 & \mb  1 \\
\mb 0 & \mb 0 & \mb 1 &\mb  0\\
\mb 0 &-\mb 1 &\mb  0 &\mb  0\\
\mb -1 & \mb 0 & \mb 0 & \mb 0\end{pmatrix}, \quad

\bs \gamma_{2} &= & \begin{pmatrix}\mb 0 & \mb 0 & \mb 0 &-\mb i\\
\mb 0 & \mb 0 & \mb i & \mb 0\\
\mb 0 &\mb  i & \mb 0 & \mb 0\\
-\mb i & \mb 0 &\mb  0 &\mb  0\end{pmatrix}, \quad

\bs \gamma_{3}= \begin{pmatrix}\mb 0 & \mb 0 &\mb  1 & \mb 0\\
\mb 0 & \mb 0 &\mb  0 &-\mb  1\\
-\mb 1 & \mb 0 & \mb 0 &\mb  0\\
\mb 0 & \mb 1 &\mb  0 & \mb  0\end{pmatrix}
\end{eqnarray}
furthermore one defines a matrix by means the product of all the 4 matrices above
\begin{equation}
\bs \gamma_{5}= i  \bs \gamma_{0} \bs \gamma_{1} \bs \gamma_{2} \bs \gamma_{3}
\end{equation}
''Weyl (chiral) representation''
\begin{equation}
\bs{\gamma}_{0}= -i \left(\begin{matrix}\mb 0 & \mb 1\\
\mb 1 & \mb 0\end{matrix}\right),\quad\bs{\gamma}_{m}=\left(\begin{matrix}\mb 0 & \bs{\sigma}_{m}\\
-\bs{\sigma}_{m} & \mb 0\end{matrix}\right) \quad m= 1,2,3
\end{equation}
or explicitely
\begin{equation}
\bs \gamma_{0} = \begin{pmatrix}\mb 1 &\mb  0 & \mb  0 & \mb 0\\
\mb 0 & \mb 1 &\mb  0 &\mb  0 \\
\mb 0 & \mb 0 &- \mb 1 &\mb  0\\
\mb 0 & \mb 0 & \mb 0 &-\mb 1\end{pmatrix}
\end{equation}
\begin{eqnarray}
\bs \gamma_{1} =  \begin{pmatrix}\mb  0 & \mb  0 &\mb  0 & \mb  1 \\
\mb 0 & \mb 0 & \mb 1 &\mb  0\\
\mb 0 &-\mb 1 &\mb  0 &\mb  0\\
\mb -1 & \mb 0 & \mb 0 & \mb 0\end{pmatrix}, \quad

\bs \gamma_{2} &= & \begin{pmatrix}\mb 0 & \mb 0 & \mb 0 &-\mb i\\
\mb 0 & \mb 0 & \mb i & \mb 0\\
\mb 0 &\mb  i & \mb 0 & \mb 0\\
-\mb i & \mb 0 &\mb  0 &\mb  0\end{pmatrix}, \quad

\bs \gamma_{3}= \begin{pmatrix}\mb 0 & \mb 0 &\mb  1 & \mb 0\\
\mb 0 & \mb 0 &\mb  0 &-\mb  1\\
-\mb 1 & \mb 0 & \mb 0 &\mb  0\\
\mb 0 & \mb 1 &\mb  0 & \mb  0\end{pmatrix}
\end{eqnarray}

Given gauge fields $h_\mu^A (\mb{x})$ and a gauge field tensor $\mb{F}_{\mu\nu} (\mb{x})$ defined by
\[
\mb{F}_{\mu\nu}(\mb{x}) \equiv F_{\mu\nu}^A(\mb{x}) \mb{E}_A
\]
where $\mb{E}_A$ are the basis vectors of the underlying algebra (or in case of a group its generators) the components of the field strength tensor $ F_{\mu\nu}^A(\mb{x})$ are constructed as follows:
\[
F_{\mu\nu}^A(\mb{x}) = \partial_\mu h_\nu^A (\mb{x}) - \partial_\nu h_\mu^A (\mb{x}) + C_{BC}^A h_\mu^B (\mb{x}) h_\nu^C (\mb{x})
\]
with $C_{BC}^A$ the [[structure constants|Structure Constants]] of the algebra (or the generators of the group respectively).

Unlike [[GR|General Relativity]], in which the metric plays the fundamental role, gauge-theoretic approaches to gravity including spinorial degrees of freedom require the introduction of  orthonormal frames of reference, i.e. [[tetrads|Tetrad]].

!!!!Historical developments
* Utiyama (1956): Gauge group: [[Lorentz group|Lorentz Transformation]] $SO(1,3)$.
* [[Kibble, Sciama|Einstein-Cartan-Sciama-Kibble Theory]] (1961/62):  Gauge group: [[Poincaré group|Poincaré Transformation]] $ISO(1,3)$.
* [[MacDowell and Mansouri|MacDowell-Mansouri Theory]]: Gauge group: [[De Sitter group|De Sitter Space]].
* Lasenby, Doran, Gull: [[GTG]], a gauge theory of gravity based on the [[(Clifford)-spacetime algebra|Spacetime Algebra]] and geometric calculus.

Papers:
* [[On the Gauge Aspects of Gravity - F. Gronwald, F. W. Hehl|http://arxiv.org/PS_cache/gr-qc/pdf/9602/9602013v1.pdf]] {{t100Cite{[[pct. 129|http://scholar.google.de/scholar?hl=de&lr=&cites=10614705260219585310]]}}}
* [[The Gauge Treatment of Gravity - D. Ivanenko, G. Sardanashvily|http://d.scribd.com/docs/1a20pii30gr5gsf7o4tq.pdf]] [[local|papers/1a20pii30gr5gsf7o4tq.pdf]] [[pct. 70|http://scholar.google.de/scholar?cites=17052178446235642366&hl=de&as_sdt=2000]]
* [[Curve It, Gauge It, or Leave it? Practical Underdetermination in Gravitational Theories - H. Lyre, T. O. Eynck|http://philsci-archive.pitt.edu/archive/00000514/00/Cgl.pdf]] [[local|papers/Cgl.pdf]] [[pct. 8|http://scholar.google.de/scholar?cites=15362596553086915092&hl=de&as_sdt=2000]]
* [[On Foundations of Poincaré - Gauge Theory of Gravity - B. N.Frolov|http://arxiv.org/PS_cache/gr-qc/pdf/0507/0507103v1.pdf]] [[pct. 3|http://scholar.google.de/scholar?cites=10787058953442112313&hl=de&as_sdt=2000]]
* [[Gauging the Twisted Poincaré Symmetry as Noncommutative Theory of Gravitation - M. Chaichian, M. Oksanen, A. Tureanu, G. Zet|http://arxiv.org/PS_cache/arxiv/pdf/0807/0807.0733v1.pdf]]
* [[Gauge Theory of Gravitation and General Relativity - G. Zet|http://www.infim.ro/rrp/2005_57_4/09-623-647.pdf]] pct. 0

Links:
* [[WIKIPEDIA - Geomerical Frustration|http://en.wikipedia.org/wiki/Geometrical_frustration]]
* [[Geomerical Frustration - Physics Today 02/2006|http://www.physics.rutgers.edu/grad/681/GFrustration_physics.today.pdf]]

>Einstein’s “general relativity,” ... has two central ideas: (1) Spacetime geometry “tells” mass-energy how to move; and (2) mass-energy “tells” spacetime geometry how to curve. ... the way spacetime tells mass-energy how to move is automatically obtained from the Einstein field equation by using the identity of Riemannian geometry, known as the Bianchi identity, which tells us that the covariant divergence of the Einstein tensor is zero.
> I. Ciufolini, J. A. Wheeler - Gravitation and Inertia

The [[Riemannian geometry|Riemann Space]] underlying Einstein’s theory can be formulated either in terms of the [[metric|Metric Tensor]] $g_{\mu\nu}$ or a frame field ([[vielbein|Tetrad]]) ${h_\mu}^a$.

Videos:
* [[Einstein's Theory (lecture 1 - 12) - L. Susskind|http://www.youtube.com/view_play_list?p=6C8BDEEBA6BDC78D]]
* [[Caltech's Physics: Gravitational Waves - A Web-Based Course|http://elmer.tapir.caltech.edu/ph237/]]
* [[50 Years of the Cauchy Problem in General Relativity|http://fanfreluche.math.univ-tours.fr/Cauchy2.html]]

* [[The Sequence of the Human Genome - J. C. Venter et al.|http://www.upch.edu.pe/facien/dbmbqf/gorjeda/cursos/geneticaavanzada%202007/articulos/Science-2001-venter-hgs.pdf]] {{t1000Cite{[[pct. 7321|http://scholar.google.de/scholar?cites=7202406481006653037&hl=de]]}}}

The ''Geodesic Equation'' describes the trajectory $\bs \gamma$ with the shortest path length between two points $a$ and $b$.
It can be attained by varying the path length of possible paths from $a$ to $b$ and finding the minimum.
The path length $l(\bs \gamma)$ of a curve $\bs \gamma$ is given by
\[
l(\bs \gamma) = \int_a^b ds = \int_a^b \sqrt {g_{\mu\nu}(\bs \gamma(s)) \dot {\mb \gamma}^\mu (s) \dot {\mb{\gamma}}^\nu(s)} ds
\]
This can be expressed in terms of an [[action|Action Principle]] $S$ as
\[
S(\bs \gamma, \dot{\bs \gamma}) = \int_a^b L (\bs \gamma , \dot{\bs \gamma}) ds
\]
with
\[
L(\bs \gamma, \dot{\bs \gamma}) = \int_a^b \sqrt {g_{\mu\nu}(\bs \gamma (s)) \dot \gamma^\mu(s) \dot \gamma^\nu(s)} ds
\]

Finding the path with minimum length therefore is equivalent to minimizing the action, which results in the Lagrange equations, i.e.
\[
{\partial L\over\partial \gamma^\rho} - {d\over ds}{\partial L\over\partial \dot{\gamma}^\rho} = 0
\]
Inserting (the square of) our Lagrangian, we get
\[
\frac{\partial g_{\mu\nu}(\gamma)}{\partial \gamma^\rho} \dot{\gamma}^\mu \dot{\gamma}^\nu - 2 {d\over ds} \left (g_{\rho \nu} (\bs \gamma) \dot{\gamma}^\nu \right )= 0
\]
assuming that the metric tensor is symmetric.

Carrying out the total differentiation yields:
\[
\frac{\partial g_{\mu\nu}(\bs \gamma)}{\partial \gamma^\rho} \dot{\gamma}^\mu \dot{\gamma}^\nu - 2 \frac{\partial g_{\rho\nu}(\bs \gamma)}{\partial \gamma^\mu}\dot{\gamma}^\mu \dot{\gamma}^\nu - 2 g_{\rho\nu} (\bs \gamma) \ddot{\gamma}^\nu = 0
\]

Splitting up the second term and renaming indices leads to
\[
\left (\frac{\partial g_{\mu\nu}(\bs\gamma)}{\partial \gamma^\rho} - \frac{\partial g_{\rho\nu}(\mb{\gamma})}{\partial \gamma^\mu} - \frac{\partial g_{\rho\mu}(\bs \gamma)}{\partial \gamma^\nu} \right ) \dot{\gamma}^\mu \dot{\gamma}^\nu - 2 g_{\rho\nu} (\bs \gamma) \ddot{\gamma}^\nu = 0
\]

!!!!Examples

''Point particle in special relativity:''
\[
S = - m c^2 \int_{C} \, d \tau
\]
This action is invariant under reparametrizations of $\tau$. One can fix this invariance by different gauge fixings (e.g. static gauge, light-front gauge).

''Point particle in general relativity:''
In General Relativity the ''Geodesic Equation'' describes the trajectory $x^\mu(\tau)$ of a point particle (without spin) - the ''Geodesic'' - under the action of gravitation:
\[
a^{\lambda}(\tau) + \Chr{\lambda}{\mu \nu} u^{\mu}(\tau) u^{\nu}(\tau) = 0
\]
with $  \Chr{\lambda}{\mu \nu} $ the [[Christoffel connection|Levi-Civita Connection]] and $\tau$ proper time.

''Spinning particle in special relativity:''
The geodesic is a straight line which is not influenced by the particle's spin.

''Spinning particle in general relativity:''
If spin is considered (but without spin precession) the equation has to be modified by an additional force term which yields one of the [[Mathisson-Papapetrou equations|Mathisson-Papapetrou Equations]]
\[
a^{\lambda} + \Chr{\lambda}{\mu \nu} u^{\mu}(\tau) u^{\nu} = \frac{1}{2m} R^\lambda_{\rho\sigma\omega} S^{\sigma\omega} u^{\rho}
\]
Hence the spinning particle does not follow a geodesic any more.

If furthermore a (constant) electromagnetic field $F_{\mu\nu}$ is added one gets the following equation which is a special case of the [[Dixon-Souriau equations|Dixon-Souriau Equations]]:
\[
a^{\lambda} + \Chr{\lambda}{\mu \nu} u^{\mu}(\tau) u^{\nu} (\tau)= (\frac{1}{2} R^\lambda_{\rho\sigma\omega} S^{\sigma\omega}  + e F_{\lambda\rho}) \frac {u^{\rho}}{m}
\]

''Geometrodynamics'' is the study of curved empty space and the evolution of this geometry with time according to [[Einstein’s equations of motion|Einstein Field Equations]].
The sources of curvature are conceived however differently in geometrodynamics and in the usual theory of relativity. In the latter any warping of the Riemannian space-time manifold is due to masses and fields of non-geometric origin. In geometrodynamics by contrast only those masses and fields are considered which can be built out of geometry itself.

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if (config.options.txtIncrementalSearchMin===undefined) config.options.txtIncrementalSearchMin=3;

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	listMaxSize: 10,
	listHeading: 'Found %0 matching title%1...',
	searchItem: "Search for '%0'...",
	handler:
	function(place,macroName,params,wikifier,paramString,tiddler) {
		var quiet	=params.contains("quiet");
		var showlist	=params.contains("showlist");
		var search	=params.contains("search");
		params = paramString.parseParams("anon",null,true,false,false);
		var instyle	=getParam(params,"inputstyle","");
		var liststyle	=getParam(params,"liststyle","");
		var filter	=getParam(params,"filter","");
		var html=this.html;
		var keyevent=window.event?"onkeydown":"onkeypress"; // IE event fixup for ESC handling
		html=html.replace(/%keyevent%/g,keyevent);
		html=html.replace(/%search%/g,search);
		html=html.replace(/%quiet%/g,quiet);
		html=html.replace(/%showlist%/g,showlist);
		html=html.replace(/%display%/g,showlist?'block':'none');
		html=html.replace(/%position%/g,showlist?'static':'absolute');
		html=html.replace(/%instyle%/g,instyle);
		html=html.replace(/%liststyle%/g,liststyle);
		html=html.replace(/%filter%/g,filter);
		if (config.browser.isIE) html=this.IEtableFixup.format([html]);
		var span=createTiddlyElement(place,'span');
		span.innerHTML=html; var form=span.getElementsByTagName("form")[0];
		if (showlist) this.fillList(form.list,'',filter,search,0);
	},
	html:
	'<form onsubmit="return false" style="display:inline;margin:0;padding:0">\
		<input name=gotoTiddler type=text autocomplete="off" accesskey="G" style="%instyle%"\
			title="Enter title text... ENTER=goto, SHIFT-ENTER=search for text, DOWN=select from list"\
			onfocus="this.select(); this.setAttribute(\'accesskey\',\'G\');"\
			%keyevent%="return config.macros.gotoTiddler.inputEscKeyHandler(event,this,this.form.list,%search%,%showlist%);"\
			onkeyup="return config.macros.gotoTiddler.inputKeyHandler(event,this,%quiet%,%search%,%showlist%);">\
		<select name=list style="display:%display%;position:%position%;%liststyle%"\
			onchange="if (!this.selectedIndex) this.selectedIndex=1;"\
			onblur="this.style.display=%showlist%?\'block\':\'none\';"\
			%keyevent%="return config.macros.gotoTiddler.selectKeyHandler(event,this,this.form.gotoTiddler,%showlist%);"\
			onclick="return config.macros.gotoTiddler.processItem(this.value,this.form.gotoTiddler,this,%showlist%);">\
		</select><input name="filter" type="hidden" value="%filter%">\
	</form>',
	IEtableFixup:
	"<table style='width:100%;display:inline;padding:0;margin:0;border:0;'>\
		<tr style='padding:0;margin:0;border:0;'><td style='padding:0;margin:0;border:0;'>\
		%0</td></tr></table>",
	getItems:
	function(list,val,filter) {
		if (!list.cache || !list.cache.length || val.length<=config.options.txtIncrementalSearchMin) {
			// starting new search, fetch and cache list of tiddlers/shadows/tags
			list.cache=new Array();
			if (filter.length) {
				var fn=store.getMatchingTiddlers||store.getTaggedTiddlers;
				var tiddlers=store.sortTiddlers(fn.apply(store,[filter]),'title');
			} else
				var tiddlers=store.getTiddlers("title","excludeLists");
			for(var t=0; t<tiddlers.length; t++) list.cache.push(tiddlers[t].title);
			if (!filter.length) {
				for (var t in config.shadowTiddlers) list.cache.pushUnique(t);
				var tags=store.getTags();
				for(var t=0; t<tags.length; t++) list.cache.pushUnique(tags[t][0]);
			}
		}
		var found = [];
		var match=val.toLowerCase();
		for(var i=0; i<list.cache.length; i++)
			if (list.cache[i].toLowerCase().indexOf(match)!=-1) found.push(list.cache[i]);
		return found;
	},
	getItemSuffix:
	function(t) {
		if (store.tiddlerExists(t)) return "";  // tiddler
		if (store.isShadowTiddler(t)) return " (shadow)"; // shadow
		return " (tag)"; // tag
	},
	fillList:
	function(list,val,filter,search,key) {
		if (list.style.display=="none") return; // not visible... do nothing!
		var indent='\xa0\xa0\xa0';
		var found = this.getItems(list,val,filter); // find matching items...
		found.sort(); // alpha by title
		while (list.length > 0) list.options[0]=null; // clear list
		var hdr=this.listHeading.format([found.length,found.length==1?"":"s"]);
		list.options[0]=new Option(hdr,"",false,false);
		for (var t=0; t<found.length; t++) list.options[list.length]=
			new Option(indent+found[t]+this.getItemSuffix(found[t]),found[t],false,false);
		if (search)
			list.options[list.length]=new Option(this.searchItem.format([val]),"*",false,false);
		list.size=(list.length<this.listMaxSize?list.length:this.listMaxSize); // resize list...
		list.selectedIndex=key==38?list.length-1:key==40?1:0;
	},
	keyProcessed:
	function(ev) { // utility function
		ev.cancelBubble=true; // IE4+
		try{event.keyCode=0;}catch(e){}; // IE5
		if (window.event) ev.returnValue=false; // IE6
		if (ev.preventDefault) ev.preventDefault(); // moz/opera/konqueror
		if (ev.stopPropagation) ev.stopPropagation(); // all
		return false;
	},
	inputEscKeyHandler:
	function(event,here,list,search,showlist) {
		if (event.keyCode==27) {
			if (showlist) { // clear input, reset list
				here.value=here.defaultValue;
				this.fillList(list,'',here.form.filter.value,search,0);
			}
			else if (list.style.display=="none") // clear input
				here.value=here.defaultValue;
			else list.style.display="none"; // hide list
			return this.keyProcessed(event);
		}
		return true; // key bubbles up
	},
	inputKeyHandler:
	function(event,here,quiet,search,showlist) {
		var key=event.keyCode;
		var list=here.form.list;
		var filter=here.form.filter;
		// non-printing chars bubble up, except for a few:
		if (key<48) switch(key) {
			// backspace=8, enter=13, space=32, up=38, down=40, delete=46
			case 8: case 13: case 32: case 38: case 40: case 46: break; default: return true;
		}
		// blank input... if down/enter... fall through (list all)... else, and hide or reset list
		if (!here.value.length && !(key==40 || key==13)) {
			if (showlist) this.fillList(here.form.list,'',here.form.filter.value,search,0);
			else list.style.display="none";
			return this.keyProcessed(event);
		}
		// hide list if quiet, or below input minimum (and not showlist)
		list.style.display=(!showlist&&(quiet||here.value.length<config.options.txtIncrementalSearchMin))?'none':'block';
		// non-blank input... enter=show/create tiddler, SHIFT-enter=search for text
		if (key==13 && here.value.length) return this.processItem(event.shiftKey?'*':here.value,here,list,showlist);
		// up or down key, or enter with blank input... shows and moves to list...
		if (key==38 || key==40 || key==13) { list.style.display="block"; list.focus(); }
		this.fillList(list,here.value,filter.value,search,key);
		return true; // key bubbles up
	},
	selectKeyHandler:
	function(event,list,editfield,showlist) {
		if (event.keyCode==27) // escape... hide list, move to edit field
			{ editfield.focus(); list.style.display=showlist?'block':'none'; return this.keyProcessed(event); }
		if (event.keyCode==13 && list.value.length) // enter... view selected item
			{ this.processItem(list.value,editfield,list,showlist); return this.keyProcessed(event); }
		return true; // key bubbles up
	},
	processItem:
	function(title,here,list,showlist) {
		if (!title.length) return;
		list.style.display=showlist?'block':'none';
		if (title=="*")	{ story.search(here.value); return false; } // do full-text search
		if (!showlist) here.value=title;
		story.displayTiddler(null,title); // show selected tiddler
		return false;
	}
}
//}}}

!!!!Interpretations
* Gravitation represents a "metrical elasticity" of space which is brought about by quantum fluctuations of the vacuum (Andrei Sakharov).
* Space-time is a crystal with dislocations and disclinations ([["world crystal"|World Crystal]]) which has undergone a quantum phase transition to a nematic phase by a condensation of dislocations.
* Space-time is a medium which can be described as a [[Fermi system|Fermi System]].

Papers:
* [[Quantum Phase Transitions and the Breakdown of Classical General Relativity - G. Chapline|http://arxiv.org/PS_cache/gr-qc/pdf/0012/0012094v1.pdf]]  {{t100Cite{[[pct. 104|http://scholar.google.de/scholar?cites=4549488117690173279&hl=de]]}}}
* [[Nonholonomic Mapping Principle for Classical and Quantum Mechanics in Spaces with Curvature and Torsion - H. Kleinert|http://arxiv.org/PS_cache/gr-qc/pdf/0203/0203029v1.pdf]] [[local|papers/0203029v1.pdf]][[pct. 27|http://scholar.google.de/scholar?cites=5964503436918444234&hl=de]]

The ''Gravitino'' is the conjectured [[supersymmetric|Supersymmetry]] partner of the graviton.
Its action is given by
\begin{equation}
\mathcal L= ? \frac{i}{2}  \Psi_\mu^* \gamma^{[\mu} \gamma^\nu \gamma^{\lambda]} \partial_\nu \Psi_\lambda
\end{equation}

Papers:
* [[Gravi-Weak Unification - F. Nestia, R. Percacci|http://arxiv.org/PS_cache/arxiv/pdf/0706/0706.3307v2.pdf]]

The composition of relativistics velocities can be described by algebraic structures called a ''Gyrogroups'' which were introduced by A. A. Ungar. Gyrogroups are noncommutative and nonassociatve which is related to Thomas precession in special theory of relativity.

In 1975, Rudolf Haag, Jan T. Łopuszański and Martin Sohnius published a proof (''Haag-Łopuszański-Sohnius Theorem'') which shows, that by weakening the assumptions of the [[Coleman-Mandula theorem|Coleman-Mandula Theorem]] allowing both commuting and anticommuting symmetry generators, there is a nontrivial extension of the [[Poincaré algebra|Poincaré Transformation]], namely the [[supersymmetry algebra|Supersymmetry]].

Journals:
* All Possible Generators of Supersymmetries of the S Matrix - R. Haag, J. T. Łopuszański, M. Sohnius {{t500Cite{[[jct. 829|http://scholar.google.de/scholar?cites=14850286862044094598&as_sdt=2005&sciodt=2000&hl=de]]}}}

Given an order $n$, one has the following examples of [[Hadamard matrices|Hadamard Matrix]]:

!!!!!n = 1
~~
|+|
~~
!!!!!n = 2
~~
|+|+|
|+|-|
~~
!!!!!n = 4
~~
|+|+|+|+|
|+|-|+|-|
|+|+|-|-|
|+|-|-|+|
~~
flipping the second and the fourth column one gets:
~~
|+|+|+|+|
|+|-|+|-|
|+|-|-|+|
|+|+|-|-|
~~
which represents the [[sign matrix|Sign Tables]] of an antisymmetric [[multiplication table|Multiplication Tables]] and corresponds to the multiplication table of the right handed [[quaternions|Quaternion]].
If instead on one flips the third and the fourth column one gets the sign matrix of the multiplication table of the left handed quaternions:
~~
|+|+|+|+|
|+|-|-|+|
|+|+|-|-|
|+|-|+|-|
~~
!!!!!n = 8
~~
|+|+|+|+|+|+|+|+|
|+|-|+|-|+|-|+|-|
|+|+|-|-|+|+|-|-|
|+|-|-|+|+|-|-|+|
|+|+|+|+|-|-|-|-|
|+|-|+|-|-|+|-|+|
|+|+|-|-|-|-|+|+|
|+|-|-|+|-|+|+|-|
~~
[[Automorphism group|Automorphism]]: order  $2^{10} \cdot 3 \cdot 7 = 21.504$.

With the following map of the columns $(1 \to 1, 2 \to 6, 3 \to 8, 4 \to 7, 5 \to 5, 6 \to 4, 7 \to 3, 8 \to 2)$ one gets the table
~~
|+|+|+|+|+|+|+|+|
|+|-|+|-|+|-|-|+|
|+|-|-|+|+|+|-|-|
|+|+|-|-|+|-|+|-|
|+|-|-|-|-|+|+|+|
|+|+|-|+|-|-|-|+|
|+|+|+|-|-|+|-|-|
|+|-|+|+|-|-|+|-|
~~
which is the sign matrix of the multiplication table of the [[octonions|Octonion]] obtained by classical [[Cayley-Dickson doubling|Cayley-Dickson Doubling]].
(Notice however that by flipping the columns we have changed the determinant from $2^{12}$ to $-2^{12}$, which happens if two rows (or columns) are switched).
By appropriately flipping rows and columns one can get all $240$ non-equal sign tables associated with the [[480 different octonion algebras|480 Octonion Multiplication Tables]].
Furthermore, Monte\-Carlo simulations suggest that there are all in all $2.640 = 11\cdot 240$ different possible $8$-dimensional sign tables when requiring
* the number of $+1$- and $-1$-entries is equal for each row and column (except for the border-row and -column). I.e. the matrices are Hadamard matrices,
* they are normalized,
* the diagonal consists of $-1$ entries (except for the one element of the border), i.e. it corresponds to a non-split algebra,
* the matrices are antisymmetric.
(Unfortunately I am lacking any explanation of this result yet).

The algorithm also reproduces the $2$ different sign tables in case of the quaternions, given above.

Furthermore this matrix is Hadamard equivalent to a representation of the [[Fano plane|Fano Plane]].

!!!!!n = 16
The five distinct Hadamard matrices of order $16$ can be taken to be:
~~
|+|+|+|+|+|+|+|+|+|+|+|+|+|+|+|+|
|+|-|+|-|+|-|+|-|+|-|+|-|+|-|+|-|
|+|+|-|-|+|+|-|-|+|+|-|-|+|+|-|-|
|+|-|-|+|+|-|-|+|+|-|-|+|+|-|-|+|
|+|+|+|+|-|-|-|-|+|+|+|+|-|-|-|-|
|+|-|+|-|-|+|-|+|+|-|+|-|-|+|-|+|
|+|+|-|-|-|-|+|+|+|+|-|-|-|-|+|+|
|+|-|-|+|-|+|+|-|+|-|-|+|-|+|+|-|
|+|+|+|+|+|+|+|+|-|-|-|-|-|-|-|-|
|+|-|+|-|+|-|+|-|-|+|-|+|-|+|-|+|
|+|+|-|-|+|+|-|-|-|-|+|+|-|-|+|+|
|+|-|-|+|+|-|-|+|-|+|+|-|-|+|+|-|
|+|+|+|+|-|-|-|-|-|-|-|-|+|+|+|+|
|+|-|+|-|-|+|-|+|-|+|-|+|+|-|+|-|
|+|+|-|-|-|-|+|+|-|-|+|+|+|+|-|-|
|+|-|-|+|-|+|+|-|-|+|+|-|+|-|-|+|
~~
[[Automorphism group|Automorphism]]: order $ 2^{15} \cdot 3^2 \cdot 5 \cdot 7 = 10.321.920$.
Determinant: $2^{32}$.
Symmetric, contrary to all the other matrices listed in the following.
This matrix is Hadamard equivalent to a representation of the [[Fano tetrahedron|Fano Spaces]].
~~
|+|+|+|+|+|+|+|+|+|+|+|+|+|+|+|+|
|+|-|+|-|+|-|+|-|+|-|+|-|+|-|+|-|
|+|+|-|-|+|+|-|-|+|+|-|-|+|+|-|-|
|+|-|-|+|+|-|-|+|+|-|-|+|+|-|-|+|
|+|+|+|+|-|-|-|-|+|+|+|+|-|-|-|-|
|+|-|+|-|-|+|-|+|+|-|+|-|-|+|-|+|
|+|+|-|-|-|-|+|+|+|+|-|-|-|-|+|+|
|+|-|-|+|-|+|+|-|+|-|-|+|-|+|+|-|
|+|+|+|+|+|+|+|+|-|-|-|-|-|-|-|-|
|+|-|+|-|+|-|-|+|-|+|-|+|-|+|+|-|
|+|+|-|-|+|+|-|-|-|-|+|+|-|-|+|+|
|+|-|-|+|+|-|+|-|-|+|+|-|-|+|-|+|
|+|+|+|+|-|-|-|-|-|-|-|-|+|+|+|+|
|+|-|+|-|-|+|+|-|-|+|-|+|+|-|-|+|
|+|+|-|-|-|-|+|+|-|-|+|+|+|+|-|-|
|+|-|-|+|-|+|-|+|-|+|+|-|+|-|+|-|
~~
[[Automorphism group|Automorphism]]: order $2^{15}\cdot 3^2 = 294.912$.
Determinant: $-2^{32}$.
Hadamard equivalent to its transpose.

~~
|+|+|+|+|+|+|+|+|+|+|+|+|+|+|+|+|
|+|-|+|-|+|-|+|-|+|-|+|-|+|-|+|-|
|+|+|-|-|+|+|-|-|+|+|-|-|+|+|-|-|
|+|-|-|+|+|-|-|+|+|-|-|+|+|-|-|+|
|+|+|+|+|-|-|-|-|+|+|+|+|-|-|-|-|
|+|-|+|-|-|+|-|+|+|-|+|-|-|+|-|+|
|+|+|-|-|-|-|+|+|+|+|-|-|-|-|+|+|
|+|-|-|+|-|+|+|-|+|-|-|+|-|+|+|-|
|+|+|+|+|+|+|+|+|-|-|-|-|-|-|-|-|
|+|+|+|+|-|-|-|-|-|-|-|-|+|+|+|+|
|+|+|-|-|+|-|+|-|-|-|+|+|-|+|-|+|
|+|+|-|-|-|+|-|+|-|-|+|+|+|-|+|-|
|+|-|+|-|+|-|-|+|-|+|-|+|-|+|+|-|
|+|-|+|-|-|+|+|-|-|+|-|+|+|-|-|+|
|+|-|-|+|+|+|-|-|-|+|+|-|-|-|+|+|
|+|-|-|+|-|-|+|+|-|+|+|-|+|+|-|-|
~~
[[Automorphism group|Automorphism]]: order $2^{14} \cdot 3 = 49.152$.
Determinant: $2^{32}$.
Hadamard equivalent to its transpose.
~~
|+|+|+|+|+|+|+|+|+|+|+|+|+|+|+|+|
|+|-|+|-|+|-|+|-|+|-|+|-|+|-|+|-|
|+|+|-|-|+|+|-|-|+|+|-|-|+|+|-|-|
|+|-|-|+|+|-|-|+|+|-|-|+|+|-|-|+|
|+|+|+|+|-|-|-|-|+|+|+|+|-|-|-|-|
|+|-|+|-|-|+|-|+|+|-|+|-|-|+|-|+|
|+|+|-|-|-|-|+|+|+|+|-|-|-|-|+|+|
|+|-|-|+|-|+|+|-|+|-|-|+|-|+|+|-|
|+|+|+|+|+|+|+|+|-|-|-|-|-|-|-|-|
|+|+|+|-|+|-|-|-|-|-|-|+|-|+|+|+|
|+|+|-|+|-|-|-|+|-|-|+|-|+|+|+|-|
|+|+|-|-|-|+|+|-|-|-|+|+|+|-|-|+|
|+|-|+|+|-|+|-|-|-|+|-|-|+|-|+|+|
|+|-|+|-|-|-|+|+|-|+|-|+|+|+|-|-|
|+|-|-|+|+|-|+|-|-|+|+|-|-|+|-|+|
|+|-|-|-|+|+|-|+|-|+|+|+|-|-|+|-|
~~
[[Automorphism group|Automorphism]]: order $ 2^{12} \cdot 3 \cdot 7 = 86.016$.
Determinant: $-2^{32}$
Transpose equals the next matrix in the list.
~~
|+|+|+|+|+|+|+|+|+|+|+|+|+|+|+|+|
|+|-|+|-|+|-|+|-|+|+|+|+|-|-|-|-|
|+|+|-|-|+|+|-|-|+|+|-|-|+|+|-|-|
|+|-|-|+|+|-|-|+|+|-|+|-|+|-|+|-|
|+|+|+|+|-|-|-|-|+|+|-|-|-|-|+|+|
|+|-|+|-|-|+|-|+|+|-|-|+|+|-|-|+|
|+|+|-|-|-|-|+|+|+|-|-|+|-|+|+|-|
|+|-|-|+|-|+|+|-|+|-|+|-|-|+|-|+|
|+|+|+|+|+|+|+|+|-|-|-|-|-|-|-|-|
|+|-|+|-|+|-|+|-|-|-|-|-|+|+|+|+|
|+|+|-|-|+|+|-|-|-|-|+|+|-|-|+|+|
|+|-|-|+|+|-|-|+|-|+|-|+|-|+|-|+|
|+|+|+|+|-|-|-|-|-|-|+|+|+|+|-|-|
|+|-|+|-|-|+|-|+|-|+|+|-|-|+|+|-|
|+|+|-|-|-|-|+|+|-|+|+|-|+|-|-|+|
|+|-|-|+|-|+|+|-|-|+|-|+|+|-|+|-|
~~
[[Automorphism group|Automorphism]]: order $ 2^{12} \cdot 3 \cdot 7 = 86.016$.
Determinant: $-2^{32}$
Transpose equals the former matrix in the list.
This matrix is Hadamard equivalent (but not equal) to the [[sign matrix|Sign Tables]] of the [[sedenions|Sedenion]] (see MAGMA example below).

//Question: Are there any algebraic counterparts for the other three $16$-dimensional Hadamard matrices ?//

!!!![[MAGMA|http://magma.maths.usyd.edu.au/calc/]]^^[[Help|MAGMA]]^^ examples
* [[Code File|code/MAGMAHadamardMatrices.txt]]
* [[Handbook of MAGMA Functions, Chapter 121 - Hadamard Matrices|code/MAGMAHadamardMatrices.pdf]]

The ''Heisenberg Algebra'' in quantum mechanics relates position- and momentum-operators and is defined by the following commutation-relations: 
\begin{eqnarray} 
[\hat x^i, \hat p^j] &= & i \hbar \delta^{ij} \\
[\hat p^i, \hat p^j] &= &0 
\end{eqnarray}

>For many years whenever I got into a different topic I found out who was behind the scene, and sure enough, it was Hermann Weyl.
> - Michael Atiyah [1] -

Links: 
* [[[1] An Interview with Michael Atiyah|http://kryakin.com/files/Atiyah.pdf]]
* [[WIKIPEDIA - Hermann Weyl|http://en.wikipedia.org/wiki/Hermann_Weyl]]
* [[Weylmann.com|http://www.weylmann.com/]]

The ''Heterotic String Theory'' is the only [[string theory|Superstring Theory]] with solely closed strings. In ten-dimensional space-time it is equipped with $\mathcal N = 1$ [[supersymmetry|Supersymmetry]] and an [[E8]]$\times$[[E8]] or [[SO(32)]] gauge group. Only for these two gauge groups one gets a cancellation of [[anomalies|Anomaly]].

The heterotic string is derived from the 26-dimensional bosonic string in that its excitations are split up into ''left-movers'' and ''right-movers'':

Left movers:
26-dimensional, bosonic, 16 dimensions compactified.
480 generators of [[E8]]$\times$[[E8]] or [[SO(32)]].

Right movers:
10 dimensional superstring with bosonic and fermionic degrees of freedom related by $\mathcal N = 1$ (local) supersymmetry.

In the low energy limit one gets the following effective action which modifies Einstein gravity:
\[
S = \int dx^4 \sqrt{-g} \; e^{-2\Phi} (R + 12\partial_\mu \Phi \partial^\mu \Phi ?  \frac{1}{2\cdot 3!} H_{\mu\nu\sigma}H^{\mu\nu\sigma})
\]
with $H_{\mu\nu\sigma}$ the antisymmetric [[Kalb-Ramond|Kalb-Ramond Field]]- or axion-field which can be decomposed according to:
\[
H_{\mu\nu\sigma} = \partial_{[\mu} B_{\nu\sigma]} + (\Omega_Y)_{\mu\nu\sigma} + (\Omega_L)_{\mu\nu\sigma}
\]
$\Omega_Y$ and $\Omega_L$ are Yang\-Mills- and Lorentz\-Chern Simons terms respectively.

Papers:
* [[Fermionic Subspaces of the Bosonic String - A. Chattaraputi, F. Englert, L. Houart, A. Taorminak|http://arxiv.org/PS_cache/hep-th/pdf/0212/0212085v1.pdf]] [[local|papers/0212085v1.pdf]] [[pct. 5|http://scholar.google.de/scholar?cites=3936724868441634729&hl=de]]
* [[Grand Unification in the Heterotic Brane World - P. K. S. Vaudrevange|http://arxiv.org/PS_cache/arxiv/pdf/0812/0812.3503v1.pdf]] pct. 0

!!!! An essay ...
<html><center><img src="images/pyramide-hierarchie.jpg" style="width: 395px; "/></center></html>
For the world to be structured the way it is, a big seperation of the orders of the grades of the world polyvector is required.
If we assume that Planck's constant determines the separation between the vector- and the bivector grades (between relativity and quantum mechanics/spin), it becomes clear, that if this constant were not very small, classical physics and quantum mechanics would be blurred and we would not observe a distinct classical world.
If we assume that the values of elementary charges determine the separation between the second and third grades, then it follows, that if this value were much bigger, the charges were also much bigger and the world would be much more clumped together.
A similar problem arises if one considers the separation of the third and the fourth grade. One might speculate that the cosmological constant - which is of the order $10^{-120}$ - is the relevant value for their separation. If it were much bigger, the universe would not have expanded to a size comparable to what we see today.
Furthermore if the seperations would generally be smaller, higher grades (5 and above) might play a significant role if they existed.
If one assumes that they do in fact exist, given the actual constants of nature, they can be neglected as they are expected to be of order $10^{-160}$ and smaller.

Summarizing the things just said heuristically in terms of a world polyvector $\Phi$, it would look sth. like this:
\begin{eqnarray}
\Phi &=&c \mb e_\mu + \hbar \mb e_{\mu\nu}  + e \mb e_{\mu\nu\rho}  + \Lambda \mb e_{\mu\nu\rho\sigma} + \Omega \mb e_{\mu\nu\rho\sigma\tau} + \ldots  \\
&\approx& \mb e_\mu + 10^{-40} \mb e_{\mu\nu} + 10^{-80} \mb e_{\mu\nu\rho}  + 10^{-120} \mb e_{\mu\nu\rho\sigma} + 10^{-160} \mb e_{\mu\nu\rho\sigma\tau} + \ldots \\
&\approx& \mb e_\mu + l_P \mb e_{\mu\nu} + l_P^2 \mb e_{\mu\nu\rho}  + l_p^3 \mb e_{\mu\nu\rho\sigma} +  l_p^4 \mb e_{\mu\nu\rho\sigma\tau} + \ldots
\end{eqnarray}
with $c$ the speed of light, $\hbar$ Planck's constant, $e$ the elementary charge, $\Lambda$ the cosmological constant and $\Omega$ a constant, which would be a signature of a 5-th dimension. // QUESTION: Could the Bolzmann constant be involved here ? //

In the last step we have assumed that the separations are of the order of a multiple of Planck's length. Due to dimensionality reasons, to be able to add two adjacent grades, one must relate them by a dimension of length. As the natural and most fundamental length scale is the Planck length, it is chosen.
The representation in term of a polyvector also suggest that all the constants in nature can be derived from one fundamental unit which here is assumed to be the [[planck length|Planck Units]].
As we are dealing with ratios of the order $10^{40}$, the [[large number hypothesis|Large Number Hypothesis]] might be related to polyvector physics.

// QUESTION: Could the scalar part which has been omitted here be the mass of the universe which is about $10^{55}g \approx 10^{40}m_P$ (Eddington ?) //

© by Markus Maute, 2009

Papers:
* [[The Hierarchy Problem and New Dimensions at a Millimeter - N. Arkani–Hamed, S. Dimopoulos, G. Dvali|http://arxiv.org/PS_cache/hep-ph/pdf/9803/9803315v1.pdf]]

Links:
* [[Stanford Encyclopedia of Philosophy|http://plato.stanford.edu/entries/spacetime-holearg/]]

The ''Howe–Tucker String Action'' is equivalent to the [[Nambu Goto action|Dirac-Nambu-Goto Action]]. It is invariant under Weyl rescaling of the world metric and as a consequence, the string classical energy–momentum tensor has vanishing trace.

Papers:
* [[A Critical Analysis of the Hydrino Model - A. Rathke|http://arxiv.org/PS_cache/quant-ph/pdf/0505/0505150v1.pdf]] [[pct. 8|http://scholar.google.de/scholar?cites=13240601268371920337&hl=de]]

Links:
* [[BlackLight Power Inc.|http://www.blacklightpower.com/]]

/***
|Name|ImageSizePlugin|
|Source|http://www.TiddlyTools.com/#ImageSizePlugin|
|Version|1.2.1|
|Author|Eric Shulman|
|License|http://www.TiddlyTools.com/#LegalStatements|
|~CoreVersion|2.1|
|Type|plugin|
|Description|adds support for resizing images|
This plugin adds optional syntax to scale an image to a specified width and height and/or interactively resize the image with the mouse.
!!!!!Usage
<<<
The extended image syntax is:
{{{
[img(w+,h+)[...][...]]
}}}
where ''(w,h)'' indicates the desired width and height (in CSS units, e.g., px, em, cm, in, or %). Use ''auto'' (or a blank value) for either dimension to scale that dimension proportionally (i.e., maintain the aspect ratio). You can also calculate a CSS value 'on-the-fly' by using a //javascript expression// enclosed between """{{""" and """}}""". Appending a plus sign (+) to a dimension enables interactive resizing in that dimension (by dragging the mouse inside the image). Use ~SHIFT-click to show the full-sized (un-scaled) image. Use ~CTRL-click to restore the starting size (either scaled or full-sized).
<<<
!!!!!Examples
<<<
{{{
[img(100px+,75px+)[images/meow2.jpg]]
}}}
[img(100px+,75px+)[images/meow2.jpg]]
{{{
[<img(34%+,+)[images/meow.gif]]
[<img(21% ,+)[images/meow.gif]]
[<img(13%+, )[images/meow.gif]]
[<img( 8%+, )[images/meow.gif]]
[<img( 5% , )[images/meow.gif]]
[<img( 3% , )[images/meow.gif]]
[<img( 2% , )[images/meow.gif]]
[img(  1%+,+)[images/meow.gif]]
}}}
[<img(34%+,+)[images/meow.gif]]
[<img(21% ,+)[images/meow.gif]]
[<img(13%+, )[images/meow.gif]]
[<img( 8%+, )[images/meow.gif]]
[<img( 5% , )[images/meow.gif]]
[<img( 3% , )[images/meow.gif]]
[<img( 2% , )[images/meow.gif]]
[img(  1%+,+)[images/meow.gif]]
{{tagClear{
}}}
<<<
!!!!!Revisions
<<<
2009.02.24 [1.2.1] cleanup width/height regexp, use '+' suffix for resizing
2009.02.22 [1.2.0] added stretchable images
2008.01.19 [1.1.0] added evaluated width/height values
2008.01.18 [1.0.1] regexp for "(width,height)" now passes all CSS values to browser for validation
2008.01.17 [1.0.0] initial release
<<<
!!!!!Code
***/
//{{{
version.extensions.ImageSizePlugin= {major: 1, minor: 2, revision: 1, date: new Date(2009,2,24)};
//}}}
//{{{
var f=config.formatters[config.formatters.findByField("name","image")];
f.match="\\[[<>]?[Ii][Mm][Gg](?:\\([^,]*,[^\\)]*\\))?\\[";
f.lookaheadRegExp=/\[([<]?)(>?)[Ii][Mm][Gg](?:\(([^,]*),([^\)]*)\))?\[(?:([^\|\]]+)\|)?([^\[\]\|]+)\](?:\[([^\]]*)\])?\]/mg;
f.handler=function(w) {
	this.lookaheadRegExp.lastIndex = w.matchStart;
	var lookaheadMatch = this.lookaheadRegExp.exec(w.source)
	if(lookaheadMatch && lookaheadMatch.index == w.matchStart) {
		var floatLeft=lookaheadMatch[1];
		var floatRight=lookaheadMatch[2];
		var width=lookaheadMatch[3];
		var height=lookaheadMatch[4];
		var tooltip=lookaheadMatch[5];
		var src=lookaheadMatch[6];
		var link=lookaheadMatch[7];

		// Simple bracketted link
		var e = w.output;
		if(link) { // LINKED IMAGE
			if (config.formatterHelpers.isExternalLink(link)) {
				if (config.macros.attach && config.macros.attach.isAttachment(link)) {
					// see [[AttachFilePluginFormatters]]
					e = createExternalLink(w.output,link);
					e.href=config.macros.attach.getAttachment(link);
					e.title = config.macros.attach.linkTooltip + link;
				} else
					e = createExternalLink(w.output,link);
			} else
				e = createTiddlyLink(w.output,link,false,null,w.isStatic);
			addClass(e,"imageLink");
		}

		var img = createTiddlyElement(e,"img");
		if(floatLeft) img.align="left"; else if(floatRight) img.align="right";
		if(width||height) {
			var x=width.trim(); var y=height.trim();
			var stretchW=(x.substr(x.length-1,1)=='+'); if (stretchW) x=x.substr(0,x.length-1);
			var stretchH=(y.substr(y.length-1,1)=='+'); if (stretchH) y=y.substr(0,y.length-1);
			if (x.substr(0,2)=="{{")
				{ try{x=eval(x.substr(2,x.length-4))} catch(e){displayMessage(e.description||e.toString())} }
			if (y.substr(0,2)=="{{")
				{ try{y=eval(y.substr(2,y.length-4))} catch(e){displayMessage(e.description||e.toString())} }
			img.style.width=x.trim(); img.style.height=y.trim();
			config.formatterHelpers.addStretchHandlers(img,stretchW,stretchH);
		}
		if(tooltip) img.title = tooltip;

		// GET IMAGE SOURCE
		if (config.macros.attach && config.macros.attach.isAttachment(src))
			src=config.macros.attach.getAttachment(src); // see [[AttachFilePluginFormatters]]
		else if (config.formatterHelpers.resolvePath) { // see [[ImagePathPlugin]]
			if (config.browser.isIE || config.browser.isSafari) {
				img.onerror=(function(){
					this.src=config.formatterHelpers.resolvePath(this.src,false);
					return false;
				});
			} else
				src=config.formatterHelpers.resolvePath(src,true);
		}
		img.src=src;
		w.nextMatch = this.lookaheadRegExp.lastIndex;
	}
}

config.formatterHelpers.addStretchHandlers=function(e,stretchW,stretchH) {
	e.title=((stretchW||stretchH)?'DRAG=stretch/shrink, ':'')
		+'SHIFT-CLICK=show full size, CTRL-CLICK=restore initial size';
	e.statusMsg='width=%0, height=%1';
	e.style.cursor='move';
	e.originalW=e.style.width;
	e.originalH=e.style.height;
	e.minW=Math.max(e.offsetWidth/20,10);
	e.minH=Math.max(e.offsetHeight/20,10);
	e.stretchW=stretchW;
	e.stretchH=stretchH;
	e.onmousedown=function(ev) { var ev=ev||window.event;
		this.sizing=true;
		this.startX=!config.browser.isIE?ev.pageX:(ev.clientX+findScrollX());
		this.startY=!config.browser.isIE?ev.pageY:(ev.clientY+findScrollY());
		this.startW=this.offsetWidth;
		this.startH=this.offsetHeight;
		return false;
	};
	e.onmousemove=function(ev) { var ev=ev||window.event;
		if (this.sizing) {
			var s=this.style;
			var currX=!config.browser.isIE?ev.pageX:(ev.clientX+findScrollX());
			var currY=!config.browser.isIE?ev.pageY:(ev.clientY+findScrollY());
			var newW=(currX-this.offsetLeft)/(this.startX-this.offsetLeft)*this.startW;
			var newH=(currY-this.offsetTop )/(this.startY-this.offsetTop )*this.startH;
			if (this.stretchW) s.width =Math.floor(Math.max(newW,this.minW))+'px';
			if (this.stretchH) s.height=Math.floor(Math.max(newH,this.minH))+'px';
			clearMessage(); displayMessage(this.statusMsg.format([s.width,s.height]));
		}
		return false;
	};
	e.onmouseup=function(ev) { var ev=ev||window.event;
		if (ev.shiftKey) { this.style.width=this.style.height=''; }
		if (ev.ctrlKey)  { this.style.width=this.originalW; this.style.height=this.originalH; }
		this.sizing=false;
		clearMessage();
		return false;
	};
	e.onmouseout=function(ev) { var ev=ev||window.event;
		this.sizing=false;
		clearMessage();
		return false;
	};
}
//}}}

<<include "trajectory2.html">>

/***
|''Name:''|abego.IncludePlugin|
|''Version:''|1.0.1 (2007-04-30)|
|''Type:''|plugin|
|''Source:''|http://tiddlywiki.abego-software.de/#IncludePlugin|
|''Author:''|Udo Borkowski (ub [at] abego-software [dot] de)|
|''Documentation:''|[[IncludePlugin Documentation|http://tiddlywiki.abego-software.de/#%5B%5BIncludePlugin%20Documentation%5D%5D]]|
|''Community:''|([[del.icio.us|http://del.icio.us/post?url=http://tiddlywiki.abego-software.de/index.html%23IncludePlugin]]) ([[Support|http://groups.google.com/group/TiddlyWiki]])|
|''Copyright:''|&copy; 2007 [[abego Software|http://www.abego-software.de]]|
|''Licence:''|[[BSD open source license (abego Software)|http://www.abego-software.de/legal/apl-v10.html]]|
|''~CoreVersion:''|2.1.3|
|''Browser:''|Firefox 1.5.0.9 or better; Internet Explorer 6.0|
***/
/***
This plugin's source code is compressed (and hidden). Use this [[link|http://tiddlywiki.abego-software.de/archive/IncludePlugin/Plugin-Include-src.1.0.0.js]] to get the readable source code.
***/
///%
if(!window.abego){window.abego={};}var invokeLater=function(_1,_2,_3){return abego.invokeLater?abego.invokeLater(_1,_2,_3):setTimeout(_1,_2);};abego.loadFile=function(_4,_5,_6){var _7=function(_8,_9,_a,_b,_c){return _8?_5(_a,_b,_9):_5(undefined,_b,_9,"Error loading %0".format([_b]));};if(_4.search(/^((http(s)?)|(file)):/)!=0){if(_4.search(/^((.\:\\)|(\\\\)|(\/))/)==0){_4="file://"+_4;}else{var _d=document.location.toString();var i=_d.lastIndexOf("/");_4=_d.substr(0,i+1)+_4;}_4=_4.replace(/\\/mg,"/");}loadRemoteFile(_4,_7,_6);};abego.loadTiddlyWikiStore=function(_f,_10,_11,_12){var _13=function(_14,_15){if(_12){_12(_14,"abego.loadTiddlyWikiStore",_15,_f,_11);}};var _16=function(_17,_18){var _19=_18.indexOf(startSaveArea);var _1a=_18.indexOf("<!--POST-BODY-END--"+">");var _1b=_18.lastIndexOf(endSaveArea,_1a==-1?_18.length:_1a);if((_19==-1)||(_1b==-1)){return config.messages.invalidFileError.format([_f]);}var _1c="<html><body>"+_18.substring(_19,_1b+endSaveArea.length)+"</body></html>";var _1d=document.createElement("iframe");_1d.style.display="none";document.body.appendChild(_1d);var doc=_1d.document;if(_1d.contentDocument){doc=_1d.contentDocument;}else{if(_1d.contentWindow){doc=_1d.contentWindow.document;}}doc.open();doc.writeln(_1c);doc.close();var _1f=doc.getElementById("storeArea");_17.loadFromDiv(_1f,"store");_1d.parentNode.removeChild(_1d);return null;};var _20=function(_21){_13("Error when loading %0".format([_f]),"Failed");_10(undefined,_f,_11,_21);return _21;};var _22=function(_23){_13("Loaded %0".format([_f]),"Done");_10(_23,_f,_11);return null;};var _24=function(_25,_26,_27,_28){if(_25===undefined){_20(_28);return;}_13("Processing %0".format([_f]),"Processing");var _29=config.messages.invalidFileError;config.messages.invalidFileError="The file '%0' does not appear to be a valid TiddlyWiki file";try{var _2a=new TiddlyWiki();var _2b=_16(_2a,_25);if(_2b){_20(_2b);}else{_22(_2a);}}catch(ex){_20(exceptionText(ex));}finally{config.messages.invalidFileError=_29;}};_13("Start loading %0".format([_f]),"Started");abego.loadFile(_f,_24,_11);};(function(){if(abego.TiddlyWikiIncluder){return;}var _2c="waiting";var _2d="loading";var _2e=1000;var _2f=-200;var _30=-100;var _31=-300;var _32;var _33=[];var _34={};var _35=[];var _36;var _37=[];var _38;var _39=function(){if(_32===undefined){_32=config.options.chkUseInclude===undefined||config.options.chkUseInclude;}return _32;};var _3a=function(url){return "No include specified for %0".format([url]);};var _3c=function(){var _3d=_35;_35=[];if(_3d.length){for(var i=0;i<_37.length;i++){_37[i](_3d);}}};var _3f;var _40=function(){if(_36!==undefined){clearInterval(_36);}_3f=0;var _41=function(){abego.TiddlyWikiIncluder.sendProgress("","","Done");};_36=setInterval(function(){_3f++;if(_3f<=10){return;}clearInterval(_36);_36=undefined;abego.TiddlyWikiIncluder.sendProgress("Refreshing...","","");refreshDisplay();invokeLater(_41,0,_2f);},1);};var _42=function(_43){var _44;for(var i=0;i<_33.length;i++){var _46=abego.TiddlyWikiIncluder.getStore(_33[i]);if(_46&&(_44=_43(_46,_33[i]))){return _44;}}};var _47=function(){if(!window.store){return invokeLater(_47,100);}var _48=store.fetchTiddler;store.fetchTiddler=function(_49){var t=_48.apply(this,arguments);if(t){return t;}if(config.shadowTiddlers[_49]!==undefined){return undefined;}if(_49==config.macros.newTiddler.title){return undefined;}return _42(function(_4b,url){var t=_4b.fetchTiddler(_49);if(t){t.includeURL=url;}return t;});};if(_33.length){_40();}};var _4e=function(){if(!window.store){return invokeLater(_4e,100);}var _4f=store.getTiddlerText("IncludeList");if(_4f){wikify(_4f,document.createElement("div"));}};var _50=function(_51){var _52=function(){var _53=store.forEachTiddler;var _54=function(_55){var _56={};var _57;var _58=function(_59,_5a){if(_56[_59]){return;}_56[_59]=1;if(_57){_5a.includeURL=_57;}_55.apply(this,arguments);};_53.call(store,_58);for(var n in config.shadowTiddlers){_56[n]=1;}_56[config.macros.newTiddler.title]=1;_42(function(_5c,url){_57=url;_5c.forEachTiddler(_58);});};store.forEachTiddler=_54;try{return _51.apply(this,arguments);}finally{store.forEachTiddler=_53;}};return _52;};var _5e=function(_5f,_60){return _5f[_60]=_50(_5f[_60]);};abego.TiddlyWikiIncluder={};abego.TiddlyWikiIncluder.setProgressFunction=function(_61){_38=_61;};abego.TiddlyWikiIncluder.getProgressFunction=function(_62){return _38;};abego.TiddlyWikiIncluder.sendProgress=function(_63,_64,_65){if(_38){_38.apply(this,arguments);}};abego.TiddlyWikiIncluder.onError=function(url,_67){displayMessage("Error when including '%0':\n%1".format([url,_67]));};abego.TiddlyWikiIncluder.hasPendingIncludes=function(){for(var i=0;i<_33.length;i++){var _69=abego.TiddlyWikiIncluder.getState(_33[i]);if(_69==_2c||_69==_2d){return true;}}return false;};abego.TiddlyWikiIncluder.getIncludes=function(){return _33.slice();};abego.TiddlyWikiIncluder.getState=function(url){var s=_34[url];if(!s){return _3a(url);}return typeof s=="string"?s:null;};abego.TiddlyWikiIncluder.getStore=function(url){var s=_34[url];if(!s){return _3a(url);}return s instanceof TiddlyWiki?s:null;};abego.TiddlyWikiIncluder.include=function(url,_6f){if(!_39()||_34[url]){return;}var _70=this;_33.push(url);_34[url]=_2c;var _71=function(_72,_73,_74,_75){if(_72===undefined){_34[url]=_75;_70.onError(url,_75);return;}_34[url]=_72;_35.push(url);invokeLater(_3c);};var _76=function(){_34[url]=_2d;abego.loadTiddlyWikiStore(url,_71,null,_38);};if(_6f){invokeLater(_76,_6f);}else{_76();}};abego.TiddlyWikiIncluder.forReallyEachTiddler=function(_77){var _78=function(){store.forEachTiddler(_77);};_50(_78).call(store);};abego.TiddlyWikiIncluder.getFunctionUsingForReallyEachTiddler=_50;abego.TiddlyWikiIncluder.useForReallyEachTiddler=_5e;abego.TiddlyWikiIncluder.addListener=function(_79){_37.push(_79);};abego.TiddlyWikiIncluder.addListener(_40);if(config.options.chkUseInclude===undefined){config.options.chkUseInclude=true;}config.shadowTiddlers.AdvancedOptions+="\n<<option chkUseInclude>> Include ~TiddlyWikis (IncludeList | IncludeState | [[help|http://tiddlywiki.abego-software.de/#%5B%5BIncludePlugin%20Documentation%5D%5D]])\n^^(Reload this ~TiddlyWiki to make changes become effective)^^";config.shadowTiddlers.IncludeState="<<includeState>>";var _7a=function(e,_7c,_7d){if(!anim||!abego.ShowAnimation){e.style.display=_7c?"block":"none";return;}anim.startAnimating(new abego.ShowAnimation(e,_7c,_7d));};abego.TiddlyWikiIncluder.getDefaultProgressFunction=function(){setStylesheet(".includeProgressState{\n"+"background-color:#FFCC00;\n"+"position:absolute;\n"+"right:0.2em;\n"+"top:0.2em;\n"+"width:7em;\n"+"padding-left:0.2em;\n"+"padding-right:0.2em\n"+"}\n","abegoInclude");var _7e=function(){var e=document.createElement("div");e.className="includeProgressState";e.style.display="none";document.body.appendChild(e);return e;};var _80=_7e();var _81=function(_82){removeChildren(_80);createTiddlyText(_80,_82);_7a(_80,true,0);};var _83=function(){invokeLater(function(){_7a(_80,false,_2e);},100,_30);};var _84=function(_85,_86,_87,url,_89){if(_87=="Done"||_87=="Failed"){_83();return;}if(_86=="abego.loadTiddlyWikiStore"){_3f=0;if(_87=="Processing"){_81("Including...");}}else{_81(_85);}};return _84;};abego.TiddlyWikiIncluder.setProgressFunction(abego.TiddlyWikiIncluder.getDefaultProgressFunction());config.macros.include={};config.macros.include.handler=function(_8a,_8b,_8c,_8d,_8e,_8f){_8c=_8e.parseParams("url",null,true,false,true);var _90=parseInt(getParam(_8c,"delay","0"));var _91=_8c[0]["url"];var _92=getFlag(_8c,"hide",false);if(!_92){createTiddlyText(createTiddlyElement(_8a,"code"),_8d.source.substring(_8d.matchStart,_8d.nextMatch));}for(var i=0;_91&&i<_91.length;i++){abego.TiddlyWikiIncluder.include(_91[i],_90);}};config.macros.includeState={};config.macros.includeState.handler=function(_94,_95,_96,_97,_98,_99){var _9a=function(){var s="";var _9c=abego.TiddlyWikiIncluder.getIncludes();if(!_9c.length){return "{{noIncludes{\nNo includes or 'include' is disabled (see AdvancedOptions)\n}}}\n";}s+="|!Address|!State|\n";for(var i=0;i<_9c.length;i++){var inc=_9c[i];s+="|{{{"+inc+"}}}|";var t=abego.TiddlyWikiIncluder.getState(inc);s+=t?"{{{"+t+"}}}":"included";s+="|\n";}s+="|includeState|k\n";return s;};var _a0=function(){removeChildren(div);wikify(_9a(),div);if(abego.TiddlyWikiIncluder.hasPendingIncludes()){invokeLater(_a0,500,_31);}};var div=createTiddlyElement(_94,"div");invokeLater(_a0,0,_31);};var _a2=Tiddler.prototype.isReadOnly;Tiddler.prototype.isReadOnly=function(){return _a2.apply(this,arguments)||this.isIncluded();};Tiddler.prototype.isIncluded=function(){return this.includeURL!=undefined;};Tiddler.prototype.getIncludeURL=function(){return this.includeURL;};var _a3={getMissingLinks:1,getOrphans:1,getTags:1,reverseLookup:1,updateTiddlers:1};for(var n in _a3){_5e(TiddlyWiki.prototype,n);}var _a5=function(){if(abego.IntelliTagger){_5e(abego.IntelliTagger,"assistTagging");}};var _a6=function(){if(config.macros.forEachTiddler){_5e(config.macros.forEachTiddler,"findTiddlers");}};_47();invokeLater(_4e,100);invokeLater(_a5,100);invokeLater(_a6,100);})();
//%/

<<includeState>>

> There are $10^{11}$ stars in the galaxy. That used to be a huge number. But it's only a hundred billion. It's less than the national deficit! We used to call them astronomical numbers. Now we should call them economical numbers.
>- Richard P. Feynman
{{floatright { <html><span style="padding-left:10px"><iframe name="content" src="http://www.markus-maute.de/trajectory/advert_sky.html" width=180 height=620 scrolling="no"></iframe> </html> }}} 
A simple inflationary model with a very clear physical motivation was proposed by Alan Guth in 1981. This was based on the theory of supercooling during the cosmological phase transitions. Even though this scenario did not work, it played a profound role in the development of inflationary cosmology since it contained a very clear explanation how inflation may solve the major cosmological problems.

!!!! Chaotic Inflation
Chaotic inflation resolved all problems of foregoing inflationary models. It predicts that inflation may begin even if there was no thermal equilibrium in the early universe, and it may occur even in the theories with simplest potentials such as $V(\phi) \sim \phi^2$. But it is not limited to theories with polynomial potentials: chaotic inflation occurs in any theory where the potential has a sufficiently flat region, which allows the existence of the slow-roll regime.

A realistic value for the initial mass of the universe is about $3 \cdot 10^{−6}$ in [[Planck units|Planck Units]]. Therefore, the total amount of inflation achieved is of the order $10^{10^{10}}$. The total duration of inflation in this model is about $10^{-30}$ seconds.
Even if the initial size is as small as the Planck size $l_P \sim 10^{-33}$ cm, after $10^{-30}$ seconds of inflation the universe acquires a huge size of $10^{10^{10}}$ cm ! This number is model-dependent, but __in all realistic models the size of the universe after inflation appears to be many orders of magnitude greater than the size of the part of the universe which we can see now__, having a size of about $10^{28}$ cm. The consequence is an [["inflationary" multiverse|Multiverse]].

Papers:
* [[Inflationary Cosmology - A. Linde|http://arxiv.org/PS_cache/arxiv/pdf/0705/0705.0164v2.pdf]] {{t1000Cite{[[pct. 101|http://scholar.google.de/scholar?cites=8331324851624463474&as_sdt=2005&sciodt=2000&hl=de]]}}}

Videos:
* [[Cosmic Inflation and the Accelerating Universe - A. Guth 1|http://www.youtube.com/watch?v=HwCCMHH378Q&feature=related]] [[2|http://www.youtube.com/watch?v=lFkGTzMm7lQ&feature=related]]
* [[Inflationary Cosmology - A. Guth - 2|http://www.youtube.com/watch?v=IQUqRJJ24GQ&feature=related]]

/***
|Name|InlineJavascriptPlugin|
|Source|http://www.TiddlyTools.com/#InlineJavascriptPlugin|
|Documentation|http://www.TiddlyTools.com/#InlineJavascriptPluginInfo|
|Version|1.9.5|
|Author|Eric Shulman|
|License|http://www.TiddlyTools.com/#LegalStatements|
|~CoreVersion|2.1|
|Type|plugin|
|Description|Insert Javascript executable code directly into your tiddler content.|
''Call directly into TW core utility routines, define new functions, calculate values, add dynamically-generated TiddlyWiki-formatted output'' into tiddler content, or perform any other programmatic actions each time the tiddler is rendered.
!!!!!Documentation
>see [[InlineJavascriptPluginInfo]]
!!!!!Revisions
<<<
2009.04.11 [1.9.5] pass current tiddler object into wrapper code so it can be referenced from within 'onclick' scripts
2009.02.26 [1.9.4] in $(), handle leading '#' on ID for compatibility with JQuery syntax
|please see [[InlineJavascriptPluginInfo]] for additional revision details|
2005.11.08 [1.0.0] initial release
<<<
!!!!!Code
***/
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<html><center><img src="images/integralOctonions.jpg" style="width: 257px; "/></center></html>
The ''Integral Octonions'' (a.k.a. ''Integer Cayley Numbers'', ''Integral Octaves'', ''Octavians'') are a discrete non-associative ring based on the [[octonion algebra|Octonion]] $\mathbb O$. They define a [[maximal order|Maximal Order]] within the octonions and can be regarded as a natural generalisation of the ring of integers $\mathbb Z$, as is the case for the [[Gaussian integers|Gaussian Integer]] and the [[Hurwitz integers|Hurwitz Integer]].

The integral octonions form a lattice inside $\mathbb O$ which is a rescaled [[E8 lattice|E8 Lattice]].

A special subset are the ''Octavian Units'' which are [[integral elements|Integral Elements]] having inverses that are again integral octonions. There are $240$ such elements and they correspond to the vertices of the inner shell of the $E_8$-lattice. After rescaling with $\sqrt 2$ they are identical with the $E_8$-roots.

!!!!Construction
Given a non-split octonion algebra obtained by a classical [[Cayley-Dickson doubling|Cayley-Dickson Doubling]], one possible representation of the $240$ octavian units is as follows (see [1]) :
\begin{eqnarray}
&& \mb r = \pm \mb e_0 \\
&&  \mb c_{i} = \pm \mb e_i, \;\, i  = 1, \ldots, 7 \\
&&\mb q_1 \equiv \frac 12  (\pm \mb e_0 \pm \mb e_2 \pm \mb e_3 \pm \mb e_5), \;\;\,  \mb q_1^\bot \equiv \frac 12  (\pm \mb e_1 \pm \mb e_4 \pm \mb e_6 \pm \mb e_7) \\
&&\mb q_2 \equiv \frac 12  (\pm \mb e_0 \pm \mb e_1 \pm \mb e_3 \pm \mb e_6), \;\;\,  \mb q_2^\bot \equiv \frac 12  (\pm \mb e_2 \pm \mb e_4 \pm \mb e_5 \pm \mb e_7) \\
&&\mb q_3 \equiv \frac 12  (\pm \mb e_0 \pm \mb e_1 \pm \mb e_2 \pm \mb e_7), \;\;\,  \mb q_3^\bot \equiv\frac 12  (\pm \mb e_3 \pm \mb e_4 \pm \mb e_5 \pm \mb e_6) \\
&&\mb q_4 \equiv \frac 12  (\pm \mb e_0 \pm \mb e_5 \pm \mb e_6 \pm \mb e_7), \;\;\, \mb q_4^\bot \equiv \frac 12  (\pm \mb e_1 \pm \mb e_2 \pm \mb e_3 \pm \mb e_4) \\
&&\mb q_5 \equiv \frac 12  (\pm \mb e_0 \pm \mb e_1 \pm \mb e_4 \pm \mb e_5), \;\;\,  \mb q_5^\bot \equiv \frac 12  (\pm \mb e_2 \pm \mb e_3 \pm \mb e_6 \pm \mb e_7) \\
&&\mb q_6 \equiv \frac 12  (\pm \mb e_0 \pm \mb e_2 \pm \mb e_4 \pm \mb e_6), \;\;\,  \mb q_6^\bot \equiv \frac 12  (\pm \mb e_1 \pm \mb e_3 \pm \mb e_5 \pm \mb e_7) \\
&&\mb q_7 \equiv \frac 12  (\pm \mb e_0 \pm \mb e_3 \pm \mb e_4 \pm \mb e_7), \;\;\, \mb q_7^\bot \equiv \frac 12  (\pm \mb e_1 \pm \mb e_2 \pm \mb e_5 \pm \mb e_6) \\
\end{eqnarray}
The set of units therefore decomposes according to $240 = 8\cdot 2 + 14 \cdot 2^4 = 16 + 2\cdot 56 + 2\cdot 56 = 16 + 112 + 112$.

Computer simulations show that $\langle \mb q_i|\mb q_i^\bot\rangle = 0$ holds irrespective of sign combinations.

(Notice that for a given pair $\mb q_i$, $\mb q_i^\bot$, every index $0, \ldots, 7$ occurs exactly once. I.e. the indices of $\mb q_i$ and $\mb q_i^\bot$ are complementary (or "dual") to one another. As $\mb e_0$ is contained in all the $\mb q_i$'s, one has a duality ${}^*$ between $3$ and $4$ indices, e.g. $(235)^* = (1467)$).

This construction fails in case of the [[split octonion algebras|Split Octonion]].

!!!!! Relationship with [[Steiner systems|Steiner System]]
The elements $\mb q_1, \ldots, \mb q_7$ can also be obtained from the [[Steiner triple system|Steiner Triple System]] $STS(7) = \{\{1,2,7\},\{1,3,6\},\{1,4,5\},\{2,3,4\},\{2,5,6\},\{3,5,7\},\{4,6,7\}\}$ which is equivalent to the [[Fano plane|Fano Planes - Classification]] $15$ of class $1$ (for details see: [[classification of Fano planes|Fano Planes - Classification]]).
<html><center><img src="images/E8Fano.jpg" style="width: 155px; "/></center></html>
Furthermore the elements $\mb q_1, \ldots, \mb q_7, \mb q_1^\bot, \ldots, \mb q_7^\bot$  are in $1:1$-correspondence with a [[SQS(8)-Steiner quadruple system|Steiner Quadruple System]]. (Given the $7$ triples of a $STS$, the $\mb q_i^\bot$ are determined by its extension to the respective $SQS$).

There are $30$ inequivalent representations of the Fano plane and hence one has $30$ variants of the construction of the octavian units given above. However only for $7$ of them one obtains a closed algebra. This corresponds to the known fact that there exist $7$ different [[maximal orders|Maximal Order]] of the octonions.
Besides the construction described above, the other $6$ constructions are based on the Fano planes $2,3,4,7$ and $12$ of class $1$, according to the classification mentioned.

The situation is more involved if one considers all [[480 different octonion algebras|480 Octonion Multiplication Tables]]. (We restrict ourselves to the non-split case here):
These fall into $30$ classes, each one represented by a different Fano plane. (The $7$ lines of the respective Fano plane are the $7$ associative triples of the associated algebra).
Among algebras within a class, multiplication tables are equal modulo [[signs|Sign Tables]] of their [[structure constants|Structure Constants]]. It turns out that given a class, the maximal orders are identical for all $16$ algebras therein.
For different classes however they are different. Yet the number of maximal orders is the same for all $30$ classes and hence for all $480$ different octonion algebras.

The following table shows the maximal orders for the $30$ octonion algebras based on the different Fano planes. The notation of the Fano planes is given by ($\langle class\rangle$, $\langle number \rangle$):
~~
|!Algebra |!Maximal orders|
|(1,1) |(2,1), (2,2), (2,3), (2,4), (2,7), (2,12), (2,15)|
|(1,2) |(2,1), (2,2), (2,3), (2,5), (2,9), (2,10), (2,14)|
|(1,3) |(2,1), (2,2), (2,3), (2,6), (2,8), (2,11), (2,13)|
|(1,4) |(2,1), (2,4), (2,5), (2,6), (2,7), (2,11), (2,14)|
|(1,5) |(2,2), (2,4), (2,5), (2,6), (2,8), (2,10), (2,15)|
|(1,6) |(2,3), (2,4), (2,5), (2,6), (2,9), (2,12), (2,13)|
|(1,7) |(2,1), (2,4), (2,7), (2,8), (2,9), (2,10), (2,13)|
|(1,8) |(2,3), (2,5), (2,7), (2,8), (2,9), (2,11), (2,15)|
|(1,9) |(2,2), (2,6), (2,7), (2,8), (2,9), (2,12), (2,14)|
|(1,10)|(2,2), (2,5), (2,7), (2,10), (2,11), (2,12), (2,13)|
|(1,11)|(2,3), (2,4), (2,8), (2,10), (2,11), (2,12), (2,14)|
|(1,12)|(2,1), (2,6), (2,9), (2,10), (2,11), (2,12), (2,15)|
|(1,13)|(2,3), (2,6), (2,7), (2,10), (2,13), (2,14), (2,15)|
|(1,14)|(2,2), (2,4), (2,9), (2,11), (2,13), (2,14), (2,15)|
|(1,15)|(2,1), (2,5), (2,8), (2,12), (2,13), (2,14), (2,15)|
|(2,1) |(1,1), (1,2), (1,3), (1,4), (1,7), (1,12), (1,15)|
|(2,2) |(1,1), (1,2), (1,3), (1,5), (1,9), (1,10), (1,14)|
|(2,3) |(1,1), (1,2), (1,3), (1,6), (1,8), (1,11), (1,13)|
|(2,4) |(1,1), (1,4), (1,5), (1,6), (1,7), (1,11), (1,14)|
|(2,5) |(1,2), (1,4), (1,5), (1,6), (1,8), (1,10), (1,15)|
|(2,6) |(1,3), (1,4), (1,5), (1,6), (1,9), (1,12), (1,13)|
|(2,7) |(1,1), (1,4), (1,7), (1,8), (1,9), (1,10), (1,13)|
|(2,8) |(1,3), (1,5), (1,7), (1,8), (1,9), (1,11), (1,15)|
|(2,9) |(1,2), (1,6), (1,7), (1,8), (1,9), (1,12), (1,14)|
|(2,10)|(1,2), (1,5), (1,7), (1,10), (1,11), (1,12), (1,13)|
|(2,11)|(1,3), (1,4), (1,8), (1,10), (1,11), (1,12), (1,14)|
|(2,12)|(1,1), (1,6), (1,9), (1,10), (1,11), (1,12), (1,15)|
|(2,13)|(1,3), (1,6), (1,7), (1,10), (1,13), (1,14), (1,15)|
|(2,14)|(1,2), (1,4), (1,9), (1,11), (1,13), (1,14), (1,15)|
|(2,15)|(1,1), (1,5), (1,8), (1,12), (1,13), (1,14), (1,15)|
~~
Some observations:
* Given an algebra, the $7$ maximal orders stem exclusively from one of the $2$ classes of Fano planes.
* The class of the Fano plane class on which the algebra is based is always opposite to the ones of its maximal orders. I.e. the construction given above always fails if one chooses the Steiner system based on the the $7$ associative triples ("quaternionic roots") of the algebra.
* All in all there are $210 = 30\cdot 7$ maximal orders for the $30$ Fano planes. A closer inspection of the table reveals that each Fano plane occurs exactly $7$ times. I.e. the Fano planes are equidistributed among the maximal orders. In other words, given a Fano plane, one can always find $7\cdot 16 = 112$ algebras that have it as maximal order. For the other $368$ cases the $240$ integral elements do not close under the respective octonion product.

As a consequence of the above one has $3.360 = 480 \cdot 7 = 30\cdot 112$ different constructions of the octavian units and hence of the $E_8$-root system. (An observation: This number divides the order of the $E_8$-Weyl group. Is this an accident ?)

!!!!!Relationship with [[codes|Blockcode]]
The $\mb E_8$-lattice can also be expressed in terms of [[Reed-Muller-|Reed-Muller Code]], [[Hadamard-|Hadamard Code]] or [[Hamming-|Hamming Code]] codes as follows:
\begin{eqnarray}
\mathbb E_8 &= & 2 \mathbb Z^8 + [8,4,4] \\
& =& 2 \mathbb Z^8 + \operatorname{RM}(1,3) \\
& = & 2 \mathbb Z^8 + \operatorname{Had}(3) \\
&=& 2 \mathbb Z^8 + \operatorname{Ham_E}(3)
\end{eqnarray}

If we take the inner shell of this lattice and rescale it appropriately we get the octavian units.
The $r$ and $c_i$ in the construction above correspond with the $\mathbb Z^8$ sublattice and the $q_i$ and $q^\bot_i$ with the sublattice generated by the code.

The  $q_i$ and $q^\bot_i$ can be mapped to the $14$ weight-$4$ codewords as follows:
\begin{eqnarray}
&& q_1 = [10110100], \; \bar q^\bot_1 = [01001011], \\
&& q_2 = [11010010], \; \bar q^\bot_2 = [00101101], \\
&& q_3 = [11100001], \; \bar q^\bot_3 = [00011110], \\
&& q_4 = [10000111], \; \bar q^\bot_4 = [01111000], \\
&& q_5 = [11001100], \; \bar q^\bot_5 = [00110011], \\
&& q_6 = [10101010], \; \bar q^\bot_6 = [01010101], \\
&& q_7 = [10011001], \; \bar q^\bot_7 = [01100110], \\
\end{eqnarray}
For the scalar product in $\mathbb F_2$-space one also has $\langle \mb q_i|\mb q_i^\bot\rangle = 0$.

Using the $q_i$'s, replacing the $0$'s by $-1$'s and adding the weight-$8$ word of the code one can create the following $8\times 8$-matrix
\[
\begin{pmatrix} q_1\\ q_2\\q_3\\q_4\\q_5\\q_6\\q_7\\q_8\end{pmatrix} = \begin{pmatrix}
1&-1&1&1&-1&1&-1&-1\\
1&1&-1&1&-1&-1&1&-1\\
1&1&1&-1&-1&-1&-1&1\\
1&-1&-1&-1&-1&1&1&1\\
1&1&-1&-1&1&1&-1&-1\\
1&-1&1&-1&1&-1&1&-1\\
1&-1&-1&1&1&-1&-1&1\\
1&1&1&1&1&1&1&1 \end{pmatrix}
\]
which is a [[Hadamard Matrix|Hadamard Matrix]]. (See Magma example below). In case of $8$ dimensions this matrix is unique up to automorphisms of the order $21.504 = 2^{10} \cdot 3 \cdot 7$ which is a divisor of the order of the $E_8$-Weyl group $ \operatorname{ord}(Aut(E_8))/21.504 = 32.400$. (Is this accidential ?)
A permutation of rows or columns leads to an equivalent Hadamard matrix. In terms of the code this means that the three relevant parameters $n$, $k$ and $d$ remain unchanged. Analogous arguments apply to the $\bar{\mb q_i}$'s.
Notice that the [[(minimal) Hamming distance|Hamming Distance]] of $4$ translates into the fact that for a given triad of the Fano plane at least two elements must be different. In terms of designs that means that two blocks must be different in at least two points. (In fact they differ in exactly two points).

!!!!!Roots of unity
One can group the integral octonions according to their multiplicative [[order|Order]] $m$:
\begin{eqnarray}
m = 1:&& \mb e_0 \\
m = 2:&& -\mb e_0 \\
m = 3: &&\frac 12  (\mb e_0 \pm \mb e_i \pm \mb e_j \pm \mb e_k) \\
m = 4: && \pm \mb e_i\\
           &&   \frac 12  (\pm \mb e_i \pm \mb e_j \pm \mb e_k \pm \mb e_l) \\
m = 6: && \frac 12  (-\mb e_0 \pm \mb e_i \pm \mb e_j \pm \mb e_k)
\end{eqnarray}
or in other words, given elements are an $m$-th ''Root Of Unity''.
This way the number $240$ splits up according to $240 = 1 + 1 + 56 + 14+ 112 + 56$.

Computer experiments show that for any two integral octonions $\mb o_1$ and $\mb o_2$ one has
\[
\mb o_1 \mb o_2^m = \mb o_1
\]
with $m \in \{1,2,3,4,6\}$.
This generalizes the multiplicative orders given above, which are reproduced in the case $\mb o_1=\mb o_2$.

One has $114$ sixth roots of unity given by the $\mb r$'s and the $\mb q_i$'s. (I.e. the sixth roots of unity are exactly those integral octonions that contain the identity element).
They fall into two classes with $57$ elements each, those with $\mb r^3 = \mb q_i^3 = 1$ and those with $\mb r^3 = \mb q_i^3 = -1$.
The $\mb q_i$'s are also referred to as ''Brandt Transformers'' in literature due to a theorem by Brandt stating:
The map$ \mb X \rightarrow \mb A^{-1} \mb X \mb A$ is an [[automorphism|Automorphism]] if and only if $\mb A$ is a sixth root of unity.

!!!!! [[D8|Checkerboard Lattice]] (= [[SO(16)]]) construction
Alternatively one can construct the octavian units building upon the set $R$ of $112$ roots of the group $SO(16)$.
This requires an adequate change of basis of the octonion algebra.
The elements of the new basis $\mb e'_i$ (which are [[simple roots|Simple Root]] of $SO(16$)) can be chosen as follows:
\begin{eqnarray}
\mb e'_0 \equiv (\mb e_0 + \mb e_4), \quad \mb e'_4 \equiv (\mb e_0 - \mb e_4) \\
\mb e'_1 \equiv (\mb e_1 + \mb e_5), \quad \mb e'_5 \equiv (\mb e_1 - \mb e_5) \\
\mb e'_2 \equiv (\mb e_2 + \mb e_6), \quad \mb e'_6 \equiv (\mb e_2 - \mb e_6) \\
\mb e'_3 \equiv (\mb e_3 + \mb e_7), \quad \mb e'_7 \equiv (\mb e_3 - \mb e_7) \\
\end{eqnarray}
or, written in a more compact form:
\[
\mb e'_i \equiv (\mb e_i + \mb e_{i+4}), \quad \mb e'_{i+4} \equiv (\mb e_i - \mb e_{i+4}), \quad i = 0,1,2,3
\]
With them the set of roots of $SO(16)$ is given by:
\[
R = \pm \mb e'_i \pm \mb e'_j,  \quad i, j = 0,\ldots 7, \;\,  i<j
\]
That is, one picks all possible $2$-element sets out of a set of $8$ elements. There are $28 = \large {8 \choose 2}$ possibilities to do this. Combining this with the $4$ sign combinations one gets all the $112$ roots.
Notice that $R$ is not closed under octonion multiplication.

$R$ can be extended to the full set of $E_8$-roots by adjoining those $128$ elements
\[
\sum_{i =0}^7 \pm \mb e'_i
\]
which have an odd number of "-" signs.

!!!![[Isotopies|Isotopy]]
The group of isotopies is isomorphic to $Spin^+_8 (\mathbb F_2) = 2^2 O_8^+(2)$ with $O_8^+(2)$ the finite, simple [[orthogonal group|Orthogonal Group]] over $\mathbb F_2$. It is generated by the triple of maps $(\mb L_{\mb A}, \mb R_{\mb A}, \mb B_{\mb A})$ with $\mb L_{\mb A}$ and $\mb R_{\mb A}$ [[left- and right translations|Left- and Right Translation]] and $\mb B_{\mb A}$ a bimultiplication, given by $\mb B_{\mb A}(\mb X) = \mb A^{-1} \mb X \mb A^{-1}$.

!!!![[SAGE|http://www.sagenb.org/]]^^[[Help|Sage]]^^ examples
{{{
RM = gap.ReedMullerCode(1,3)
N = gap.Elements(RM)
gap.Size(N)
Aut = RM.AutomorphismGroup()
gap.Size(Aut)
gap.Elements(RM)
gap.WeightDistribution(RM)
}}}

!!!![[MAGMA|http://magma.maths.usyd.edu.au/calc/]]^^[[Help|MAGMA]]^^ examples
{{{
R := MatrixRing(Integers(), 8);

H := R!
[1,-1,1,1,-1,1,-1,-1,
1,1,-1,1,-1,-1,1,-1,
1,1,1,-1,-1,-1,-1,1,
1,-1,-1,-1,-1,1,1,1,
1,1,-1,-1,1,1,-1,-1,
1,-1,1,-1,1,-1,1,-1,
1,-1,-1,1,1,-1,-1,1,
1,1,1,1,1,1,1,1];

IsHadamard(H);
}}}

Papers:
* [[Ideals in the Intgral Octaves - D. Allcock|http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.22.8580&rep=rep1&type=pdf]] [[pct. 6|http://scholar.google.de/scholar?cites=2431900507781036899&hl=de]]
* [[Beyond Ideals in the Dickson Ring of Integral Octonions - F. Chaitin-Chatelin|http://www.umcs.maine.edu/~chaitin/f7.pdf]] [[pct. 3|http://scholar.google.de/scholar?cites=10547392125336397789&hl=de]]
* [[Hyperbolic Weyl Groups and the Four Normed Division Algebras - A. J. Feingold, A. Kleinschmidt, H. Nicolai|http://arxiv.org/PS_cache/arxiv/pdf/0805/0805.3018v1.pdf]] [[pct. 1|http://scholar.google.de/scholar?hl=de&lr=&cites=6176913252844757445&um=1&ie=UTF-8&ei=OjV8Stn9Cdi1sgaT6_zYAg&sa=X&oi=science_links&resnum=1&ct=sl-citedby]]
* [[Prime Factorization of Integral Cayley Octaves - P. Rehm|http://archive.numdam.org/ARCHIVE/AFST/AFST_1993_6_2_2/AFST_1993_6_2_2_271_0/AFST_1993_6_2_2_271_0.pdf]] [[pct. 7|http://scholar.google.de/scholar?cites=5391424943379139762&hl=de&as_sdt=2000]]
* [[Cayley Orders - A. M. Cohen, G. Nebe, W. Plesken|http://archive.numdam.org/ARCHIVE/CM/CM_1996__103_1/CM_1996__103_1_63_0/CM_1996__103_1_63_0.pdf]] [[pct. 2|http://scholar.google.de/scholar?cites=15837980437847642931&hl=de&as_sdt=2000]]
* [[Hyperbolic Weyl Groups and the four Normed Division Algebras - A. J. Feingold, A. Kleinschmidt, H. Nicolai|http://aps.arxiv.org/PS_cache/arxiv/pdf/0805/0805.3018v1.pdf]] [[pct. 1|http://scholar.google.de/scholar?cites=6176913252844757445&hl=de]]
* [[Integral Octonions and E8 - M. Koca|http://streaming.ictp.trieste.it/preprints/P/86/224.pdf]] pct 0
* [[Factorization and Congruence in the Arithmetics of Cayley's Algebra (1989) - by P. J. C. Lamont|http://journals.cambridge.org/action/displayFulltext?type=1&fid=5052048&jid=GMJ&volumeId=33&issueId=02&aid=5052040]] pct. 0

Theses:
* [[Ganzzahlige Oktonionen - T. Quade|http://www.quadi.de/~thomas/diplom/Diplom.pdf]] [[local|theses/GanzzahligeOktonionen.pdf]] - Related website: [[Thomas Mathe-Seite|http://www.quadi.de/~thomas/mathe.htm]]

Google Books:
* [[[1] The Beauty of Geometry: Twelve Essays - H. S. M. Coxeter (chapt. 2)|http://books.google.com/books?id=beTjmcibCH8C&pg=PA21&lpg=PA21&dq=The+Beauty+of+Geometry+integral+cayley+numbers&source=bl&ots=kRPl6CTZZ1&sig=SIz34DSqcVyLopq-LeyiYwtvv2c&hl=de&ei=fUuOSvzuBJGe_AbOoo32DQ&sa=X&oi=book_result&ct=result&resnum=1#v=onepage&q=&f=false]] [[local|google_books/TheBeautyOfGeometry.pdf]] [[bct. 7|http://scholar.google.de/scholar?cites=13159578209779229825&hl=de]] brl. 10

Links:
* [[WIKIPEDIA - Intelligent Design|http://en.wikipedia.org/wiki/Intelligent_design]]
* [[A Designer Universe? - S. Weinberg|http://www.physlink.com/education/essay_weinberg.cfm]]

Videos:
* [[Was the Universe Designed?|http://www.counterbalance.net/cqinterv/design-body.html]]

''Isospin'' is not an exact [[symmetry|Symmetries]]. In [[QCD]], for example, it is approximatively exact only at low energies.

Papers:
* [[Isospin and Local Space-Time Rotations - J. G. Valatin|http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.cmp/1103841070]] pct. 0

[[JHyperComplex|http://www.jhypercomplex.com]] is a Java API for doing hypercomplex computations (both numerical and algebraic) being developed by the author of this Wiki.

It is the result of realizing that when doing calculations with hypercomplex numbers (e.g. quaternions, octonions) classically with paper and pencil one often runs into the the same stupid, mechanistic, boring and hence error-prone calculations.
Furthermore there are things one cannot do this way due to them being too complex. Furthermore for larger algebras (which are very interesting in respect their applications in physics !), playing around and experimenting is not feasible any more.

In the meantime \JHyperComplex has become quite a potent research tool (unique of its kind) and has yielded quite a few interesting results.

This WIKI in parts is a byproduct of the development of this software and contains some results obtained by it.

There is a lot more that could said about \JHyperComplex. If you have questions, please contact me [[here|Welcome]].

If you don't believe that hypercomplex numbers are interesting, you should check out another piece of software I have written, namely  [[HyperFract|http://www.HyperFract.com]].

<html><center><img src="images/JohnHortonConway.jpg " style="width: 165px; "/></center></html>
Papers:
* [[The Free Will Theorem - J. Conway, S. Kochen|http://arxiv.org/PS_cache/quant-ph/pdf/0604/0604079v1.pdf]] [[pct. 34|http://scholar.google.de/scholar?cites=16215323905036932989&hl=de]]
* [[The Strong Free Will Theorem - J. Conway, S. Kochen|http://arxiv.org/PS_cache/arxiv/pdf/0807/0807.3286v1.pdf]] [[pct. 3|http://scholar.google.com/scholar?hl=de&lr=&cites=10339603623079889885&um=1&ie=UTF-8&ei=BWi_SuPuKtH-_AbssfGCAQ&sa=X&oi=science_links&resnum=3&ct=sl-citedby]]

Videos:
* [[Princeton University: Free Will Lecture Series|http://www.princeton.edu/WebMedia/lectures/]]

Books:
* [[On Numbers and Games|books/JHConway_OnNumbersAndGames.djvu]] {{t500Cite{[[bct. 624|http://scholar.google.de/scholar?cites=11910834121902210311&hl=de]]}}}

The (abelian) ''Kalb\-Ramond Field'' or ''Axion Field'' $B_{\mu\nu}$ is a two-form field which appears in the low energy limit of [[string theory|Superstring Theory]], in [[quantum gravity|Quantum Gravity]] and in several other frameworks in particle physics. Most attempts to incorporate mass to gauge field models in four dimensions take into account this object added to a one form gauge field.

(In string theory) the axion field  also shows up in the context of [[3-cocycles|3-Cocycle]] which are related to a violation of the [[Jacobi identity|Jacobian]], leading to a nonassociative algebra.

The ''Kemmer Equation'' describes a massive particle with spin 1 and was first derived in 1931 by Kemmer.
Its is a Dirac type equation but involves matrices obeying a different scheme of commutation rules. The theory can be developed in strikingly close correspondence to Dirac’s electron theory; practically all the definitions of physical quantities like spin, magnetic moment etc. have their exact counterpart.

The ''Klein Gordon Equation'' describes the dynamics of a scalar field $\Phi(\mb x)$ with mass m and is given by:
\[
(\partial_\mu\partial^\mu + m^2) \Phi (\mb x) \equiv (\square + m^2) \Phi (\mb x) = 0
\]
In case that $m=0$ one gets the [[D'Alembert equation|D'Alembert Equation]].

The ''Kochen\-Specker Theorem'' (''KS Theorem''), formulated in 1967, states that there cannot be any hidden variables in a non-contextual quantum mechanical model.  Besides Bell's theorem the KS theorem is a fundamental "no go theorem" for hidden variable theories in quantum mechanics.

The proof of the KS theorem is notoriously complex.

Papers:
* [[The Problem of Hidden Variables in Quantum Mechanics - S. Kochen, E. P. Specker|http://www.hep.princeton.edu/~mcdonald/examples/QM/kochen_iumj_17_59_68.pdf]] [[local|papers/kochen_iumj_17_59_68.pdf]]  {{t500Cite{[[pct. 843|http://scholar.google.de/scholar?cites=18441803985553364076&hl=de&as_sdt=2000]]}}} - The original paper on the subject.

Links:
* [[Stanford Encyclopedia of Philosophy - The Kochen-Specker Theorem|http://plato.stanford.edu/entries/kochen-specker/]]
* [[Webpage of Karl Svozil|http://tph.tuwien.ac.at/~svozil/]]

Videos:
* [[The Paradox of Kochen and Specker - John Conway|http://www.princeton.edu/WebMedia/flash/lectures/20090330_conway_free_will.shtml]]

The ''Korteweg-de Vries Equation'' (''\KdV Equation'' for short) is a nonlinear partial differential equation of the form
\[
u_t - 6 u u_{xx} + u_{xxx} = 0
\]
The equation was first written down by Korteweg and de Vries in 1895 in connection with the evolution of long water waves down canals of rectangular cross section. One solution of the equation leads to a mathematical representation of [[solitons|Soliton]], which were observed for the first time in 1834 in water canals by John Scott Russell.
The \KdV-equation also arises in plasma physics, in the study of an harmonic lattices, and in the propagation of waves in elastic rods.

Links: 
* [[WIKIPEDIA - Korteweg–de Vries Equation|http://en.wikipedia.org/wiki/Korteweg%E2%80%93de_Vries_equation]]

If $\mathcal H$ is a subgroup of a finite group $\mathcal G$, then the [[order|Order]] of $\mathcal H$ divides the order of $\mathcal G$.

The complex pole in the photon propagator in QED is known as the ''Landau Ghost''.

The presence of the Landau ghost is not taken as a serious drawback of QED since the momentum scale at which it appears is far from measurable and, at this scale QED should probably be modified to include other electroweak effects. However, for hadronic models the ghosts are a problem, since they appear in the meson and nucleon propagators at a scale as low as the nucleon mass.

The presence of ghost poles in the propagators violates basic theorems of local quantum field theory, and the ghosts are physically unacceptable because they correspond to eigenstates of the system with complex energies and probabilities.

Videos:
* [[The Six Billion Dollar Experiment|http://www.dailymotion.com/video/xaa0qo_the-six-billion-dollar-experiment-h_tech]]

The ''Large Number Hypothesis'' alludes to the fact that several (dimensionless) ratios of physical constants are of the order of $10^{40}$, a multiple or a simple power of it.
<html><center><img src="images/large_numbers.jpg" style="width: 540px; "/></center></html>
Links:
* [[Wikipedia, Dirac Large Number Hypothesis|http://en.wikipedia.org/wiki/Dirac_large_numbers_hypothesis]]

''Lattice Gas Automata'' (''LGA'') or ''Lattice Gas Cellular Automata'' (''LGCA'') are [[cellular automata|Cellular Automaton]] that allow for the simulation of fluid flows. From the LGCA, it is possible to derive the macroscopic Navier\-Stokes equations.

Any numerical evolution of a discretized partial differential equation can be interpreted as the evolution of some LGA. In the continuous time and space limit such a [[cellular automaton|Cellular Automaton]] mimics the behavior of the partial differential equation.

In 2 dimensions, models based on square lattices lack rotational invariance. This problem can be cured by models based on hexagonal lattices.

In 3 dimensions, the only space filling regular polytope, the cube, lacks rotational invariance. The only other polytopes that come into consideration, the dodecahedron and the icosahedron on the other hand are not space filling. In order to come up with suitable models for three dimensions one considers emdeddings in higher dimensional spaces.

For a quantum version of LGA see: [[quantum cellular automata|Quantum Cellular Automaton]].

Links:
* [[WIKIPEDIA - Lattice Gas Automaton|http://en.wikipedia.org/wiki/Lattice_gas_automaton]]

/***
|''Name:''|LaunchApplicationPlugin|
|''Author:''|Lyall Pearce|
|''Source:''|http://www.Remotely-Helpful.com/TiddlyWiki/LaunchApplication.html|
|''License:''|[[Creative Commons Attribution-Share Alike 3.0 License|http://creativecommons.org/licenses/by-sa/3.0/]]|
|''Version:''|1.4.0|
|''~CoreVersion:''|2.3.0|
|''Requires:''| |
|''Overrides:''| |
|''Description:''|Launch an application from within TiddlyWiki using a button|
!!!!!Usage
<<<
{{{<<LaunchApplication "buttonLabel" "tooltip" "application" ["arguments" ...]>>}}}
{{{<<LaunchApplicationButton "buttonLabel" "tooltip" "application" ["arguments" ...]>>}}}
{{{<<LaunchApplicationLink "buttonLabel" "tooltip" "application" ["arguments" ...]>>}}}
* buttonLabel is anything you like
* tooltip is anything you like
* application is a path to the executable (which is Operating System dependant)
* arguments is any command line arguments the application requires.
* You must supply relative path from the location of the TiddlyWiki OR a fully qualified path
* Forward slashes works fine for Windows

{{{<<LaunchApplication...>>}}} functions the same as {{{<<LaunchApplicationButton...>>}}}

eg.

{{{
<<LaunchApplicationButton "Emacs" "Linux Emacs" "file:///usr/bin/emacs">>
}}}
<<LaunchApplicationButton "Emacs" "Linux Emacs" "file:///usr/bin/emacs">>

{{{
<<LaunchApplicationLink "LocalProgram" "Program relative to Tiddly html file" "localDir/bin/emacs">>
}}}
<<LaunchApplicationLink "LocalProgram" "Program relative to Tiddly html file" "localDir/bin/emacs">>

{{{
<<LaunchApplicationButton "Open Notepad" "Text Editing" "file:///e:/Windows/notepad.exe">>
}}}
<<LaunchApplicationButton "Open Notepad" "Text Editing" "file:///e:/Windows/notepad.exe">>

{{{
<<LaunchApplicationLink "C Drive" "Folder" "file:///c:/">>
}}}
<<LaunchApplicationLink "C Drive" "Folder" "file:///c:/">>


!!!!!Revision History
* 1.1.0 - leveraged some tweaks from from Bradly Meck's version (http://bradleymeck.tiddlyspot.com/#LaunchApplicationPlugin) and the example text.
* 1.2.0 - Make launching work in Linux too and use displayMessage() to give diagnostics/status info.
* 1.3.0 - execute programs relative to TiddlyWiki html file plus fix to args for firefox.
* 1.3.1 - parameters to the macro are properly parsed, allowing dynamic paramters using {{{ {{javascript}} }}} notation.
* 1.4.0 - updated core version and fixed empty tooltip and added launch link capability

<<<
***/
//{{{
version.extensions.LaunchApplication = {major: 1, minor: 4, revision: 0, date: new Date(2007,12,29)};
config.macros.LaunchApplication = {};
config.macros.LaunchApplicationButton = {};
config.macros.LaunchApplicationLink = {};

function LaunchApplication(appToLaunch,appParams) {
    if(! appToLaunch)
	return;
    var tiddlyBaseDir = self.location.pathname.substring(0,self.location.pathname.lastIndexOf("\\")+1);
    if(!tiddlyBaseDir || tiddlyBaseDir == "") {
	tiddlyBaseDir = self.location.pathname.substring(0,self.location.pathname.lastIndexOf("/")+1);
    }
    // if Returns with a leading slash, we don't want that.
    if(tiddlyBaseDir.substring(0,1) == "/") {
	tiddlyBaseDir = tiddlyBaseDir.substring(1);
    }
    if(appToLaunch.indexOf("file:///") == 0) // windows would have C:\ as the resulting file
    {
	tiddlyBaseDir = "";
	appToLaunch = appToLaunch.substring(8);
    }

    if (config.browser.isIE) {
	// want where the tiddly is actually located, excluding tiddly html file

	var theShell = new ActiveXObject("WScript.Shell");
	if(theShell) {
            // the app name may have a directory component, need that too
	    // as we want to start with current working dir as the location
	    // of the app.
	    var appDir = appToLaunch.substring(0, appToLaunch.lastIndexOf("\\"));
	    if(! appDir || appDir == "") {
		appDir = appToLaunch.substring(0, appToLaunch.lastIndexOf("/"));
	    }
	    appParams = appParams.length > 0 ? " \""+appParams.join("\" \"")+"\"" : "";
	    try {
		theShell.CurrentDirectory = decodeURI(tiddlyBaseDir + appDir);
		var commandString = ('"' +decodeURI(tiddlyBaseDir+appToLaunch) + '" ' + appParams);
		pluginInfo.log.push(commandString);
	        theShell.run(commandString);
	    } catch (e) {
		displayMessage("LaunchApplication cannot locate/execute file '"+tiddlyBaseDir+appToLaunch+"'");
		return;
	    }
	} else {
	    displayMessage("LaunchApplication failed to create ActiveX component WScript.Shell");
	}
    } else { // Not IE
	// want where the tiddly is actually located, excluding tiddly html file
	netscape.security.PrivilegeManager.enablePrivilege("UniversalXPConnect");
        var file = Components.classes["@mozilla.org/file/local;1"].createInstance(Components.interfaces.nsILocalFile);
        var launchString;
	try { // try linux/unix format
            launchString = decodeURI(tiddlyBaseDir+appToLaunch);
	    file.initWithPath(launchString);
	} catch (e) {
	    try { // leading slash on tiddlyBaseDir
                launchString = decodeURI("/"+tiddlyBaseDir+appToLaunch);
		file.initWithPath(launchString);
	    } catch (e) {
		try { // try windows format
		    launchString = decodeURI(appToLaunch).replace(/\//g,"\\");
		    file.initWithPath(launchString);
		} catch (e) {
		    try { // try windows format
			launchString = decodeURI(tiddlyBaseDir+appToLaunch).replace(/\//g,"\\");
			file.initWithPath(launchString);
		    } catch (e) {
			displayMessage("LaunchApplication cannot locate file '"+launchString+"' : "+e);
			return;
		    } // try windows mode
		} // try windows mode
	    }; // try with leading slash in tiddlyBaseDir
	}; // try linux/unix mode
	try {
	    if (file.isFile() && file.isExecutable()) {
		displayMessage("LaunchApplication executing '"+launchString+"' "+appParams.join(" "));
		var process = Components.classes['@mozilla.org/process/util;1'].createInstance(Components.interfaces.nsIProcess);
		process.init(file);
		process.run(false, appParams, appParams.length);
	    }
	    else
	    {
		displayMessage("LaunchApplication launching '"+launchString+"' "+appParams.join(" "));
		file.launch(); // No args available with this option
	    }
	} catch (e) {
	    displayMessage("LaunchApplication cannot execute/launch file '"+launchString+"'");
	}
    }
};

config.macros.LaunchApplication.handler = function (place,macroName,params,wikifier,paramString,tiddler) {
    // 0=ButtonText, 1=toolTip, 2=AppToLaunch, 3...AppParameters
    if (params[0] && (params[1] || params[1] == "") && params[2]) {
        var theButton = createTiddlyButton(place, getParam(params,"buttonText",params[0]), getParam(params,"toolTip",params[1]), onClickLaunchApplication);
        theButton.setAttribute("appToLaunch", getParam(params,"appToLaunch",params[2]));
        params.splice(0,3);
        theButton.setAttribute("appParameters", params.join(" "));
        return;
    }
}
config.macros.LaunchApplicationButton.handler = function (place,macroName,params,wikifier,paramString,tiddler) {
    config.macros.LaunchApplication.handler (place,macroName,params,wikifier,paramString,tiddler);
}

config.macros.LaunchApplicationLink.handler = function (place,macroName,params,wikifier,paramString,tiddler) {
    // 0=ButtonText, 1=toolTip, 2=AppToLaunch, 3...AppParameters
    if (params[0] && (params[1] || params[1] == "") && params[2]) {
        //var theLink = createExternalLink(place, getParam(params,"buttonText",params[0]));
        var theLink = createTiddlyButton(place, getParam(params,"buttonText",params[0]), getParam(params,"toolTip",params[1]), onClickLaunchApplication,"link");
        theLink.setAttribute("appToLaunch", getParam(params,"appToLaunch",params[2]));
        params.splice(0,3);
        theLink.setAttribute("appParameters", params.join(" "));
        return;
    }
}

function onClickLaunchApplication(e) {
	var theAppToLaunch = this.getAttribute("appToLaunch");
	var theAppParams = this.getAttribute("appParameters").readMacroParams();
	LaunchApplication(theAppToLaunch,theAppParams);
}

//}}}

The 6 elementary particles electron, electron-neutrino, muon, muon-neutrino, tauon and tauon-neutrino are called ''Leptons''. Leptons are subject to the electro-weak and gravitational force.

The ''Lewis\-Tolman Lever Paradox'' (or ''Right\-Angle Lever Paradox'') is one of the first paradoxes of special relativity proposed in 1909.

Papers:
* [[Right Angle Lever Paradox - J. C. Nickerson, R. T. McAdory|http://polaris.deas.harvard.edu/galileo/images/material/1469/351/reltorque.pdf]] [[pct. 2|http://scholar.google.de/scholar?cites=14416921559306520612&hl=de]]
* [[The Lewis-Tolman Lever Paradox - J. W. Butler|http://www.physics.princeton.edu/~mcdonald/examples/mechanics/butler_ajp_38_360_70.pdf]] [[pct. 1|http://scholar.google.de/scholar?cites=7742882603248677411&hl=de]]
* [[Covariant Formulation of Hooke's Law - O. Gron|http://www.physics.princeton.edu/~mcdonald/examples/mechanics/gron_ajp_49_28_81.pdf]] pct. 0
* [[The Lack of Rotation in a Moving Right Angle Lever - J. Franklin|http://arxiv.org/PS_cache/arxiv/pdf/0805/0805.1196v2.pdf]] pct. 0
* [[Relativistic Angular Momentum - N. Menicucci|http://panda.unm.edu/Courses/Finley/P495/TermPapers/relangmom.pdf]] pct. 0

{{center{[img(518px+, )[images/lissi2.jpg]]}}}

{{center{[img(643px+, )[images/lissi1.jpg]]}}}
!!!!Properties
* [[Lie brackets|Commutator]] between [[root vectors|Root Vector]] correspond to vector addition between their roots, and to interactions between particles.
* Each root vector corresponds to a type of elementary particle.

!!!!A personal point of view
My understanding of the problems with the model is, that it seems to violate the [[Coleman-Mandula theorem|Coleman-Mandula Theorem]] and a comparable theorem by J. Mather and W. Thurston (see [1]), as gravity is part of the $E_8$-connection. (Although Lisi claims that this is not so if considering [[SO(4,1)|De Sitter Space]] instead of $SO(3,1)$). Still, I believe, the idea of identifying elementary particles with the roots of [[E8]] is extremely appealing and beautiful and it has already been described by other authors as well. (A hunch of mine is, that there is a deep relationship between particle states and the [[octonionic X-product|X-Product]]).

My suggestion how one could maybe cure the model is to embed $E_8$ in a larger algebra (which is certainly not a classical simple [[Lie group|Lie Group]] any more) and to take the gravitational part out of the $E_8$ Lie group. It might well be a coincidence that one can fit gravity into $E_8$, as the Lorentz group is a small symmetry group that easily fits in somewhere. Furthermore, due to experiments, these days [[Lorentz violations|Lorentz Violation]] are quite of an issue, so if it will finally turn out, that the Lorentz symmetry is not an exact symmetry, there might be problems anyway.

My conclusion: It might be worth to keep an eye on Lisi's model.

Papers:
* [[An Exceptionally Simple Theory of Everything - A. G. Lisi|http://arxiv.org/PS_cache/arxiv/pdf/0711/0711.0770v1.pdf]] [[local|papers/0711.0770v1.pdf]] [[pct. 17|http://scholar.google.de/scholar?cites=9561015463132699078&hl=de]]
A response to the $E_8$-model:
* [[There is no "Theory of Everything" Inside E8 - J. Distler, S. Garibaldi|http://arxiv.org/PS_cache/arxiv/pdf/0905/0905.2658v1.pdf]] [[pct. 1|http://scholar.google.de/scholar?cites=4858571848196380677&hl=de&as_sdt=2000]] - "We analyze certain subgroups of real and complex forms of the Lie group $E_8$, and deduce that any "Theory of Everything" obtained by embedding the gauge groups of gravity and the Standard Model into a real or complex form of $E_8$ lacks certain representation theoretic properties required by physical reality."
* [[[1] Noncommutative Geometry and Physics - A. Connes|http://www.alainconnes.org/docs/einsymp.pdf]]

Links:
* [[Deferential Geometry - Garret's Personal Wiki Notebook in Theoretical Physics |http://deferentialgeometry.org/]]
* [[Garrett Lisi on Wikipedia|http://en.wikipedia.org/wiki/Antony_Garrett_Lisi]]

Presentations:
* [[The Uncontroversial Mathematics Behind Garrett Lisi's Controversial "Theory of Everything"  - A. G. Noel|http://www.math.umb.edu/~anoel/publications/hss072608.pdf]] [[local|presentations/hss072608.pdf]]

Videos:
* [[TED Talks - Garrett Lisi on his Theory of Everything|http://www.ted.com/index.php/talks/garrett_lisi_on_his_theory_of_everything.html]]
* [[On Some Mathematics in Garrett Lisi's "E8 Theory of Everything" - B. Kostant|http://math.ucr.edu/home/baez/kostant/]]
* [[Physics Wiki (IT tools for Science)|http://pirsa.org/08090057]] - The video that kicked off this WIKI. 

Audios:
* [[A Connection with Everything - G. Lissi|http://relativity.phys.lsu.edu/ilqgs/]]
* [[Deferential Geometry - G. Lissi|http://www.matmor.unam.mx/eventos/loops07/talks/2A/]]
@@display:block;text-align:center;[img[My comments ...|images/comment.gif][Comments]]@@

The conventional ''Lorentz Group O(3,1)'' is the invariance group of Minkowski space $\mathbb{R}^{3+1}$, that is to say, the set of all linear [[automorphisms|Automorphism]] of Minkowski space that leave the (pseudo-)[[scalar product|Scalar Product]] invariant.

The Lorentz group is a $6$-dimensional non-compact [[Lie group|Lie Group]]. Its [[Lie algebra|Lie Algebra]] is isomorphic to $\mathfrak sl$$(2,\mathbb C)$ with $3$ generators $J_i$ of spacial rotations, representing the subalgebra $\mathfrak so$$(3)$ and $3$ generators $K_i$ of "spacetime-rotations" called ''Boosts'', satisfying the commutation relations:
\begin{eqnarray}
[J_i,J_j] & = &\varepsilon_{ijk} J_k \\
[K_i,K_j] & = -&\varepsilon_{ijk} J_k \\
\end{eqnarray}
Hence boosts do not form a group.

The full Lorentz group consists of $4$ disconnected spaces. Elements of a given connected space can be transformed into one another by smooth (infinitesimal) transformations, i. e. [[conjugations|Conjugation]]. Furthermore one has discrete transformations between the $4$ topologically separated pieces (conjugacy classes), namely space inversions $P$ and a time reversal $T$. (See also [[CPT-transformations|CPT-Transformations]]). The set of discrete transformations $\{1, P, T, PT\}$ forms a group which is isomorphic to the [[Klein four-group|Klein Four-group]].

The subgroup of the Lorentz group with determinant equal to $1$, ''SO(3,1)'', is called the ''Proper Lorentz Group'', also designated as ''$L_+$''. (The "+" indicates the positive sign of the determinant). It consists of the orientation preserving transformations. Its universal covering group is the group [[SL(2,C)]]. One has the isomorphism: $SO(3,1) \cong SL(2,\mathbb C)/\mathbb Z_2$.
$SO(3,1)$ is built up of of $2$ subgroups. One of them contains the identity transformation and is called ''Proper Orthochronous Lorentz Group'' or ''Restricted Lorentz Group'', also designated as ''$SO(3,1)^+$'' or ''$L_+^\uparrow$'' (where the up arrow stands for "orthochronous"). The [[quotient group|Quotient Group]] $O(1,3)/SO(1,3)^+$ is isomorphic to the Klein four-group mentioned above.

!!!!Generalisations
<html><center><img src="images/LorentzGroups.jpg" style="width: 500px; "/></center></html>

The ''M\-Algebra'' is the maximal extension of the $\mathcal{N}=1$ super-Poincaré algebra in eleven dimensions.
It is spanned by the set $G_{A}=\{J_{ab},P_a,Q_\alpha,Z_{ab},Z_{abcde}\}$, where $J_{ab}$ and $P_a$ are the generators of the [[Poincaré group|Poincaré Transformation]] and $Q_\alpha$ is a Majorana spinor supercharge with anticommutator
\begin{equation}
\{Q_{\alpha },Q_{\beta }\}=\left( C\Gamma ^{a}\right) _{\alpha \beta
}P_{a}+(C\Gamma ^{ab})_{\alpha \beta }Z_{ab}+(C\Gamma ^{abcde})_{\alpha
\beta }Z_{abcde}
\end{equation}
The charge conjugation matrix $C$ is antisymmetric, and the central charges $Z_{ab}$ and $Z_{abcde}$ are tensors under Lorentz rotations but otherwise Abelian generators. In standard eleven-dimensional supergravity, these generators correspond to the "electric" and "magnetic" charges of the $M2$ and $M5$ branes, respectively.

Papers:
* [[Poincaré Invariant Gravity with Local Supersymmetry as a Gauge Theory for the M-algebra - M. Hassaine, R. Troncoso, J. Zanelli|http://arxiv.org/PS_cache/hep-th/pdf/0306/0306258v2.pdf]] [[pct. 16|http://scholar.google.de/scholar?cites=10560539088242203663&hl=de]]
* [[On the Octonionic Superconformal M-algebra - F. Toppan|ftp://ftp2.biblioteca.cbpf.br/pub/apub/2002/nf/nf_zip/nf04502.pdf]] pct. 0

Papers:
* [[Topics in M-theory - E. Sezegin|http://arxiv.org/PS_cache/hep-th/pdf/9809/9809204v2.pdf]] [[pct. 16|http://scholar.google.de/scholar?cites=15550580083989394648&hl=de]]

!!!![[MAGMA|http://magma.maths.usyd.edu.au/calc/]]^^[[Help|MAGMA]]^^
One if the outstanding features of MAGMA is that it allows for the generation of [[lattices|Lattice]], a feature that is often missing in other computer algebra systems.

Links:
* [[MAGMA Computational Algebra System Home Page|http://magma.maths.usyd.edu.au/]]
* [[MAGMA Online Calculator|http://magma.maths.usyd.edu.au/calc/]]
* [[WIKIPEDIA - MAGMA Computer Algebra System|http://en.wikipedia.org/wiki/Magma_computer_algebra_system]]
* [[Solving Problems with MAGMA - W. Bosma, J. Cannon, C. Playoust, A. Steel|http://www.dms.auburn.edu/research/manuals/magma2.6/examples.pdf]]  [[local|lectures/SolvingProblemsWithMAGMA.pdf]] [[lct. 10|http://scholar.google.com/scholar?hl=de&lr=&cites=9486123372688473527&um=1&ie=UTF-8&ei=YuE2S46dNKfesAbnzbHSCA&sa=X&oi=science_links&resnum=10&ct=sl-citedby&ved=0CDkQzgIwCTgK]]
* [[Handbook of MAGMA Functions|http://www.msri.org/about/computing/docs/magma/]] [[local|documents/MAGMA]]
** [[Lattices|http://www.msri.org/about/computing/docs/magma/html/text826.htm]] [[local|documents/MAGMA/html/text826.htm]]
** [[Coding Theory|http://www.msri.org/about/computing/docs/magma/html/part16.htm]] [[local|documents/MAGMA/html/part16.htm]]
** [[Hadamard Matrices|http://www.msri.org/about/computing/docs/magma/html/text1517.htm]] [[local|file:///E:/Trajectory/documents/MAGMA/html/text1517.htm]]
** [[Incidence Structures and Designs|http://www.msri.org/about/computing/docs/magma/html/text1502.htm]] [[local|file:///E:/Trajectory/documents/MAGMA/html/text1502.htm]]

Examples:
* [[Applied Abstract Algebra - D. Joyner, R. Kreminski, J. Turisco|http://www.usna.edu/Users/math/wdj/book/book.html]]

Links:
* [[Maxima website|http://maxima.sourceforge.net]]

''MOND'' is a modification of Newtonian dynamics, designed to reproduce the observed ‘flat’ galaxy rotation curves using only observed distributions of visible matter and reasonable assumptions about mass/light ratios as input data. It was proposed in 1983 by Moti Milgrom.
MOND applies at the very low accelerations which occur in the outer regions of spiral galaxies and in galaxy groups.

<!--{{{-->
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The author of this WIKI ...

<html><center><img src="images/author.jpg" style="width: 341px;"/></center></html>
My ...

Philosophy: Strive for [["immortality"|Immortality]]. If you cannot make it there, you will not make it anywhere.

Passion: [["Hacking"|JHyperComplex]] the universe. 

Education: Physicist with minors in relativistic quantum field theory, mathematics and a [[thesis in astronomy|http://katalog.ub.uni-heidelberg.de/cgi-bin/titel.cgi?katkey=59805397&teil=&teil2=&start=1&pagesize=10&sess=869427d0a097b34c074d6c4fa3ad4d6c&query=markus%20Maute&vr=1&pagesize=10]] at the [[MPIA|http://www.mpia.de/Public/menu_q2e.php]] ([[Demonstrating by means of laboratory simulations, that in principle it is possible to do high resolution astronomical imaging in the near infrared light, using a glass fiber based interferometer|images/stellar_interferometer.jpg]]).

Hobby: Astrophotography, see [[Window to the Universe|http://www.markus-maute.de/universe/WindowToTheUniverse.html]] and [[Astro Photography with the Digital Camera SONY DSC-W200|http://www.markus-maute.de/sonyastro.html]].

Professional history: Software developer (with a focus on Java).  

Professional goals: A synergy of my qualifications and competence in mathematics, physics, technical and scientific writing and Java software development. 
If you know of an opportunity in this regard, I would appreciate you contacting me (e.g. via e\-mail: <html><a href="mailto:trajectory@markus-maute.de">trajectory@markus-maute.de</a></html>).

Having reached a certain age, I allow myself calling the following statements wisdom:

*{{floatright {<html><iframe name="content" src="http://www.markus-maute.de/trajectory/advert_sky.html" width=180 height=620 scrolling="no"></iframe> </html> }}} To explain a universe is one thing, to explain a universe containing life and consciousness is another thing.

* My bet: When it comes to an adequate description of the very small scales, Riemannian geometry will desperately fail.

* There are basically two different ways to build the world: Either one starts with "nothing" or with "everything". In the latter case one has to explain how one can extract "something" from "everything". In the former case one first of all has to justify why there is "something" at all and not "nothing".

* The "formula of the world" should describe why the universe is the way it is. But what describes why that formula is the way it is ?

* Quantum mechanics, as we know it, appears to be sufficient to describe the world around us. An extension to it should give us a clue as to why this description is the way it is.

* The least principle of a physical theory is consistency. The more a theory comprises, the more potent the action of this principle is. Maybe consistency is the ultimate physical principle.

* Algebra is the key to unlock the secrets of the world.

* I don't think that there is a theory of everything, but I think that there is a theory of pretty much.

* At times a physicist not just spends a lot of time doing calculations, but thinking about how one can do them not spending too much time.

* If you want to get started doing fundamental physics, Clifford algebras may serve as suitable "warm up" algebras.

* It seems, that at the moment it is easier for mankind to go down to the microcosmos than to go up to the macrocosmos. So then let's do the former first.

* Das Haag'sche Theorem sollte bei einem Physiker ein gewisses Unbehagen hervorrufen.

* I think that the state of knowledge of physics today doesn't allow for a profound interpretation of the "world".

* If you are smart enough, don't do philosophy in the first place, but algebra.

* Understanding in physics and mathematics is the kind of understanding that one can only gain by building it up step by step.

* Not knowing what a 2-cocycle is, one should better avoid talking about "the interpretation of quantum mechanics".

* Mathematics is much about having complicated thoughts about simple things.

* Understanding something pretty well means being able to extract the essence of it.

* Physics is a bit like a chess game, knowing the rules doesn't imply that one knows how to play it well.

* A times one must run into problems and solve them to gain deep insights.

* One of the most difficult things in fundamental physics is to figure out if a topic one is dealing with is irrelevant and a waste of time.

* One day someone might come up with a unified theory in physics and we will realize, that the mathematical and physical ingredients have already been around for a long time.

* Physical laws are local in nature.

* Trying to do interpretations within a conceptual framework that is not general enough is a waste of time.

* If you come up with something and are not shure about it, it's probably wrong.

* Saying something in too general a way means running into the risk that one is saying "nothing".

* Einstein and Dirac have erected two mighty pillars. Now it is up to us to put an arc on them. Doing so would be a monumental event.

* Confusion is a good starting point for doing mathematics and physics - at least if one is bothered with it.

* Probably every mathematical language has its limitations - maybe even its shortcomings.

* Intuition fails when it comes to the very large and the very small.

* The most important and frequent mathematical operation in my opinion is a "copy and paste".

* Algebra is the true entity to describe the world, geometry related to it is a matter of convenience, taking into account the human brain.

* Real physics is deep.

* Once someone has found the theory of everything, the most difficult thing is yet to come, namely to sell it to others.

* Superstring theory has been quite successful in streamlining the thinking of a generation of physicists.

* To really come to a good understanding of mathematics and physics it requires trying out a lot in vein.

* Writing sharpens your thinking/mind.

* Time isn't what prevents everything from happening at once, rather it is what makes things happen at all.

* I believe that really deep mathematical and physical truths are combinatorial in nature.

* The most successful people in everyday life are those who manage to learn the 4 basic arithmetic operations. Those who do better and those who don't are worse off.

* The internet is like a second reality from where some don't find their way back into the real world.

* Abstract mathematics sometimes is like politics, a lot is being said, often with fancy words, but in the end nothing is really "done".

* Any finite problem can be solved by a computer. As any proof is a sequence of a finite number of manipulations, the question arises whether there is anything that cannot in principle be proved by means of a computer.

* I'm suspicious about people, doing mathematics these days without the aid of a computer.

* Much at the forefront of theoretical physics these days seems to exhibits a certain lack of orientation.

* "What do you know about string theory ?" - "Not very much, I'm doing physics and have little time care about this theory."

* The "theory of everything" will probably require mathematics that is of another calibre than what we are used to.

* I believe that the inner dimensions of Kaluza and Klein are physically as real as the degrees of freedom of stress, strain, shear etc. in a rigid body.
@@display:block;text-align:center;[img[My comments ...|images/comment.gif][Comments]]@@

>Based on real experiments and computer simulations, quantum gauge theory in four dimensions is believed to have a mass gap. This is one of the most fundamental facts that makes the universe the way it is.
> - Edward Witten [1]

The ''Mass Gap'' is the difference in energy between the vacuum and the next lowest energy state. The energy of the vacuum is zero by definition, and assuming that all energy states can be thought of as particles in plane-waves, the mass gap is the mass of the lightest particle.

Theories with massless particles, like the photon or a [[Goldstone boson|Goldstone Boson]], have no mass gap.

Yang\-Mills theory is supposed to have one but a profound theoretical explanation for its existence is lacking. This is one of the seven Millennium Prize problems put forward by the Clay Mathematics Institute.

Papers:
* [[[1] The Problem of Gauge Theory - E. Witten|http://arxiv.org/PS_cache/arxiv/pdf/0812/0812.4512v3.pdf]] pct. 0

/***
|Name|MatchTagsPlugin|
|Source|http://www.TiddlyTools.com/#MatchTagsPlugin|
|Documentation|http://www.TiddlyTools.com/#MatchTagsPluginInfo|
|Version|2.0.3|
|Author|Eric Shulman|
|License|http://www.TiddlyTools.com/#LegalStatements|
|~CoreVersion|2.1|
|Type|plugin|
|Description|'tag matching' with full boolean expressions (AND, OR, NOT, and nested parentheses)|
!!!!!Documentation
> see [[MatchTagsPluginInfo]]
!!!!!Revisions
<<<
2010.03.02 2.0.3 added %6 format (tags)
| please see [[MatchTagsPluginInfo]] for additional revision details |
2008.02.28 1.0.0 initial release
<<<
!!!!!Code
***/
//{{{
version.extensions.MatchTagsPlugin= {major: 2, minor: 0, revision: 3, date: new Date(2010,3,2)};

// store.getMatchingTiddlers() processes boolean expressions for tag matching
//    sortfield (optional) sets sort order for tiddlers - default=title
//    tiddlers (optional) use alternative set of tiddlers (instead of current store)
TiddlyWiki.prototype.getMatchingTiddlers = function(tagexpr,sortfield,tiddlers) {

	var debug=config.options.chkDebug; // abbreviation
	var cmm=config.macros.matchTags; // abbreviation
	var r=[]; // results are an array of tiddlers
	var tids=tiddlers||store.getTiddlers(sortfield||"title");
	if (tiddlers && sortfield) store.sortTiddlers(tids,sortfield);
	if (debug) displayMessage(cmm.msg1.format([tids.length]));

	// try simple lookup to quickly find single tags or tags that
	// contain boolean operators as literals, e.g. "foo and bar"
	for (var t=0; t<tids.length; t++)
		if (tids[t].isTagged(tagexpr)) r.pushUnique(tids[t]);
	if (r.length) {
		if (debug) displayMessage(cmm.msg4.format([r.length,tagexpr]));
		return r;
	}
	
	// convert expression into javascript code with regexp tests,
	// so that "tag1 AND ( tag2 OR NOT tag3 )" becomes
	// "/\~tag1\~/.test(...) && ( /\~tag2\~/.test(...) || ! /\~tag3\~/.test(...) )"

	// normalize whitespace, tokenize operators, delimit with "~"
	var c=tagexpr.trim(); // remove leading/trailing spaces
	c = c.replace(/\s+/ig," "); // reduce multiple spaces to single spaces
	c = c.replace(/\(\s?/ig,"~(~"); // open parens
	c = c.replace(/\s?\)/ig,"~)~"); // close parens
	c = c.replace(/(\s|~)?&&(\s|~)?/ig,"~&&~"); // &&
	c = c.replace(/(\s|~)AND(\s|~)/ig,"~&&~"); // AND
	c = c.replace(/(\s|~)?\|\|(\s|~)?/ig,"~||~"); // ||
	c = c.replace(/(\s|~)OR(\s|~)/ig,"~||~"); // OR
	c = c.replace(/(\s|~)?!(\s|~)?/ig,"~!~"); // !
	c = c.replace(/(^|~|\s)NOT(\s|~)/ig,"~!~"); // NOT
	c = c.replace(/(^|~|\s)NOT~\(/ig,"~!~("); // NOT(
	// change tag terms to regexp tests
	var terms=c.split("~"); for (var i=0; i<terms.length; i++) { var t=terms[i];
		if (/(&&)|(\|\|)|[!\(\)]/.test(t) || t=="") continue; // skip operators/parens/spaces
		if (t==config.macros.matchTags.untaggedKeyword)
			terms[i]="tiddlertags=='~~'"; // 'untagged' tiddlers
		else
			terms[i]="/\\~"+t+"\\~/.test(tiddlertags)";
	}
	c=terms.join(" ");
	if (debug) { displayMessage(cmm.msg2.format([tagexpr])); displayMessage(cmm.msg3.format([c])); }

	// scan tiddlers for matches
	for (var t=0; t<tids.length; t++) {
	 	// assemble tags from tiddler into string "~tag1~tag2~tag3~"
		var tiddlertags = "~"+tids[t].tags.join("~")+"~";
		try { if(eval(c)) r.push(tids[t]); } // test tags
		catch(e) { // error in test
			displayMessage(cmm.msg2.format([tagexpr]));
			displayMessage(cmm.msg3.format([c]));
			displayMessage(e.toString());
			break; // skip remaining tiddlers
		}
	}
	if (debug) displayMessage(cmm.msg4.format([r.length,tagexpr]));
	return r;
}
//}}}
//{{{
config.macros.matchTags = {
	msg1: "scanning %0 input tiddlers",
	msg2: "looking for '%0'",
	msg3: "using expression: '%0'",
	msg4: "found %0 tiddlers matching '%1'",
	noMatch: "no matching tiddlers",
	untaggedKeyword: "-",
	untaggedLabel: "no tags",
	untaggedPrompt: "show tiddlers with no tags",
	defTiddler: "MatchingTiddlers",
	defTags: "",
	defFormat: "[[%0]]",
	defSeparator: "\n",
	reportHeading: "Found %0 tiddlers tagged with: '{{{%1}}}'\n----\n",
	handler: function(place,macroName,params,wikifier,paramString,tiddler) {
		var mode=params[0]?params[0].toLowerCase():'';
		if (mode=="inline")
			params.shift();
		if (mode=="report" || mode=="panel") {
			params.shift();
			var target=params.shift()||this.defTiddler;
		}
		if (mode=="popup") {
			params.shift();
			if (params[0]&&params[0].substr(0,6)=="label:") var label=params.shift().substr(6);
			if (params[0]&&params[0].substr(0,7)=="prompt:") var prompt=params.shift().substr(7);
		} else {
			var fmt=(params.shift()||this.defFormat).unescapeLineBreaks();
			var sep=(params.shift()||this.defSeparator).unescapeLineBreaks();
		}
		var sortBy="+title";
		if (params[0]&&params[0].substr(0,5)=="sort:") sortBy=params.shift().substr(5);
		var expr = params.join(" ");
		if (mode!="panel" && (!expr||!expr.trim().length)) return;
		if (expr==this.untaggedKeyword)
			{ var label=this.untaggedLabel; var prompt=this.untaggedPrompt };
		switch (mode) {
			case "popup": this.createPopup(place,label,expr,prompt,sortBy); break;
			case "panel": this.createPanel(place,expr,fmt,sep,sortBy,target); break;
			case "report": this.createReport(target,this.defTags,expr,fmt,sep,sortBy); break;
			case "inline": default: this.createInline(place,expr,fmt,sep,sortBy); break;
		}
	},
	formatList: function(tids,fmt,sep) {
		var out=[];
		for (var i=0; i<tids.length; i++) { var t=tids[i];
			var title=t.title;
			var who=t.modifier;
			var when=t.modified.toLocaleString();
			var text=t.text;
			var first=t.text.split("\n")[0];
			var desc=store.getTiddlerSlice(t.title,"description");
			desc=desc||store.getTiddlerSlice(t.title,"Description");
			desc=desc||store.getTiddlerText(t.title+"##description");
			desc=desc||store.getTiddlerText(t.title+"##Description");
			var tags=t.tags.length?'[['+t.tags.join(']] [[')+']]':'';
			out.push(fmt.format([title,who,when,text,first,desc,tags]));
		}
		return out.join(sep);
	},
	createInline: function(place,expr,fmt,sep,sortBy) {
		wikify(this.formatList(store.sortTiddlers(store.getMatchingTiddlers(expr),sortBy),fmt,sep),place);
	},
	createPopup: function(place,label,expr,prompt,sortBy) {
		var btn=createTiddlyButton(place,
			(label||expr).format([expr]),
			(prompt||config.views.wikified.tag.tooltip).format([expr]),
			function(ev){ return config.macros.matchTags.showPopup(this,ev||window.event); });
		btn.setAttribute("sortBy",sortBy);
		btn.setAttribute("expr",expr);
	},
	showPopup: function(here,ev) {
		var p=Popup.create(here); if (!p) return false;
		var tids=store.getMatchingTiddlers(here.getAttribute("expr"));
		store.sortTiddlers(tids,here.getAttribute("sortBy"));
		var list=[]; for (var t=0; t<tids.length; t++) list.push(tids[t].title);
		if (!list.length) createTiddlyText(p,this.noMatch);
		else {
			var b=createTiddlyButton(createTiddlyElement(p,"li"),
				config.views.wikified.tag.openAllText,
				config.views.wikified.tag.openAllTooltip,
				function() {
					var list=this.getAttribute("list").readBracketedList();
					story.displayTiddlers(null,tids);
				});
			b.setAttribute("list","[["+list.join("]] [[")+"]]");
			createTiddlyElement(p,"hr");
		}
		var out=this.formatList(tids," &nbsp;[[%0]]&nbsp; ","\n"); wikify(out,p);
		Popup.show();
		ev.cancelBubble=true;
		if(ev.stopPropagation) ev.stopPropagation();
		return false;
	},
	createReport: function(target,tags,expr,fmt,sep,sortBy) {
		var tids=store.sortTiddlers(store.getMatchingTiddlers(expr),sortBy);
		if (!tids.length) { displayMessage('no matches for: '+expr); return false; }
		var msg=config.messages.overwriteWarning.format([target]);
		if (store.tiddlerExists(target) && !confirm(msg)) return false;
		var out=this.reportHeading.format([tids.length,expr])
		out+=this.formatList(tids,fmt,sep);
		store.saveTiddler(target,target,out,config.options.txtUserName,new Date(),tags,{});
		story.closeTiddler(target); story.displayTiddler(null,target);
	},
	createPanel: function(place,expr,fmt,sep,sortBy,tid) {
		var s=createTiddlyElement(place,"span"); s.innerHTML=store.getTiddlerText("MatchTagsPlugin##html");
		var f=s.getElementsByTagName("form")[0];
		f.expr.value=expr; f.fmt.value=fmt; f.sep.value=sep.escapeLineBreaks();
		f.tid.value=tid; f.tags.value=this.defTags;
	}
};
//}}}
/***
//{{{
!html
<form style='display:inline;white-space:nowrap'>
<input type='text'    name='expr' style='width:50%' title='tag expression'><!--
--><input type='text'    name='fmt'  style='width:10%' title='list item format'><!--
--><input type='text'    name='sep'  style='width:5%'  title='list item separator'><!--
--><input type='text'    name='tid'  style='width:12%' title='target tiddler title'><!--
--><input type='text'    name='tags' style='width:10%' title='target tiddler tags'><!--
--><input type='button'  name='go'   style='width:8%'  value='go' onclick="
	var expr=this.form.expr.value;
	if (!expr.length) { alert('Enter a boolean tag expression'); return false; }
	var fmt=this.form.fmt.value;
	if (!fmt.length) { alert('Enter the list item output format'); return false; }
	var sep=this.form.sep.value.unescapeLineBreaks();
	var tid=this.form.tid.value;
	if (!tid.length) { alert('Enter a target tiddler title'); return false; }
	var tags=this.form.tags.value;
	config.macros.matchTags.createReport(tid,tags,expr,fmt,sep,'title');
	return false;">
</form>
!end
//}}}
***/
//{{{
// SHADOW TIDDLER for displaying default panel input form
config.shadowTiddlers.MatchTags="<<matchTags panel>>";
//}}}
//{{{
// TWEAK core filterTiddlers() for enhanced boolean matching in [tag[...]] syntax:
// use getMatchingTiddlers instead getTaggedTiddlers
var fn=TiddlyWiki.prototype.filterTiddlers;
fn=fn.toString().replace(/getTaggedTiddlers/g,"getMatchingTiddlers");
eval("TiddlyWiki.prototype.filterTiddlers="+fn);
//}}}
//{{{
// REDEFINE core handler for enhanced boolean matching in tag:"..." paramifier
// use filterTiddlers() instead of getTaggedTiddlers() to get list of tiddlers.
config.paramifiers.tag = {
	onstart: function(v) {
		var tagged = store.filterTiddlers("[tag["+v+"]]");
		story.displayTiddlers(null,tagged,null,false,null);
	}
};
//}}}

The ''Mathisson\-Papapetrou Equations'' or (''MP Equations'') are two coupled equations describing the motion of a particle with spin in a fixed gravitational background, given by:
\begin{eqnarray}
\frac{D\tilde p^\mu}{D\tau}  &=& -\frac{1}{2} {R^\mu}_{\nu\lambda\sigma} S^{\nu\lambda} u^\sigma \equiv f^\mu \\
\frac{DS^{\mu\nu}}{D\tau}   &=& \tilde{p}^{\mu}u^\nu - \tilde{p}^{\nu}u^\mu \equiv f^{\mu\nu}
\end{eqnarray}
with $\frac{D}{D\tau}$ the covariant derivative in respect to the proper time $\tau$, $u^\alpha$ the $4$-velocity, $S^{\mu\nu}$ the totally antisymmetric [[spin tensor|Spin Tensor]], $f_\mu$ the four-force acting on the particle and ${R^\mu}_{\nu\lambda\sigma}$ the [[Riemann tensor|Riemann Tensor]].

$\tilde{p}_\mu$ is called generalized momentum of the particle which in general is not aligned with the $4$-velocity and is given by
\[
\tilde{p}^{\mu} \equiv m u^\mu - \frac{DS^{\mu\nu}}{D\tau}u_\nu
\]
The first of the two equations is called ''Equation of Motion of the Particle'', the second one ''Equation Of Motion Of Spin''.
The equations can be derived by making a multipole expansion around the worldline of the particle.

Inserting the generalized momentum into the two equations one gets more explicitly
\begin{eqnarray}
\frac{D}{D\tau} \left (mu^\mu - \frac{DS^{\mu\nu}}{D\tau}u_\nu \right ) &=& -\frac{1}{2} {R^\mu}_{\nu\lambda\sigma} S^{\nu\lambda} u^\sigma  \\
\frac{DS^{\mu\nu}}{D\tau} &=& \left (m u^\mu - \frac{DS^{\mu\rho}}{D\tau}u_\rho \right ) u^\nu - \left (m u^\nu - \frac{DS^{\nu\rho}}{D\tau}u_\rho \right )u^\mu
\end{eqnarray}
hence
\begin{eqnarray}
\frac{D (m u^\mu)}{D\tau} - \frac{DS^{\mu\nu}}{D\tau} \frac{Du^\mu}{D\tau} +  \frac{D^2S^{\mu\nu}}{D\tau^2} u_\nu + \frac{1}{2} {R^\mu}_{\nu\lambda\sigma} S^{\nu\lambda} u^\sigma &=& 0\\
\frac{DS^{\mu\nu}}{D\tau} + \left (\frac{DS^{\mu\rho}}{D\tau} u^\nu - \frac{DS^{\nu\rho}}{D\tau} u^\mu \right) u_\rho    &=& 0
\end{eqnarray}
!!!!Special cases
!!!!! I.
If there is no spin precession ("spin acceleration"), i.e. $\frac{DS^{\mu\nu}}{D\tau}  = 0$, the first equation simplifies to
\[
\frac{D (m u^\mu)}{D\tau}  = -\frac{1}{2} {R^\mu}_{\nu\lambda\sigma} S^{\nu\lambda} u^\sigma \equiv f^\mu
\]
and the second one to
\[
p^{\mu}u^\nu - p^{\nu}u^\mu  =  f^{\mu\nu} = 0
\]

!!!!! II.
If in addition spin is zero, no Lorentz force acts upon it and one is left with the classical [[geodesic equation|Geodesic Equation]] of a point particle in a gravitational field:
\[
\frac{D(m u^\mu)}{D\tau}  = 0
\]
!!!! Supplementary Conditions
If the the two equations are regarded as a closed system for determining $p^\mu (\tau)$ and $S^{\mu\nu}(\tau)$, the number of equations is less then the number of unknown functions. For their full determination $3$ additional scalar supplementary conditions are required.
E.g. one introduces
the ''Corinaldesi\-Papapetrou\-Condition''
\[
S^{\mu0} u_\nu = 0
\]
or the ''Pirani Condition''
\[
S^{\mu\nu} u_\nu = 0
\]
or the ''Tulczyjew Condition''
\[
S^{\mu\nu} \tilde p_\nu = 0
\]
There seems to be no fundamental reason as to why to prefer the one or the other condition.
Therefore the most widely accepted description of spinning test particles in relativity is incomplete, at least for what concerns the arbitrary choice of supplementary conditions required to make the model compatible.

It was demonstrated by Barker and O'Connell (1974) that when keeping only second-order terms in the spin or velocity of the particle or in the gravitational radius of the centre of a gravitational source, the first two supplementary conditions lead to different non-geodesic equations of motion.

!!!!Historical
The derivation in the original papers of Mathisson and Papapetrou was done half-phenomenologically by integrating the multiple $3$-momentum of distributed matter in space and subsequently reducing it to a point and making the result covariant. A small body can be studied by a multipole expansion method: the body is equivalently described by a set of multipole (energy) moments defined along a central line. The cutoff at successive multipole orders defines a hierarchy of elementary multipole particles. The first step is the point particle (or monopole), governed by the geodesic equation of motion. The second one is the dipole ("spinning") particle with which we are concerned here.

The first general covariant derivations were given by Tulczyjew (1959), Taub (1964) and Dixon (1964; here also higher multipoles were included).

Papers:
* [[Physical Applications of a Generalized Clifford Calculus (Papapetrou Equations and Metamorphic Curvature) - W. M. Pezzaglia Jr.|http://arxiv.org/PS_cache/gr-qc/pdf/9710/9710027v1.pdf]] [[pct. 20|http://scholar.google.de/scholar?cites=17097194775030158597&hl=de&as_sdt=2000]]
* [[Particles as Wilson Lines of the Gravitational Field - L. Freidel, J. Kowalski–Glikman, A. Starodubtsev|http://arxiv.org/PS_cache/gr-qc/pdf/0607/0607014v2.pdf]] [[pct. 13|http://scholar.google.de/scholar?cites=13441903653276481488&hl=de&as_sdt=2000]]
* [[Spinning Test Particles in a Kerr Field - I - O. Semerák|http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1999MNRAS.308..863S&defaultprint=YES&filetype=.pdf]] [[local|papers/spinningTestParticleI.pdf]]  [[pct. 10|http://scholar.google.de/scholar?cites=13776176730641717923&hl=de&as_sdt=2000]] prl. 8
* [[Relativistic Motion of Spinning Particles in a Gravitational Field - C. Chicone, B. Mashhoon, B. Punsly|http://arxiv.org/PS_cache/gr-qc/pdf/0504/0504146v2.pdf]] [[pct. 7|http://scholar.google.de/scholar?cites=13830973532948109739&hl=de&as_sdt=2000]]
* [[Mathisson-Papapetrou Equations in Metric and Gauge Theories of Gravity in a Lagrangian Formulation - M. Leclerc|http://arxiv.org/PS_cache/gr-qc/pdf/0505/0505021v2.pdf]] [[pct. 5|http://scholar.google.de/scholar?cites=4973884521331268498&hl=de&as_sdt=2000]]
* [[Polydimensional Supersymmetric Principles - W. M. Pezzaglia Jr.|http://arxiv.org/PS_cache/gr-qc/pdf/9909/9909071v1.pdf]] [[pct. 4|http://scholar.google.de/scholar?cites=9266058051844227761&hl=de&as_sdt=2000]]
* [[The Plane Trajectories of Spin Particles in the Schwarzschild Field - K. Svirskas, K. Pyragas, A. Lozdiene|http://adsabs.harvard.edu/full/1988Ap&SS.149...39S  ]] [[local|papers/nph-iarticle_query.pdf]] [[pct. 2|http://scholar.google.de/scholar?cites=10111546496191620081&hl=de&as_sdt=2000]]
* [[Dirac Equations in Curved Space-Time versus Papapetrou Spinning Particles - F. Cianfrani, G. Montani|http://arxiv.org/PS_cache/arxiv/pdf/0810/0810.0447v1.pdf]] [[pct. 1|http://scholar.google.de/scholar?cites=2840262814515515661&hl=de]]
* [[Nongeodesic Motion of Spinless Particles in the Teleparallel Gravitational Wave Background - L. C. Garcia de Andrade|http://arxiv.org/PS_cache/gr-qc/pdf/0205/0205120v1.pdf]] [[pct. 1|http://scholar.google.de/scholar?cites=15631791624518052181&hl=de&as_sdt=2000]]
* [[On the Coupling Between Spinning Particles and Cosmological Gravitational Waves - I. Millillo, M. Lattanzi, G. Montani|http://arxiv.org/PS_cache/arxiv/pdf/0804/0804.0572v1.pdf]] pct. 0
* [[Deriving Mathisson - Papapetrou Equations from Relativistic Pseudomechanics - R. R. Lompay|http://arxiv.org/PS_cache/gr-qc/pdf/0503/0503054v1.pdf]] pct. 0
* [[Classical and Quantum Spins in Curved Spacetimes - A. J. Silenko|http://arxiv.org/PS_cache/arxiv/pdf/0802/0802.4443v1.pdf]] pct. 0
* [[Quasi-Maxwell Interpretation of the Spin-Curvature Coupling - Jose Natario|http://arxiv.org/PS_cache/gr-qc/pdf/0701/0701067v4.pdf]] pct. 0
* [[Mathisson’ Spinning Electron : Noncommutative Mechanics & Exotic Galilean Symmetry, 66 Years Ago - P. A. Horvathy|http://arxiv.org/PS_cache/hep-th/pdf/0303/0303099v4.pdf]] pct. 0
* [[The Papapetrou Equations and Supplementary Conditions - O. B. Karpov|http://arxiv.org/PS_cache/gr-qc/pdf/0406/0406002v2.pdf]] pct. 0
* [[Rotation and Spin in Physics - R. F. O’Connell|http://www.phys.lsu.edu/faculty/oconnell/PDFfiles/308.%20Rotation%20and%20Spin%20in%20Physics.pdf]] pct. 0
* [[Mathisson's New Mechanics: Its Aims and Realisation - W. G. Dixon|http://th-www.if.uj.edu.pl/acta/sup1/pdf/s1p0027.pdf]] pct. 0

Presentations:
* [[The Papapetrou Equation Derived as a Geodesic in a Non-holonomic Clifford Manifold - W. M. Pezzaglia Jr.|http://www.clifford.org/wpezzag/talk/1998oregon/1998oregon.pdf]] [[local|presentations/1998oregon.pdf]]
* [[Motion in Brane World Models: The Bazanski Approach - M.E.Kahil|http://www.pascos07.org/programme/talks/Kahil.pdf]] [[local|presentations/Kahil.pdf]] - With a nice compilation of the relevant formula. trl. 7

According to Julian Schwinger [1] the appropriate algebra for a quantum measurement is constrained as follows:

"We define the addition of such symbols to signify less specific selective measurements that produce a subensemble associated with any of the values in the summation, none of these being distinguished by the measurement. The multiplication of the measurement symbols represents the successive performance of measurements (read from right to left). It follows from the physical meaning of these operations that __addition is commutative and associative__, while __multiplication is associative__."

Furthermore he states:  
"But a probability is a real, nonnegative number. Hence __we shall impose an admissible restriction__ on the numbers appearing in the measurement algebra, by requiring that $\langle a'| b' \rangle$ and $\langle b' |a' \rangle$ form a pair of __complex__ conjugate __numbers__". 

This is not in general accordance with the opinion of others. 
Pascual Jordan and John von Neumann for example consider weaker algebraic constraints [2], leading to [[Jordan algebras|Jordan Algebra]] as the appropriate algebras for quantum measurements.
J. v. Neumann states: 
"Addition ($a+b$) is __commutative__ and __associative__. Jordan pointed out that a "quasi"-multiplication $a \circ b$ can be defined. ... $a \circ b$ is obviously __commutative__, but [[not necessarily associative|Nonassociative Algebra]]". 
He proceeds:
"On the other hand, an algebraic discussion will be scarcely possible, if the [[distributive law|Nondistributive Algebra]] does not hold for $a \circ b$. ...
We require distributivity merely on the basis of its truth in the present system of quantum mechanics, and its algebraic rôle in connection with the distributive law. It seems to be one of the essential features of quantum theory, although __its true (phenomenological) meaning is obscure__".

Papers:
* [[[1] The Algebra of Microscopic Measurement - J. Schwinger|http://www.ncbi.nlm.nih.gov/pmc/articles/PMC222753/]] [[local|papers/pnas00197-0092.pdf]] [[pct. 18|http://scholar.google.de/scholar?cites=9329205194451544274&hl=de&as_sdt=2000]]
* [[[2] On an Algebraic Generalization of the Quantum Mechanical Formalism. I. (1936) - J. v. Neumann|http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN477674380_0043&DMDID=DMDLOG_0045]] [[local|papers/AlgebraicGeneralisationOfQMI.pdf]] [[pct. 5|http://scholar.google.de/scholar?cites=8458537790388091765&as_sdt=2005&sciodt=2000&hl=de]]
* [[Über eine Klasse Nichtassoziativer Hyperkomplexer Algebren - P. Jordan|http://gdz.sub.uni-goettingen.de/dms/load/img/?IDDOC=64263]] pct. 0

Abstracts: 
* [[Über die Multiplikation Quantenmechanischer Größen - P. Jordan|http://www.springerlink.com/content/v0th72087k872828/]] [[pct. 32|http://scholar.google.de/scholar?cites=9336359917466676733&as_sdt=2005&sciodt=2000&hl=de]]

{{center{[img(527px+, )[images/spacetimes.jpg]]}}}
Presentations:
* [[Equations of Motion in the Gauge Gravity Models|http://www.fuw.edu.pl/~krp/mathisson/wyklady/Obukhov.pdf]]
* [[Status and Prospects of Non-Riemannian Cosmology - D. Puetzfeld|http://folk.uio.no/dirkpu/files/COSMO04s.pdf]]

An $n$-dimensional ''Metric Affine Space'', commonly denoted $(\mathbb L_n,g)$, is a smooth manifold with independent [[metric|Metric Tensor]] structure and linear [[connection|Connection]] (therefore the wording "metric affine"). In general, such spaces posses nontrivial [[curvature|Curvature]], [[torsion|Torsion]] and [[nonmetricity|Non-Metricity Tensor]].

!!!!Applications
* Liquid crystals,
* dislocated metals,
* lattice defects,
* [[metric affine gravity (MAG)|Metric Affine Gravity]].

!!!!Geometrical interpretations
In $3$-dimensional continuum theory
* __torsion__ is related to __line defects__ ([[dislocations|Dislocation]]) which cause spin moment stress.
* __Non-metricity__ is related to densities of __point defects__ which cause double-stress ("hyperstress") without moment. Just as ordinary stress is the analogue of the (Hilbert) energy-momentum density, hyperstress finds its field theoretical image in the densities of [[hypermomentum|Hypermomentum]] which is the sum of spin-, dilation- and  [[shear|Shear]]-currents. This decomposition corresponds to the Lorentz (rotation) group, dilation (isotropic volume scaling), and anisotropic shear deformation with fixed volume.

Papers: 
* [[Metric-Affine Manifold (2004) - A. Kleyn|http://cdsweb.cern.ch/record/710823/files/ext-2004-014.ps.gz]] [[pct. 1|http://scholar.google.com/scholar?hl=de&lr=&cites=18079762928780859198&um=1&ie=UTF-8&sa=X&ei=aGRQTMvUMNiUOMWfxa4B&ved=0CDoQzgIwBA]]

A [[connection|Connection]] is said to be ''Metric Compatible'' if the covariant derivative of the metric vanishes, i.e.
\[
D_\rho g_{\mu\nu} (\mb x) = g_{\mu\nu;\rho}(\mb x) = 0
\]
If this condition is satisfied for every point $\mb x$ in a manifold, this fact is also expressed as: "the manifold possesses ''Metricity''" or the "''Metric Postulate'' (or ''Riemann Constraint'') is satisfied".
Furthermore the relation above also goes under the name ''Ricci Lemma''.

For a ''metric compatible'' manifold the [[nonmetricity tensor|Non-Metricity Tensor]] vanishes
\[
-Q_{\mu\nu\sigma} = D_\mu g_{\nu\sigma}(x) =  0
\]
A geometrical interpretation of metricity of a manifold is, that [[parallel transport|Parallel Transport]] conserves lengths. 

Metric compatibility does not imply that the [[torsion tensor|Torsion]] is zero.

!!!!Some thoughts 
* Giving up metricity means giving up the "rigidity" of the light cone, something we are so much used to from GR. Exceptions are maps from the light cone to itself ([[conformal maps|Conformal Group]]), i.e. rescalings of the light cone.
* As the light cone is related to massless gauge bosons, one is tempted to speculate if "deformations" of the light cone could offer a mass generating mechnism (maybe an alternative to the [[Higgs mechanism|Higgs Mechanism]]). 
* Non-metricity introduces non-linearities, which might be interesting at small scales close to the Planck scale. If this is required for a field theory to be [[renormalizable|Renormalization]], trying to quantize classical [[GR|General Relativity]] which is based on the metricity condition (even [[EC-theory|Einstein-Cartan Theory]] is so) would be a hopeless endeavour. This might also indicate as to why [[fourth order theories of gravity|Fourth Order Theory]] perform better when it comes to renormalizability.
* Although we do not know how spacetime looks like at small scales, it nevertheless might we worth trying to see if it is possible to come up with a plausible model, exhibiting non-metricity. For such a model to be realistic, it is indispensable that causality strictly holds on scales that have already been probed experimentally (i.e. above a certain scale). One idea (see also [[organic universe|Organic Universe]]) is, that spacetime consists of small subunits, which replicate, leading to the growth of spacetime, i.e. an inflating universe. If this replication process takes place stochastically, locally it can induce inhomogeneities (defects of spacetime). These inhomogeneities can be interpreted as violations of metricity. (Which can be understood as violations of [[microcausality|Microcausality]], an important issue in [[quantum gravity|Quantum Gravity]]). For these fluctuations to be of interest they must be above the Planck scale, as at the Planck scale, assuming that standard quantum mechanics can be applied there, quantum fluctuations render spacetime fuzzy anyway. On the large scale these inhomogeneities should cancel out (in the same way as do atomic and nuclear spins in the case of large, massive objects), leading to a description in terms of general relativity in the limit with a nearly "perfect" light cone (and no net spin, concerning our analogy). If this scenario were true, it could offer an alternative to the [["big desert"|Big Desert]]. 

!!!!Examples
The following [[metric affine spaces|Metric Affine Space]] $(\mathbb L_n,g)$ are metric compatible: 
* $\mathbb U_n$ = [[Einstein-Cartan space|Riemann-Cartan Space]],
* $\mathbb A_n$ = [[Weitzenböck (teleparallel) space|Weitzenböck Connection]],
* $\mathbb V_n$ = [[Riemann space|Riemann Space]],
* $\mathbb M_n$ = [[Minkowski space|Lorentz Group]].
{{center{[img(392px+, )[images/metricity.jpg]]}}}


<html><center><iframe name="content" src="http://www.markus-maute.de/trajectory/advert_60.html" width=51% height=86></iframe></center></html>Papers:
* [[Aspects of Nonmetricity in Gravity Theories - R. F. Sobreiro, V. J. Vasquez Otoya|http://arxiv.org/PS_cache/arxiv/pdf/0711/0711.0020v2.pdf]] [[local|papers/0711.0020v2.pdf]]  pct. 0

If the Arkani\-Hamed\-Dimopoulos\-Dvali model of extra dimensions is true, the [[LHC|Large Hadron Collider]], which operates at energies on the order of $10^3$ \GeV, will be able to produce microscopic black holes.

The notation in the following is kept short and is hence a bit "sloppy" for the sake of getting the key ideas better across. (In particular factors were left out).

Given a spinor $\bs \Psi$ satisfying the [[Dirac equation|Dirac Equation]] $\bs \partial \bs \Psi = m \bs \Psi$, we consider a rotated spinor $\bs \Psi'$, given by
\[
\bs \Psi' = \mb R \bs \Psi
\]
where $\mb R$ is a grade $2$ [[polyvector|Polyvector Space]] ([[rotor|Clifford Geometric Algebra]]).

Assuming the validity of the Leibnitz rule, we get
\[
\bs \partial \bs \Psi' = (\bs \partial \mb R)  \bs \Psi + \mb R (\bs \partial \bs \Psi)
\]
and inserting the Dirac equation
\[
\bs \partial \bs \Psi' = (\bs \partial \mb R) \mb R^{-1} \bs \Psi' +  m \bs \Psi'
\]
I. e. the Dirac equation for the transformed spinor has acquired an additional term, namely $(\bs \partial \mb R) \mb R^{-1} \bs \Psi'$.
The rotor can in general be expressed as
\[
\mb R = \pm \exp (\mb B)
\]
with $\mb B$ a bivector.
Thus for the additional term we get
\[
(\bs \partial \mb R) \mb R^{-1} \bs \Psi' = (\bs \partial \mb B) \bs \Psi'
\]
As $\partial \mb B$ acting on $\bs \Psi'$ must return a spinor (the mass term is a spinor), $\mb R^{-1} \bs \partial \mb R$ has to be a rotor, i.e. a grade $2$ object.

To make the Dirac equation invariant under a spinor rotation, we introduce a covariant derivative $\mb D$ by adding an adequate grade-$2$ term $\bs  \Omega$ to the derivative such that it cancels the additional term from above upon acting on a spinor. I.e.
\begin{eqnarray}
\mb D \equiv \bs \partial + \bs  \Omega
\end{eqnarray}
It is clear that $\mb D$ is multigraded. This should demonstrate a certain superiority of the geometric algebra concept over the conventional approach.

$\mb \Omega$ is constrained in that it has to satisfy a transformation rule, we derive in the following:

Given two covariant derivatives $\mb D$ and $\mb D'$ acting on spinors $\bs \Psi$ and $\bs \Psi'$ that are related by $\bs \Psi' = \mb R \bs \Psi$, one has
\begin{eqnarray}
\mb D' \bs \Psi' = m \bs \Psi' & =& m \mb R \bs \Psi = \mb R (\mb D \bs \Psi) \\
\bs \partial \bs \Psi' + \bs \Omega' \bs \Psi' &=& \mb R (\bs \partial \bs \Psi+ \bs \Omega \bs \Psi) \\
(\bs \partial \mb R) \bs \Psi + \mb R \bs \partial  \bs \Psi + \bs \Omega' \mb R \bs \Psi& = &\mb R \bs \partial \bs \Psi + \mb R \bs \Omega \bs \Psi
\end{eqnarray}
Therefore
\[
\bs \Omega'  = \mb R  \bs \Omega \mb R^{-1} - (\bs \partial \mb R) \mb R^{-1}
\]
which is the well known transformation law for gauge fields.

!!!!!Some remarks
* We have only regarded the spinor for a fixed  spacetime point. I.e. the covariant derivative is based on this consideration.
* As $\bs \Omega$ is a rotor, it must be element of a Lie group. Therefore any gauge field should be representable in terms of Lie algebras, given the assumptions above. Prima facie any Lie group will do. The interesting question that arises is why in fact all the gauge symmetries are restricted to the groups $U(1)$, $SU(2)$ and $SU(3)$.

!!!!Examples
!!!![[Electrodynamics]]
In electrodynamics we have a $U(1)$-symmetry. This can be expressed in terms of the rotator as follows:
\[
\mb R(\mb x) = \exp(\chi (\mb x) \mb B)
\]
with $\chi (\mb x)$ a scalar function (a.k.a [[gauge function|Gauge Transformation]]), depending on the spacetime point $\mb x = (t,\vec x)$ (which we'll suppress henceforward) and $\mb B$ a constant bivector blade ($2$-blade).
Therefore
\[
(\bs \partial \mb R) \mb R^{-1}  = \bs \partial \chi(\mb x) \mb B
\]
This is the term we have to cancel, so we set
\[
\bs \Omega = -\bs \partial \chi(\mb x) \mb B
\]
In component notation this reads
\[
\Omega_\mu = -\partial_\mu \chi(\mb x) \equiv -A_\mu
\]
For the transformation of $A_\mu$ the formula from above gives us
\[
\Omega'_\mu = \Omega_\mu - \partial_\mu \chi
\]
This is the known transformation property of the electromagnetic vector potential $A_\mu$ which justifies the identification of $\Omega_\mu$ with $-A_\mu$.
I.e. a Dirac spinor is only determined up to a phase factor $\chi(\mb x)$ which however is spacetime dependent (i.e. we have local gauge invariance).

Hence for the Dirac equation we get
\[
\mb D \bs \Psi = \bs \partial \bs \Psi - \mb A \bs \Psi  = m \bs \Psi
\]
Adding the electromagnetic field to the Dirac equation in this way is known as ''Minimal Coupling''.


Lectures:
* [[Physical Applications of Geometric Algebra. Lecture 14|http://www.mrao.cam.ac.uk/~clifford/ptIIIcourse/lectures/lect14.ps.gz]]

Reformulate physics and mathematics and recast it in a (coherent and consistent) form, that at least oneself can understand.

Why ?
* A lot of redundancies. Things are rediscovered over and over again and often it is not realised that they are the same. This corresponds to the problem of different representations for the same thing and different "brain wirings" of different authors, prefering different representations.
* One cannot learn mathematics and physics, one has to discover it.
* A lot is copied and pasted by authors, who do not really understand what the true meaning of the content is. This is even worse when things are copied over and over again. Errors accumulate and propagate. (Therefore it is no wonder, that original works are often way more joyful to read).
* A lot is too abstract, too far from an application. A priorisation of the relevance of mathematical structures in respect to their applicability in physics is needed. Classification of structures in mathematics is essential but one deals a lot with practically uninteresting structures (one can easily get lost).
* Reading and writing (or WIKIing these days) is better what concerns understanding than just reading.

The ''Multiverse'', believe it or not ...

Videos:
* [[Alex Vilenkin interview about the Multiverse part 1|http://www.youtube.com/watch?v=3t46oqx08VE&feature=related]] [[part 2|http://www.youtube.com/watch?v=TPipTsM33lA&feature=related]] [[part 3|http://www.youtube.com/watch?v=7Te_KIPZjSM&feature=response_watch]]

<html><center><img src="images/math_clock.jpg" style="width: 280px; "/></center></html>

Papers:
* [[My Favorite Integer Sequences N. J. A. Sloane|http://www.research.att.com/~njas/doc/sg.pdf]] [[pct. 9|http://scholar.google.com/scholar?hl=de&lr=&cites=15076201712377063024&um=1&ie=UTF-8&ei=2A2lSpX8B5m4sgaHtoXTBA&sa=X&oi=science_links&resnum=1&ct=sl-citedby]]

Links:
* [[Cognitive Theoretic Model of the Universe (CMTU) - C. Langan|http://www.megafoundation.org/CTMU/]] - How a man with an alledged I.Q. of around 200 "sees" the universe.
* [[Ben Goertzel Essays|http://www.goertzel.org/essays.htm]]
* [[University of Toronto Mathematics Network - Question Corner and Discussion Area|http://www.math.toronto.edu/mathnet/questionCorner/qc.ps]]
* [[Articles by S. M. Phillips|http://www.smphillips.8m.com/html/articles.html]] - Interesting stuff, but to be taken with a grain of salt. ("Octonion Algebra is isomorphic to E8 Lie Algebra").
* [[Bitmaps for a Digital Theory of Everything - R. Aschheim|http://www.cs.indiana.edu/~dgerman/2008midwestNKSconference/rasch.pdf]]
* [[Strings and Loops in Event Symmetric Space-Time - P. Gibbs|http://arxiv.org/PS_cache/hep-th/pdf/9407/9407136v1.pdf]]
* [[Rafiki Inc.|http://www.codefun.com/]]
* [[The Cellular Universe website - C. Ranzan|http://www.cellularuniverse.org/]]
* [[Tony Smith's Homepage, 240 Thoughts|http://www.valdostamuseum.org/hamsmith/SWTxt.html]]
* [[Verman University Mathematical Quotations Server|http://math.furman.edu/~mwoodard/mqs/mquot.shtml]]
* [[God does not play Dice|http://www.god-does-not-play-dice.net/]]
* [[Gennady I. Shipov|http://www.shipov.com/]] - Torsion, warp drives and all that ... 
* [[The Orientation Congruent Algebra and the Native Exterior Calculus of Twisted Differential Forms|http://felicity.freeshell.org/math/index.htm#vis-tw-objs]]
* [[viXra.org|http://www.vixra.org/]] - The alternative arXiv.

{{center{//This is a [[Draft]] !//}}}
Henceforward I outline a suggested unifying definition of curvature, called ''$n$-Curvature''. It is an attempt to describe [[curvature|Curvature]] in a very generic sense, i.e. independent of a concrete algebraic realisation. One therefore expects [[structure constants|Structure Constants]] to appear, that, once one decides for a concrete algebra are to be replaced by concrete values. This concept can be regarded as part of the [[P-space|Polyvector Space]] concept where the concrete algebra is "stripped off" in the beginning to do geometry on a more conceptual level. 

One of the main motivations for $n$-curvature is to be able to go beyond [[group manifolds|Lie Group Manifold]] and classical [[Riemannian geometry|Riemann Space]], i.e. to have a formalism available that allows one for the description of [[loop-manifolds|Quasigroup Manifold]] and to do [[nonassociative differential geometry|Nonassociative Differential Geometry]] and to harness the techniques developed to do [[nonassociative physics|Nonassociative Physics]]. 
Certainly one of the consistency checks this framework has to stand is that it reproduces Riemannian geometry in its entirety. Furthermore, as its development is partially guided by [[3-web theory|3-Web]], it should also reproduce results of this theory. Actually $n$-curvature can be seen as a reformulation of [[(hexagonal) three-web theory|Hexagonal 3-Web]] with the aim to better apply it to physics.

!!!!1-curvature
A one-parameter curve $\Phi(\tau)$ has $1$-curvature. This is given by the first derivative $\frac{\Phi(\tau+ d\tau)- \Phi(\tau)}{d\tau} \equiv \bs \partial_\tau \Phi(\tau)$. If we are on a $n$-dimensional manifold we have $n$-linearly independent curves, we parametrize with $x^\mu$ and hence $n$ independent $1$-curvatures given by $\bs \partial_{x^\mu} \Phi(\mb x)$. We define $\bs \partial_{x^\mu} \equiv  \bs  \partial_\mu \equiv \mb e_\mu(\mb x)$. The $\bs \partial_{x^\mu}$, or equivalently $ \mb e_\mu(\mb x)$, define our local frame. 

Note that we do not specify a coordinate representation for the $\mb e_\mu$, as we will not need this, i.e. we work coordinate-free. It is important to realise that the index $\mu$ is not a coordinate index, rather it serves for numbering the basis elements. As we are assuming a non-orthogonal frame here, we use Greek indices for doing so.

For the first order change of $\Phi$, i.e. assuming that the manifold has only $1$-curvature, we have the differential 
\begin{eqnarray}
\bs d \Phi(\mb x) & = & dx^1 \frac{\bs \partial \Phi(\mb x)}{\bs \partial x^1} + \ldots + dx^n \frac{\bs \partial \Phi(\mb x)}{\bs \partial x^n}  \\
&=&  (dx^1 \bs \partial_{x^1} + \ldots + dx^n \bs \partial_{x^n}) \Phi(\mb x) \\
&=&  (dx^1 \mb e_1 + \ldots + dx^n \mb e_n) \Phi(\mb x) \\
&\equiv& (\bs d x^1 \partial_1 + \ldots + \bs d x^n \partial_n) \Phi(\mb x)
\end{eqnarray}
and the the operator $\bs d$ is given by
\begin{eqnarray}
\bs d & =& dx^1 \mb e_1 + \ldots + dx^n \mb e_n \\
&\equiv& \bs  d x^1 \partial_1 + \ldots + \bs dx^n \partial_n
\end{eqnarray}
Changing the coordinates according to $\mb x \rightarrow \mb x'$, we get 
\begin{eqnarray}
\bs d &=& dx'^i \frac{\partial x^1}{\partial x'^1} \mb e_1 + \ldots + dx'^i \frac{\partial x^n}{\partial x'^i} \mb e_n \\ 
 &=& dx'^i(\mb J_{\mb x'})_{ij} \mb e_j 
\end{eqnarray}
with $\mb J_{\mb x'}$ the [[Jacobi matrix|Jacobi Matrix]], corresponding to the transformation.

$\bs d \Phi$ is a [[1-form|Differential Form]] and also denoted $\bs \omega$ or in the context of physics $\mb A$, representing a gauge potential (called a gauge $1$-form or $1$-form field).

!!!!!Example
The $S^2$-sphere with radius $r$. We take $r = \Phi (\phi, \theta)$, with the two angles $\phi$, $\theta$, i.e. in every point of the sphere we have two independent $1$-curvatures, having equal absolute value.

!!!![[2-Curvature]]
!!!![[3-Curvature]]
!!!!4-curvature 
We define the grade-$4$ differential by
\begin{eqnarray}
\bs d &= &dx^\mu \mb e_\mu + dx^{\mu\nu} \mb e_\mu \mb e_\nu + dx^{\mu\nu\rho} (\mb e_\mu \mb e_\nu) \mb e_\rho + dx'^{\mu\nu\rho} \mb e_\mu (\mb e_\nu \mb e_\rho)\\
&& dx^{(1)\mu\nu\rho\sigma} ((\mb e_\mu \mb e_\nu) \mb e_\rho)\mb e_\sigma +  dx^{(2)\mu\nu\rho\sigma} \mb e_\mu ((\mb e_\nu \mb e_\rho)\mb e_\sigma) + dx^{(3)\mu\nu\rho\sigma} (\mb e_\mu \mb e_\nu) (\mb e_\rho\mb e_\sigma) + dx^{(4)\mu\nu\rho\sigma} (\mb e_\mu (\mb e_\nu \mb e_\rho))\mb e_\sigma + dx^{(5)}_{\mu\nu\rho\sigma} \mb e_\mu (\mb e_\nu (\mb e_\rho\mb e_\sigma))   
\end{eqnarray}
taking into account the $5$ possible [[association types|Association Type]] of degree $4$. 

!!!!Discrete curvature
One can conceive pushing the concept described even further to the limit as follows: 
If for example one has an algebra with structure constants which only take values of $+1$ and $-1$, one can double its multiplication table and absorb the sign in the extended basis. (An example of such a doubling are the [[quaternions|Quaternion]] and the [[quaternion group|Quaternion Group]]). In this case one "gets rid" of the structure constants. If one interprets the signs as parity, one has a reinterpretation of parity as kind of a $n$-curvature.

Papers:
* [[[1] Physical Applications of a Generalized Clifford Calculus (Papapetrou Equations and Metamorphic Curvature) - W. M. Pezzaglia Jr.|http://arxiv.org/PS_cache/gr-qc/pdf/9710/9710027v1.pdf]] [[local|papers/9710027v1.pdf]] [[pct. 20|http://scholar.google.de/scholar?cites=17097194775030158597&hl=de&as_sdt=2000]] prl. 10
@@display:block;text-align:right;font-size:12pt;font-family:Scripts;{{stretch{[img[My comments ...|images/Riemann.jpg][Comments]]}}}&nbsp;@@

Papers:
* [[Evolution of Networks - S.N. Dorogovtsev, J. F. F. Mendes|http://arxiv.org/PS_cache/cond-mat/pdf/0106/0106144v2.pdf]] {{t1000Cite{ [[pct. 2514|http://scholar.google.de/scholar?cites=7757634102586467395&hl=de]]

A manifold that does not satisfy the [[metricity condition|Metric Compatibility]] can be characterized by a so called ''Non\-Metricity Tensor'' $Q_{\mu\nu\rho}(\mb x)$ which is defined by:
\[
Q_{\mu\nu\rho}(\mb x) \equiv -D_\rho g_{\mu\nu} (\mb x) = -g_{\mu\nu;\rho} (\mb x) 
\]
In case of a $4$-dimensional manifold, $Q_{\mu\nu\rho}$ consists of $40 = 4 \cdot 10$ independent components, with the "$10$" stemming from the [[metric tensor|Metric Tensor]].

Carrying out the differentiation, assuming the validity of the Leibnitz rule, one gets
\begin{eqnarray}
D_\rho g_{\mu\nu} (\mb x) &= & \partial_\rho \langle h^a_\mu (\mb x)\mb e_a(\mb x)|h^b_\nu(\mb x) \mb e_b(\mb x)  \rangle \\
& = & \partial_\rho ( h^a_\mu (\mb x)  h^b_\nu (\mb x))  \langle \mb e_a(\mb x)|\mb e_b(\mb x) \rangle +  \langle h^a_\mu (\mb x) \partial_\rho \mb e_a(\mb x)|\mb e_\nu(\mb x)  \rangle + \langle\mb e_\mu (\mb x)| h^b_\nu (\mb x) \partial_\rho \mb e_b(\mb x) \rangle \\
& = & \partial_\rho \langle h^a_\mu (\mb x) \mb e_a| h^b_\nu (\mb x) \mb e_b  \rangle +  \langle h^a_\mu (\mb x) \Gamma^\sigma_{\rho a} \mb e_\sigma (\mb x)|\mb e_\nu(\mb x)  \rangle + \langle\mb e_\mu (\mb x)| h^b_\nu (\mb x) \Gamma^\sigma_{\rho b}\mb e_\sigma(\mb x) \rangle \\
& = & \partial_\rho g_{\mu\nu} +  \Gamma^\sigma_{\rho \mu}  g_{\sigma\nu}  + \Gamma^\sigma_{\rho \nu} g_{\mu\sigma} 
\end{eqnarray}
Due to the symmetry of the metric tensor, $Q_{\mu\nu\rho}$ is symmetric in the first two indices.

It is a remarkable fact that the [[covariant derivative|Covariant Derivative]] of the [[gamma matrices|Gamma Matrices]] does not vanish identically in a spacetime with non-metricity. This is a possible motivation for the introduction of the [[principle of local automorphism invariance|Principle of Local Automorphism Invariance]] (where, for example, nonmetricity can be expressed in terms of so called "drehbein fields").

!!!!Decomposition 
The nonmetricity tensor $Q_{\mu\nu\rho}(\mb x)$ of an $n$-dimensional manifold can be split up into its trace and trace-free parts according to 
\[
Q_{\mu\nu\rho}(\mb x) = Q_\rho(\mb x) g_{\mu\nu}(\mb x) +  Q\! \!\!\! / {}_{\mu\nu\rho}(\mb x) 
\]
with $Q_\rho = \frac 1n Q_\sigma {}^\sigma {}_\rho(\mb x)$ the ''Weyl\-Covector'' and $Q\! \!\!\! / {}_\sigma{}^\sigma {}_\rho(\mb x) = 0$.
The trace free part is a measure for the violation of local [[Lorentz invariance|Lorentz Violation]].

!!!!Special cases 
* $Q\! \!\!\! / {}_{\mu\nu\rho}(\mb x) = 0$: In this case one is dealing with a [[Weyl-Cartan geometry|Weyl Space]]. In this type of geometry angles are preserved under parallel transport whereas lengths are not. The connection of such a space (also referred to as [[Weyl connection|Weyl Connection]]) is called [[conformal|Conformal Group]]. It has the characteristic that it multiplies, under parallel transport, all the scalar products by the same factor. Consequently, null vectors remain null vectors under parallel transport. I.e. the light cone and causality are not "touched upon".  
* $Q_\rho(\mb x) g_{\mu\nu}(\mb x)=0$: If non-metricity is trace free, volumes are preserved under parallel transport. A transformation that preserves the volume is called a ''Shear''.
* $Q_{\mu\nu\rho}(\mb x)=0$: The space is metric compatible. Transformations are "rotations", preserving volumes and all scalar products.

!!!!Physical interpretation
In a general metric [[affine spacetime|Metric Affine Space]], nonmetricity appears as a field strength, side by side with curvature and torsion. In matter, the shear and dilation currents couple to nonmetricity, and they are its sources.
Within the framework of [[metric affine gravity|Metric Affine Gravity]], nonmetricity is predicted to be observable as pulsations (mass quadrupole excitations) of test matter.

!!!!Physical consequences
Nonmetricity is a __measure for the [[violation of local Lorentz invariance|Lorentz Violation]]__. In fact, for generic nonmetricity, the very concept of a light cone is lost. That is to say that there are no local Lorentz frames, i.e. there exists no conventional local limit to the flat spacetime of [[special relativity|Special Relativity]]. Therefore the classical [[equivalence principle|Equivalence Principles]] cannot be recovered.

!!!!Potential applications
Nonmetricity might be relevant 
* on very small scales of spacetime, close to the [[Planck length|Planck Units]],
* in the the early universe, during [[inflation|Inflation]], within Planck times from the "seeding" vacuum fluctuation "event". Presumably, it is then through a spontaneous breakdown of the local general affine symmetry below Planck energies, down to [[Poincaré invariance|Poincaré Transformation]], that [[special relativity|Special Relativity]] and the Riemannian metricity condition set in. Nonmetricity could provide the scalar field ([[dilaton field|Dilaton]]) necessary for inflation.

!!!!A personal remark 
Nonmetricity is closely related to the constancy of the speed of light. If one considers the full range of length scales from the Planck lenght all the way up to the size of the visible universe, only for the upper range it is save to say that the speed of light is constant, as this is covered by lots of experiments in astronomy, atomic and nuclear physics etc. Yet below length scales of what we can probe by the shortest wavelengths available today there are, say $20$ to $30$ orders, where there seems to be no reason to believe that the speed of light is constant. (It should be remembered in this context, that the outcome of the famous Michelson Morley experiment came as a surprise and was nothing one could have derived based on pure logic). The adherence to the metricity condition might explain why extrapolations of GR in the limit of infinitesimal small scales lead to singularities, which is absurd from a physics point of view. Therefore nonmetricity "kicking in" at a certain length scale could come to rescue. This could shed a totally new light on the big bang and black holes.


<html><center><iframe name="content" src="http://www.markus-maute.de/trajectory/advert_90.html" width=71% height=110></iframe></center></html>Papers:
* [[Torsion and Nonmetricity in Scalar-tensor Theories of Gravity (1993) - J. Berthias|http://arxiv.org/PS_cache/gr-qc/pdf/9303/9303013v1.pdf]] [[pct. 13|http://scholar.google.de/scholar?cites=4201920672165426479&as_sdt=2005&sciodt=2000&hl=de]]
* [[Gravity as Nonmetricity - A. Poltorak|http://www.poltorak.com/files/gravity-as-nonmetricity-web.pdf]] [[pct. 2|http://scholar.google.de/scholar?cites=11563956491227275167&as_sdt=2005&sciodt=2000&hl=de]]
* [[Non Metric Mass (1998) - M. D. Roberts|http://cdsweb.cern.ch/record/375402/files/9812091.pdf]]  [[pct. 2|http://scholar.google.com/scholar?hl=de&lr=&cites=10103920091255961112&um=1&ie=UTF-8&ei=zJRVTOLqHeeiOPrUwZ4O&sa=X&oi=science_links&ct=sl-citedby&resnum=2&ved=0CCIQzgIwAQ]]

Theses:
* [[The Quantisation of Spin Gauge Theory (2002) - A. Geitner|http://kops.ub.uni-konstanz.de/volltexte/2002/875/pdf/geitner.pdf]] [[local|theses/geitner.pdf]] 

Lectures:
* [[Notes on Differential Geometry - M. Visser|http://www.mcs.vuw.ac.nz/courses/MATH464/2008T1/Lecture-Notes/notes.pdf]] [[local|lectures/notes.pdf]] lrl. 9 

Google books: 
* [[Geometry, Spinors, and Applications - D. J. Hurley, M. A. Vandyck|http://books.google.com/books?id=dyxLHCe8VXsC&printsec=frontcover&dq=spinors+hurley&source=bl&ots=xJXb-Hg6UZ&sig=su0Y9RWcCebGQYKAI0Nv-aoxByk&hl=de&ei=_rxRTL_CHYePOPjhvfgE&sa=X&oi=book_result&ct=result&resnum=1&ved=0CCAQ6AEwAA#v=onepage&q&f=false]] [[bct. 14|http://scholar.google.de/scholar?cites=8181905652266614362&as_sdt=2005&sciodt=2000&hl=de]] TRD

> ... the usual assumption in noncommutative geometry, including Alain Connes' version, that differential forms should be associative, appears to be too strong.
> - S. Majid [1] -

In ordinary quantum mechanics the Schrödinger and Heisenberg pictures are equivalent, which is not true in nonassociative quantum mechanics. Indeed, whilst the concept of a [[Hilbert space|Hilbert Space]] fails for nonassociative algebras, the Heisenberg approach still can be realized. Hence, constructing a theory of nonassociative quantum mechanics one must give up the conventional Hilbert space approach and look for a generalization based on the Heisenberg description, maybe only in terms of a density matrix.

Papers:
* [[Tensor Model and Dynamical Generation of Commutative Nonassociative Fuzzy Spaces - N. Sasakura|http://arxiv.org/PS_cache/hep-th/pdf/0606/0606066v5.pdf]] [[pct. 6|http://scholar.google.de/scholar?cites=6507915836773940075&hl=de]]
* [[Nonassociative Algebras and Nonperturbative Field Theory for Hierarchical Models - A. Pordt, C. Wieczerkowski|http://arxiv.org/PS_cache/hep-lat/pdf/9406/9406005v1.pdf]] [[pct. 2|http://scholar.google.de/scholar?cites=8302213333222678112&as_sdt=2005&sciodt=2000&hl=de]]
* [[The Poincaré Group in a Semisemidirect Product with a Non-associative Algebra with Representations that include Particles and Quarks - F. E. Schroeck, Jr.|http://www.math.du.edu/data/preprints/m0816.pdf]] [[pct. 1|http://scholar.google.de/scholar?cites=3242823591165102418&hl=de]]
!!!!!Gauge Theory
* [[[1] Gauge Theory on Nonassociative Spaces - S. Majid|http://arxiv.org/PS_cache/math/pdf/0506/0506453v1.pdf]] [[local|papers/0506453v1.pdf]] [[pct. 9|http://scholar.google.de/scholar?cites=1591564176579341259&hl=de]] - prl. 9
* [[Analytic Loops and Gauge Fields - E.K. Loginov|http://arxiv.org/PS_cache/hep-th/pdf/0109/0109206v1.pdf]] [[pct. 3|http://scholar.google.de/scholar?cites=12535028054047114180&hl=de]]
!!!!! QM
* [[Geometry of the Three-qubit State, Entanglement and Division Algebras - B. A. Bernevig, H.-D. Chen|https://netfiles.uiuc.edu/hdchen/www/pdf/JPA8325.pdf]] [[pct. 30|http://scholar.google.de/scholar?cites=14257895914684491581&hl=de]]
* [[A Non-associative Quantum Mechanics - V. Dzhunushaliev|http://arxiv.org/PS_cache/hep-th/pdf/0502/0502216v5.pdf]] [[pct. 6|http://scholar.google.de/scholar?cites=5117351471756551342&hl=de]]
* [[Emergent Time from Non-Associative Quantum Theory - J. Köplinger, V. Dzhunushaliev, M. Gogberashvili|http://www.fqxi.org/data/essay-contest-files/Koeplinger_JKVDMGFQXiTime20.pdf?phpMyAdmin=0c371ccdae9b5ff3071bae814fb4f9e9]] - pct. 0
!!!!! [[QFT|Nonassociative Quantum Field Theory]]
!!!!!QCD
* [[Quark State Confinement as a Consequence of the Extension of the Bose-Fermi Recoupling to SU(3) Colour - W. P. Joyce|http://arxiv.org/PS_cache/hep-th/pdf/0306/0306256v1.pdf]] [[pct. 3|http://scholar.google.de/scholar?cites=17228381713554108384&hl=de]]
* [[A New Approach to Chromodynamics - M. J. Hayashi|http://www.slac.stanford.edu/cgi-wrap/getdoc/slac-pub-1936.pdf]] pct. 0

Links:
* [[Octonionic physics|Octonionic Physics]]

Presentations:
* [[Nonassociative Quantum Field Theory - M. Liebmann|http://physik.uni-graz.at/itp/doktoratskolleg/talks/Liebmann20080109.pdf]]

Google Books:
* [[Time, Quantum, and Information (chapter: "Ur Theory and Space-Time Structure" - D. R. Finkelstein) - L. Castell, O. Ischebeck|http://books.google.de/books?hl=de&lr=&id=0xvBwotTuTEC&oi=fnd&pg=PA397&dq=%22international+journal+of+theoretical+Physics%22+%22cayley+Dickson+algebras%22&ots=DvkhB_FIoT&sig=3p7uoxE56byy5r5RfzWaoLKXw6I]] [[local|google_books/TimeQuantumAndInformation.pdf]] pct. 0

See also: [[quantum field theory|Quantum Field Theory]].

Papers:
* [[One-loop Unitarity of Scalar Field Theories on Poincaré Invariant Commutative Nonassociative Spacetimes - Y. Sasai, N. Sasakura|http://arxiv.org/PS_cache/hep-th/pdf/0604/0604194v2.pdf]] [[pct. 8|http://scholar.google.de/scholar?cites=15008719768523439994&hl=de]]
* [[Particle Scattering in Nonassociative Quantum Field Theory - V. D. Dzhunushaliev|http://arxiv.org/PS_cache/hep-th/pdf/9606/9606125v1.pdf]] [[pct. 5|http://scholar.google.de/scholar?cites=2693808319832489859&hl=de]]
* [[Nonperturbative Operator Quantization of Strongly Nonlinear Fields - V. Dzhunushaliev|http://arxiv.org/PS_cache/hep-th/pdf/0103/0103172v4.pdf]] [[pct. 5|http://scholar.google.de/scholar?cites=4059072563260696001&hl=de]]

In ''Noncommutative [[Quantum Field Theory]]'' it is assumed that the spacetime coordinates do not commute, i.e.
\[
[\hat x^{\mu}, \hat x^{\nu}] \equiv i \Theta^{\mu\nu}  \ne 0
\]
This relation adds to those given by the [[Heisenberg algebra|Heisenberg Algebra]]. 
One gets a further uncertainty relation, given by
\[
\Delta x^{\mu} \Delta x^{\nu}  \ge \frac 12 |\Theta^{\mu\nu}|
\]
which implies a minimal scale of spacetime. In other words, spacetime if "fuzzy".

In a noncommutative spacetime with canonical commutation relations between the coordinates, [[Lorentz symmetry|Lorentz Transformation]] is violated and field theories constructed on such space-times have instead the so-called twisted Poincaré invariance.

''Nonsymmetric Gravity Theory (NGT)'' allows for antisymmetric metrics. Nonsymmetric metrics were already studied by Einstein & Straus (1946) in their search for a unified theory for gravity and electromagnetism, yet this unification was not successful. 

Links: 
* [[WIKIPEDIA - Nonsymmetric Gravitational Theory|http://en.wikipedia.org/wiki/Nonsymmetric_gravitational_theory]]

Papers:
* [[Fermions and Octonions - P. Goddard, W. Nahm, D. I. Olive, H. Ruegg, A. Schwimmer|http://www.projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.cmp/1104159976]] [[pct. 32|http://scholar.google.de/scholar?cites=10656084780159155420&hl=de]]
* [[Octonionic Quark Confinement - H. Ruegg|http://th-www.if.uj.edu.pl/acta/vol09/pdf/v09p1037.pdf]] [[pct. 2|http://scholar.google.com/scholar?hl=de&lr=&cites=16823927464145052229&um=1&ie=UTF-8&ei=NCOWSvGxO8ORsAag9aCvDQ&sa=X&oi=science_links&resnum=1&ct=sl-citedby]]
- Gravitation
* [[Geometrical Properties of an Internal Local Octonionic Space in Curved Space Time - S. Marques, C. G. Oliveira|http://lss.fnal.gov/archive/1986/pub/Pub-86-060-A.pdf]] [[local|papers/Pub-86-060-A.pdf]] [[pct. 1|http://scholar.google.de/scholar?cites=11427988464251865569&hl=de]]
* [[Natural Octonionic Generalization of General Relativity - J. Fredsted|http://arxiv.org/PS_cache/arxiv/pdf/0707/0707.0554v1.pdf]] pct. 0

Google Books:
* [[On the Role of Division, Jordan and Related Algebras in Particle Physics - F. Gursey, C. Tze, F. Gursey|http://books.google.com/books?id=sf_vpCBd65oC&printsec=frontcover&dq=octonion+%22G2+transformation%22&hl=de&source=gbs_summary_s&cad=0#PPP1,M1]]

>Die Zeiträume halbieren sich jeweils ...Die Reihe konvergiert gegen 2040, obwohl schon zwei Jahrzehnte vorher erschwingliche Maschinen schneller rechnen werden als das menschliche Hirn."
> - Jürgen Schmidhuber

''Omega Point'' is a term invented by the French Jesuit Pierre Teilhard de Chardin to describe a maximum level of complexity and consciousness towards which the universe appears to be evolving. Teilhard's term recurs in both intellectual works and popular culture, e.g. the cosmological theory proposed by the mathematical physicist Frank Tipler, or theories about an approaching [[technological singularity|http://en.wikipedia.org/wiki/Technological_singularity]] (e.g. by [[Ray Kurzweil]]).

Links:
* [[WIKIPEDIA - Frank J. Tipler|http://en.wikipedia.org/wiki/Frank_J._Tipler]]

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''An Essay ...''
<<player id=1 windows audio/organic_universe.mp3 220 45>>
The ''Organic Universe'' is a somewhat crazy idea  I like to put forward about the origin and evolution of the universe:

Let's start with the continuum which we assume is either stochastically changing (then the size doesn't matter) or practically infinite such that we can "find" any (at least not too large) structure in some "corner" of the continuum. What we need is a "seed" that can trigger and evolve into a new universe. This seed, which is assumed to be the fundamental building block of space-time and matter is conjectured to be describable as a self-replicating Von Neumann algorithm. We will only regard it on an information theoretical level, as we do not know its concrete realization, not having a theory of quantum gravity. We will call it a fundamental cell. (One can think of it as being an object with size of the order of the Planck scale). Therefore we assume that space-time is not continuous, rather it is discrete in nature. (This makes sense because it codes information. For example, when considering cosmic inflation, space-time "knows" when to inflate or not and what the current rate of inflation must be. This information is assumed to be coded in the fundamental space-time cell).

<html><center><img src="images/nematic.jpg" style="width: 314px; "/></center></html>
So we have a self-replicating cell, emdedded in a random and noisy environment. This elementary cell must be robust, otherwise mutations will easily render it incapable of replicating and therefore no large scale expansion will be possible. Therefore considering a fundamental cell that initiated our universe, it could be the result of a long evolutionary process, or we sit in a universe with an initial cell that just happened to have the right properties (antropological scenario). Anyway, if we assume that the self-replicating mechanism is not too complex, effective and minimalistic and hence not too large, the second argument also seems to be plausible. (In this context it should be mentioned that the simple rule of Conway's "life" cellular automaton is able to give rise to complex ordered patterns out of an initially disordered state, or primordial soup.)
<html><center><img src="images/inflation.gif" style="width: 378px; "/></center></html>
Now we have a robustly expanding universe. Its exponential inflation is due to the fact that the underlying process is a doubling of its fundamental cells. We will call it ''Organic Cosmic Inflation'' alluding to analogies in biology.
Two such analogies are a bacteria colony and a yeast pastry. Such systems might furthermore help us, by looking at their properties, to find analogous properties in case of our "organic" universe and to interpret them accordingly (e.g. the dymanics of its inflation or its large scale structure). In the case of biological systems the elementary cells are biological cells and any one codes the complete information of the whole system. In their case we know the concrete (physical) realization: The information is coded in the genes, the read out of the information is done by polymerases and the final information processing and messaging is carried out by biochemicals (proteins with functional units, etc.).
<html><center><img src="images/bacteriagrowth.gif" style="width: 450px; "/></center></html>
These systems can be regarded as mutation resistant self-replicating machines. (It seems that bacteria are Von Neumann automata, whereas viruses are not, as their replication is dependent on host cells). The exponential inflation of a bulk of biological cells comes to an end either due to a lack of resources, due to ageing of the cells (i.e. after a certain average number of doublings their genetic algorithms are mutated that much that the replicating mechanisms do not work any more) or because the genetic program says "stop" after a certain number of doublings, which is the case for biological organisms, reaching their grown out state.
The assertion is that the "aging" scenario could apply to space-time. In case of the first scenario it is not so clear how to interpret it in the context of our organic universe model. The third scenario seems only to be likely if our universe is a descendant of a progenitor universe because it would require a longer algorithm in its initial space-time cell. If space-time really ages, then on the long run it would erode back into the continuum (like an ice crystal which melts "back" into a liquid).

<html><center><img src="images/large_scale.jpg" style="width: 392px; "/></center></html>

Addendum:
A quite outlandish idea is that the seed of our universe was created (designed) in a foregoing universe (for whatever reason) by intelligent beings therein. The biological analogue here would be genetic manipulation of a germ cell, such that the resulting organism has certain desired properties. On the other hand it is conceivable that one day, provided we understand the elementary building blocks of space-time and are able to manipulate them, we could engineer and trigger new universes ourselves. (This however appears to be quite dangerous. Imagine initiating inflation of space-time "in front of our house").

© by Markus Maute, 2009


After having written this essay, I found the following statement in \NewScientist, March 6-12, 2010:

"Quantum mechanics tells us ... that, sooner or later, any given universe will decay spontaneously into another one with lower energy. Indeed, most cosmologists envisage our big bang as precisely such an event, during which the vacuum we live in emerged from a higher-energy vacuum that constituted a universe before ours."

This fact is new to me and it seems that the idea of our universe having a predecessor is not so oddish.
@@display:block;text-align:right;[img[My comments ...|images/comment.gif][Comments]]&nbsp;@@

A ''P\-Brane'' is a $p$-dimensional object propagating in a $D$-dimensional space-time ($D \ge p+1$) and sweeping out a $p+1$-dimensional world-volume. The action of a classical $p$-brane is not unique. One possible description is by means of the [[Dirac-Nambu-Goto action|Dirac-Nambu-Goto Action]] which represents the world volume of the membrane trajectory in space-time. Another description is based on the [[Howe-Tucker-Polyakov action|Howe-Tucker-Polyakov Action]].
The two actions are equivalent on the classical level in the sense that one gets equal equations of motion. The Dirac\-Nambu\-Goto action is an "extrinsic" description of the brane as the metric is induced by the space it is embedded in. For the Howe\-Tucker\-Polyakov action one has an "intrinsic" description as the metric does not depend on an induced metric.
!!!!Examples
$0$-brane: point-particle, worldvolume = trajectory
$1$-brane: string, worldvolume = worldsheet

Papers:
* [[Field Theory of Geometric P-Branes - Choon-Lin Ho|http://ccdb4fs.kek.jp/cgi-bin/img/allpdf?198808158]] [[pct. 4|http://scholar.google.de/scholar?cites=12474317056622001280&hl=de]]

Theses:
* [[Particle Dynamics of Branes - A. R. Ploegh|http://dissertations.ub.rug.nl/FILES/faculties/science/2008/a.r.ploegh/13-thesis.pdf]] [[local|theses/13-thesis.pdf]]
* [[Branestorming - U.Gran|http://fy.chalmers.se/~ulfgran/lic.ps.gz]] [[local|theses/lic.ps]]
* [[Field Theories of Geometric Strings, Membranes and P-Branes - Choon-Lin Ho|http://ccdb4fs.kek.jp/cgi-bin/img/allpdf?198910120]] [[local|theses/FieldTheories.pdf]]

Links:
* [[PARI/GP Website|http://pari.math.u-bordeaux.fr/]]

According to the ''Palatini Principle (or first order formalism)'', [[metric|Metric Tensor]] and [[connection|Connection]] are regarded as independent. For a Lagrangian this means that the variation is carried out for both of them independently.
For the [[Einstein Hilbert action|Einstein-Hilbert Action]] this leads - besides the [[Einstein equations|Einstein Field Equations]] when varying in respect to the [[metric|Metric Tensor]] - to a second set of equations, namely
\[
\partial_\lambda g_{\mu\nu} - \Gamma^\eta_{\lambda\mu} g_{\eta\nu} - \Gamma^\eta_{\lambda\nu} g_{\mu\eta} = 0
\]
which is the condition of [[metric compatibility|Metric Compatibility]]. These equations besides the Einstein equations can be regarded as additional field equations describing the gravitational field.

In general the Palatini principle allows the geometry to have an affine structure, that is that the associated space-time is a [[metric affine space|Metric Affine Space]]. In four dimensions, i.e. for a $(\mathbb L_4, g)$-space, there are $10$ equations for the metric tensor and $64$ for the connection.

The advantage of deriving the field equations using the Palatini method is that the geometry of space-time is less restrictive. In general the Palatini variation allows for the existence of [[torsion|Torsion]] and [[non-metricity|Non-Metricity Tensor]].

Although the principle carries the name of Palatini its introduction is usually ascribed to Einstein.

Papers:
* [[The Universality of Einstein Equations (1993) - M. Ferraris, M. Francaviglia, I. Volovich|http://arxiv.org/PS_cache/gr-qc/pdf/9303/9303007v2.pdf]] [[pct. 72|http://scholar.google.de/scholar?cites=9818932703691021574&hl=de&as_sdt=2000]]
*[[On the so-called "Palatini Method" of Variation in Covariant Gravitational Theories (1973) - A. A. El-Kholy, R. U. Sexl, H. K. Urbantke|http://archive.numdam.org/ARCHIVE/AIHPA/AIHPA_1973__18_2/AIHPA_1973__18_2_121_0/AIHPA_1973__18_2_121_0.pdf]] pct. 0
*[[Palatini Variational Principle for an Extended Einstein-Hilbert Action (1997) - H. Burton, R. B. Mann|http://arxiv.org/PS_cache/gr-qc/pdf/9711/9711003v1.pdf]] [[pct. 5|http://scholar.google.de/scholar?cites=9052327792572994752&hl=de&as_sdt=2000]]

Theses: 
* [[On the Palatini Variation and Connection Theories of Gravity - H. S. Burton (1998)|http://www.collectionscanada.gc.ca/obj/s4/f2/dsk1/tape9/PQDD_0009/NQ38225.pdf]] [[local|theses/NQ38225.pdf]] tct. 0

<html><center><img src="images/paralleltransport.jpg" style="width: 320px; "/></center></html>
!!!!Historical
The concept of parallel transport of vectors was introduced independently by Levi\-Civita and Schouten in 1917 (two years after Einstein's general relativity theory was published).

!!!!Generalisations
Instead of vectors one can also transport oriented autoparallel segments on a manifold. For details see [[geodesic loop|Geodesic Loop]]. 

See also: [[autoparallelity|Autoparallelity]].

Papers:
* [[On the Parallel Transport of Tetrad in a Riemann-Cartan Spacetime - D.-C. Chern|http://psroc.phys.ntu.edu.tw/cjp/v19/45.pdf]] [[pct. 1|http://scholar.google.de/scholar?cites=13757775929773306320&as_sdt=2005&sciodt=2000&hl=de]]

Given a set of states $\{\mb \Psi_i\}$, the ''Partition Function'' or ''State Sum'' $\mathcal Z$ is the sum over all transition amplitudes
\[
\mathcal Z = \sum_{i,j} \langle \mb \Psi_i| \mb \Psi_j \rangle
\]
In the special case that there are no "true" transitions $ \mb \Psi_i \rightarrow \mb \Psi_j$ for $i \ne j$, the state sum is the sum over all probabilities:
\[
\mathcal Z = \sum_{i} \langle \mb \Psi_i| \mb \Psi_i \rangle = \sum_{i}  |\Psi_i|^2
\]
The state sum therefore serves as the normalisation constant for transitions and probabilities of states.

!!!!Classical Partition Function
The integral over the Boltzmann factors of all phase space elements
\[
\mathcal Z_{class}(T) = \mathcal N \int dqdp \operatorname{e}{}^{-H(p,q)/kT}
\]
is called the ''Classical Partition Function'' (with $\mathcal N$ a normalisation constant). It contains all classical thermodynamic information of the system.

!!!!Quantum Statistical Partition Function
\[
\mathcal Z_{q.m.}(T) \equiv \operatorname{Tr}\left (\operatorname{e}{}^{-\hat H(p,q)/kT} \right )
\]
If $\hat H(p,q)$ is an N-particle Schrödinger\-Hamiltonian, the quantum mechanical system is referred to as ''Canonical Ensemble''.

The ''Pati\-Salam Model'' is a [[Grand Unification Theory (GUT)|GUT]] with gauge group either $SU(4) × SU(2)_L× SU(2)_R$ or $(SU(4) × SU(2)_L× SU(2)_R )/\mathbb Z_2$ and with fermions forming three families. The model includes the right-handed neutrino, which is now believed to exist (due to the observation of neutrino oscillations) and a scalar Higgs field.

>Dirac is the strangest man who ever visited my institute.
> - Niels Bohr -

Audios: 
* [[Paul Dirac Talking about the Large Numbers Hypothesis|http://www.paricenter.com/library/download/dirac01.mp3]]

The ''Pauli Matrices'' are defined by
\begin{equation}
\bs \sigma_1 = \left(\begin{matrix}0 & 1\\
1 & 0\end{matrix}\right), \quad \bs \sigma_2=\left(\begin{matrix}0 & - \mb i\\
\mb i & 0\end{matrix}\right), \quad \bs \sigma_3=\left(\begin{matrix}1 & 0\\
0 & -1\end{matrix}\right)
\end{equation}

* [[Find-pdF|http://www.find-pdf.com]]  <html>
 <div id="DIV_PREVIEW5">
      <form id="FRM_SEARCH" name="FRM_SEARCH" method="post" action="http://www.find-pdf.com/search.html">
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		</form></div></html>
* [[Pdf Search Engine|http://www.pdf-search-engine.com]] (Very comprehensive but slow) <html>
<form id="search" name="form-test" action="http://www.pdf-search-engine.com/pdf-search.php">
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                      <td width="45%"><input value="Search" style="height: 34px; background-color:#F0F0F0; font-weight:bold" class="submit" type="submit" />
</td></tr></table>
</form>
</html>
* [[Pdfgeni|http://www.pdfgeni.com/]] <html>
<div id="searchk">
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</html>

<html><center><a href ="http://pipl.com"> <b>pipl </b> - The most Comprehensive People Search on the Web </a> </center> </html>

''[[Helena Albuquerque|http://pipl.com/search/?FirstName=HELENA&LastName=ALBUQUERQUE&City=&State=&Country=&CategoryID=2&Interface=2]]''
* [[Papers on arXiv.org|http://arxiv.org/find/math/1/au:+Albuquerque_H/0/1/0/all/0/1]]

''[[Antonio Aurilia|http://pipl.com/search/?FirstName=Antonio&LastName=Aurilia&City=&State=&Country=&CategoryID=2&Interface=1]]''
* [[Papers on arXiv.org|http://arxiv.org/find/hep-th/1/au:+Aurilia_A/0/1/0/all/0/1]]

''[[Murray R. Bremner|http://pipl.com/search/?FirstName=Murray+R.&LastName=Bremner&City=&State=&Country=&CategoryID=2&Interface=1]]''
*[[Publications of Murray R. Bremner|http://math.usask.ca/~bremner/research/publications/index.html]]

''Andries Evert Brouwer''
*[[Available Preprints|http://www.win.tue.nl/~aeb/preprints.html]]

''Carlos Castro''
* [[On Dual Phase-Space Relativity, the Machian Principle and Modified Newtonian Dynamics | http://d.scribd.com/docs/2llar0kor72hx4kfy1d1.pdf]]
* [[The Euclidean gravitational action as black hole entropy, singularities, and spacetime voids | http://d.scribd.com/docs/1dhi6pq4jb0kyromue0r.pdf]]
* [[The Extended Relativity Theory in Born-Clifford Phase Spaces with a Lower and Upper Length Scales and Clifford Group Geometric Unification | http://d.scribd.com/docs/1akfy07tokc0mflej3g7.pdf]]
* [[Does Weyl’s Geometry solve the Riddle of Dark Energy ? | http://d.scribd.com/docs/o193ry6l5ji43vb3y9j.pdf]]
* [[The extended Relativity Theory in Clifford Spaces | http://d.scribd.com/docs/28qgyk8govgu41wxe7bp.pdf]]<html> &nbsp; </html>[[The Extended Relativity Theory in Clifford Spaces: Reply to a Review by W. A. Rodrigues, Jr. | http://www.ptep-online.com/index_files/2006/PP-06-05.PDF]]
* [[On Generalized Yang-Mills Theories and Extensions of the Standard Model in Clifford (Tensorial) Spaces | http://d.scribd.com/docs/1m1z3pzc1tkrm3nq8imi.pdf]]
* [[On the Noncommutative and Nonassociative Geometry of Octonionic Spacetime, Modified Dispersion Relations and Grand Unification | http://d.scribd.com/docs/1cwvb9ujudrf6omrchqv.pdf]]
*[[The Exceptional E8 Geometry of Clifford (16) Superspace and Conformal Gravity Yang-Mills Grand Unification | http://d.scribd.com/docs/vhvvtv5b7k7tp08oxzf.pdf]]
* [[The Charge-Mass-Spin Relation of Clifford Polyparticles, Kerr-Newman Black Holes and the Fine Structure Constant|http://d.scribd.com/docs/tr88t068s7zrcuq3rhv.pdf]]
*[[Polyvector Super-Poincare Algebras, M, F Theory Algebras and Generalized Supersymmetry in Clifford-Spaces| http://d.scribd.com/docs/4k77qotub4flhmxuxvf.pdf]]
*[[Moyal Deformations of Gravity Via SU(?) Gauge Theories, Branes and Topological Chern-Simons Matrix Models | http://d.scribd.com/docs/1tzzke6u1gpo3dv3lg6j.pdf]]
* [[On Chern-Simons (Super) Gravity, E8 Yang-Mills and Polyvector-Valued Gauge Theories in Clifford Spaces|http://d.scribd.com/docs/2of3an42y9k0awgm6np9.pdf]]
* [[The Clifford Space Geometry of Conformal Gravity and U(4) × U(4) Yang-Mills Unification|http://www.scribd.com/document_downloads/14363314?extension=pdf&secret_password=]]

'' Ali Hani Chamseddine''
* [[Homepage|http://sites.google.com/site/achamseddine2/home]]

''Françoise Chaitin''''-''''Chatelin''
* [[Homepage|http://www.cerfacs.fr/~chatelin/]]
* [[Elements of Hypercomputation on R and Z2 with the Dickson-Albert Inductive Process|http://www.umcs.maine.edu/~chaitin/f3.pdf]]
* [[Inductive Multiplication in Dickson Algebras|http://www.umcs.maine.edu/~chaitin/f11.pdf]]
* [[The Computing Power of Geometry|http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.37.3013&rep=rep1&type=ps]]
* [[About an Organic Logic Ruling the Continuous Evolution of SVD Measurements with Dickson Hypercomplex Numbers|http://www.cerfacs.fr/algor/reports/2007/TR_PA_07_55.pdf]]
* [[Computing Beyond Classical Logic: SVD Computation in Nonassociative Dickson Algebras|http://www.cerfacs.fr/algor/reports/2007/TR_PA_07_54.pdf]]
* [[Computing with Hypercomplex Numbers|http://www.cerfacs.fr/algor/reports/2000/TR_PA_00_69.ps.gz]] [[local|papers/TR_PA_00_69.ps]]
* [[Qualitative Computing|http://www.cerfacs.fr/algor/reports/2002/TR_PA_02_58.pdf]]
* [[Calcul Algébrique non Linéaire dans les Algèbres de Dickson|http://www.cerfacs.fr/algor/reports/2006/TR_PA_06_07.pdf]]

''Donald Chesley''
* [[Website|http://www.captaincomputersensor.net]] [[local|html/captaincomputersensor.net]]

''[[John Conway]]''

''Michael Deza''
* [[List of Publications|http://www.liga.ens.fr/~deza/papers/Lpub/lpubHyperlink.html]]

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*[[Publications|http://www.mrao.cam.ac.uk/~cjld1/pages/publications.htm]]

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* [[Papers on arXiv.org|http://arxiv.org/find/hep-th/1/au:+Gogberashvili_M/0/1/0/all/0/1]]

''Robert L. Griess, Jr''
* [[Homepage|http://www.math.lsa.umich.edu/~rlg/index.html]]

''Friedrich W. Hehl''
* [[Papers on arXiv.org|http://arxiv.org/find/gr-qc/1/au:+Hehl_F/0/1/0/all/0/1]]
* [[Papers on SPIRES|http://www.slac.stanford.edu/spires/find/hep/www?rawcmd=ea+Hehl,+Friedrich+W]]
* [[Scientific Publications of Friedrich W. Hehl|http://www.thp.uni-koeln.de/gravitation/mitarbeiter/hehllit99_65.ps]]

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* [[Geometric Calculus R & D Home Page|http://modelingnts.la.asu.edu/]]

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* [[Papers at SPIRES|http://www-spires.slac.stanford.edu/spires/find/hep/www?rawcmd=ea+Kephart,+Thomas+W]]

''Eric Lord''
* [[Homepage|http://materials.iisc.ernet.in/~lord/webfiles/eric/index.html]]

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* [[Papers on arXiv.org|http://xxx.lanl.gov/find/grp_physics/1/au:+Lyre/0/1/0/all/0/1]]

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* [[Homepage|http://www.maths.qmul.ac.uk/~majid/Welcome.html]]

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*[[Matej Pavsic's Home Page|http://www-f1.ijs.si/~pavsic/]]

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* [[Research Activities|http://www.clifford.org/~wpezzag/index.html]]

''[[Roger Penrose]]''

''Waldyr Alves Rodrigues Jr.''
*[[Website|http://www.ime.unicamp.br/~walrod/]]

''James Stasheff''
* [[Home Page|http://www.math.unc.edu/Faculty/jds/]]

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* [[Web page|http://www.ipgp.fr/~tarantola/]]

''Robert Wilson''
* [[Homepage|http://www.maths.qmw.ac.uk/~raw/]]

Links:
* [[Mathematics Genealogy Project|http://genealogy.math.ndsu.nodak.edu/]]

The ''Pioneer Anomaly'' is an unexplained constant acceleration of the Pioneer 10 and 11 spacecrafts.

Links:
* [[WIKIPEDIA - Pioneer Anomaly|http://en.wikipedia.org/wiki/Pioneer_anomaly]]

Videos:
* [[The Pioneer Anomaly (Perimeter Institute Lecture) - J. Moffat|http://streamer.perimeterinstitute.ca/mediasite/viewer/?peid=ef29e092-cb6c-4e82-9fbf-91e42a561076]]

The ''Planck Units'' are:
Length: $ l_P = \sqrt{\frac{\hbar G}{c^3}} \,$  ? $1.6 × 10^{-35}$ meters.
Time: $t_P = \frac{l_P}{c} = \sqrt{\frac{\hbar G}{c^5}} $ ? $5.4 × 10^{-44}$ seconds.
Mass: $m_P = \frac{\hbar}{t_P c^2}  = \sqrt{\frac{\hbar c}{G}}$ ? $2.2 × 10^{-8}$ kilograms.

In natural units, i.e. $c = 1$ and $\hbar = 1\,$ one has:
$ l_P = t_P = \sqrt{G}$ and $ m_P = \frac{1}{\sqrt{G}}$.

!!!!Some relationships
* The diameter of a proton is about $10^{20}$ Planck lengths.
* $1$ Planck length is the Schwarzschild radius corresponding to $1/2$ Planck mass.
* The radius of the observable universe is about $46$ billion light-years, equivalent to $4.4 \cdot 10^{26}$ meters or $2.7 \cdot 10^{61}$ Planck lengths.

See also: [[geometrical units|Geometrical Units]].

/***
''PlayerPlugin for TiddlyWiki version 1.2.x and 2.x''
^^author: Eric Shulman - ELS Design Studios
source: http://www.TiddlyTools.com/#PlayerPlugin 
license: [[Creative Commons Attribution-ShareAlike 2.5 License|http://creativecommons.org/licenses/by-sa/2.5/]]^^
status: ''ALPHA / EXPERIMENTAL''

Embed a media player in a tiddler.  

!!!!!Usage
<<<
{{{<<player [type] [URL] [width] [height] [hidecontrols]>>}}}

where ''type'' is optional, and is one of the following: ''windows'', ''realone'', ''quicktime'', or ''flash''.  If the media type is not specified, the plugin automatically renders Windows, Real, QuickTime or Flash player by matching known file extensions and/or specialized streaming-media transfer protocols (such as RTSP:).  For unrecognized media types, the plugin assumes WindowsMedia (the player with the most users... *sigh*)
<<<
!!!!!Configuration
<<<
Default player size:
width: <<option txtPlayerDefaultWidth>> height: <<option txtPlayerDefaultHeight>>
<<<
!!!!!Examples
<<<
+++[Windows Media]...
Times Square Live Webcam
{{{<<player id=1 http://www.earthcam.com/usa/newyork/timessquare/asx/tsq_stream.asx>>}}}
<<player id=1 http://www.earthcam.com/usa/newyork/timessquare/asx/tsq_stream.asx>>
===
+++[RealOne]...
BBC London: Live and Recorded news
{{{<<player id=2 http://www.bbc.co.uk/london/realmedia/news/tvnews.ram>>}}}
<<player id=2 http://www.bbc.co.uk/london/realmedia/news/tvnews.ram>>
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+++[Quicktime]...
America Free TV: Classic Comedy
{{{<<player id=3 http://www.americafree.tv/unicast_mov/AmericaFreeTVComedy.mov>>}}}
<<player id=3 http://www.americafree.tv/unicast_mov/AmericaFreeTVComedy.mov>>
===
+++[Flash]...
Asteroids arcade game
{{{<<player id=4 http://www.80smusiclyrics.com/games/asteroids/asteroids.swf 400 300>>}}}
<<player id=4 http://www.80smusiclyrics.com/games/asteroids/asteroids.swf 400 300>>
Google Video
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YouTube Video
{{{<<player id=6 flash http://www.youtube.com/v/OdT9z-JjtJk 400 300>>}}}
<<player id=6 flash http://www.youtube.com/v/OdT9z-JjtJk 400 300>>
===
<<<
!!!!!Installation
<<<
import (or copy/paste) the following tiddlers into your document:
''PlayerPlugin '' (tagged with <<tag systemConfig>>)
^^documentation and javascript for macro handling^^
<<<
!!!!!Revision History
<<<
''2006.02.26 [0.7.0]''
major re-write.  handles default params better.  create/recreate player objects via loadURL() API for use with interactive forms and scripts.
''2006.01.27 [0.6.0]''
added support for 'extra' macro params to pass through to object parameters
''2006.01.19 [0.5.0]''
Initial ALPHA release
''2005.12.23 [0.0.0]''
Started
<<<
!!!!!Credits
<<<
This feature was developed by EricShulman from [[ELS Design Studios|http:/www.elsdesign.com]].
<<<
!!!!!Code
***/

// //  macro definition

//{{{
version.extensions.player = {major: 0, minor: 8, revision: 0, date: new Date(2006,3,7)};

config.macros.player = {};
config.macros.player.html = {};
config.macros.player.handler= function(place,macroName,params) {
	var id=null;
	if (params[0].substr(0,3)=="id=") id=params.shift().substr(3);
	var type="";
	if ((params[0]=="windows")||(params[0]=="realone")||(params[0]=="quicktime")||(params[0]=="flash")) type=params.shift();
	var url=params.shift(); if (!url || !url.trim().length) url="";
	var width=params.shift();
	var height=params.shift();
	var show=(params.shift()=='hidecontrols')?"0":"1"; 
	var extras=""; while (params[0]!=undefined) extras+="<param name='"+params.shift()+"' value='"+params.shift()+"'> ";
	this.loadURL(place,id,type,url,width,height,show,extras);
}

if (config.options.txtPlayerDefaultWidth==undefined) config.options.txtPlayerDefaultWidth="100%";
if (config.options.txtPlayerDefaultHeight==undefined) config.options.txtPlayerDefaultHeight="480"; // can't use "100%"... player height doesn't stretch right :-(

config.macros.player.loadURL=function(place,id,type,url,width,height,show,extras) {

	if (id==undefined) id="tiddlyPlayer";
	if (!width) var width=config.options.txtPlayerDefaultWidth;
	if (!height) var height=config.options.txtPlayerDefaultHeight;
	if (url && (!type || !type.length)) {
		if ((url.indexOf('mms')!=-1)||(url.indexOf('.asx')!=-1)||(url.indexOf('.wvx')!=-1)||(url.indexOf('.wmv')!=-1)||(url.indexOf('.mp3')!=-1))
			var type="windows";
		else if ((url.indexOf('rtsp')!=-1)||(url.indexOf('.ram')!=-1)||(url.indexOf('.rpm')!=-1)||(url.indexOf('.rm' )!=-1)||(url.indexOf('.ra' )!=-1))
			var type="realone";
		else if ((url.indexOf('.mov')!=-1)||(url.indexOf('.qt' )!=-1))
			var type="quicktime";
		else if ((url.indexOf('.swf')!=-1)||(url.indexOf('.flv')!=-1))
			var type="flash";
	}
	if (!type) var type="none";
	if (!url) var url="";
	if (show===undefined) var show=true;
	if (!extras) var extras="";
	if (type=="none" && url.trim().length) url="<br>unrecognized media type:<br>"+url;
	if (type=="realone") height-=show?60:0; // leave room for controls
	if (type=="windows") show=show?"1":"0"; // player-specific param value
	if (type=="realone") show=show?"block":"none";
	if (type=="quicktime") show=show?"true":"false";

	// create containing div for player HTML
	// and add or replace player in TW DOM structure
	var newplayer = document.createElement("div");
	newplayer.playerType=type;
	newplayer.setAttribute("id",id+"_div");
	var existing = document.getElementById(id+"_div");
	if (existing && !place) place=existing.parentNode;
	if (!existing)
		place.appendChild(newplayer);
	else {
		if (place==existing.parentNode) place.replaceChild(newplayer,existing)
		else { existing.parentNode.removeChild(existing); place.appendChild(newplayer); }
	}

	var html='<center>'+config.macros.player.html[type];
	html=html.replace(/%i%/mg,id);
	html=html.replace(/%w%/mg,width);
	html=html.replace(/%h%/mg,height);
	html=html.replace(/%u%/mg,url);
	html=html.replace(/%s%/mg,show);
	html=html.replace(/%x%/mg,extras)+'</center>';
	newplayer.innerHTML=html;
}
//}}}

// // Player-specific API functions: isReady(id), isPlaying(id), toggleControls(id), showControls(id,flag)

//{{{
// status values:
// Windows: 0=Undefined, 1=Stopped, 2=Paused, 3=Playing, 4=ScanForward, 5=ScanReverse
//          6=Buffering, 7=Waiting, 8=MediaEnded, 9=Transitioning, 10=Ready, 11=Reconnecting
// RealOne: 0=Stopped, 1=Contacting, 2=Buffering, 3=Playing, 4=Paused, 5=Seeking
// QuickTime: 'Waiting', 'Loading', 'Playable', 'Complete', 'Error:###'
// Flash: 0=Loading, 1=Uninitialized, 2=Loaded, 3=Interactive, 4=Complete
config.macros.player.isReady=function(id)
{
	var d=document.getElementById(id+"_div"); if (!d) return false;
	var p=document.getElementById(id); if (!p) return false;
	if (d.playerType=='windows') return !((p.playState==0)||(p.playState==7)||(p.playState==9)||(p.playState==11));
	if (d.playerType=='realone') return (p.GetPlayState()>1);
	if (d.playerType=='quicktime') return !((p.getPluginStatus()=='Waiting')||(p.getPluginStatus()=='Loading'));
	if (d.playerType=='flash') return (p.ReadyState>2);
	return true;
}
config.macros.player.isPlaying=function(id)
{
	var d=document.getElementById(id+"_div"); if (!d) return false;
	var p=document.getElementById(id); if (!p) return false;
	if (d.playerType=='windows') return (p.playState==3);
	if (d.playerType=='realone') return (p.GetPlayState()==3);
	if (d.playerType=='quicktime') return (p.getPluginStatus()=='Complete');
	if (d.playerType=='flash') return (p.ReadyState<4);
	return false;
}
config.macros.player.showControls=function(id,flag) {
	var d=document.getElementById(id+"_div"); if (!d) return false;
	var p=document.getElementById(id); if (!p) return false;
	if (d.playerType=='windows') { p.ShowControls=flag; p.ShowStatusBar=flag; }
	if (d.playerType=='realone') { alert('show/hide controls not available'); }
	if (d.playerType=='quicktime')      // if player not ready, retry in one second
		{ if (this.isReady(id)) p.setControllerVisible(flag); else setTimeout('config.macros.player.showControls("'+id+'",'+flag+')',1000); }
	if (d.playerType=='flash') { alert('show/hide controls not available'); }
}
config.macros.player.toggleControls=function(id) {
	var d=document.getElementById(id+"_div"); if (!d) return false;
	var p=document.getElementById(id); if (!p) return false;
	if (d.playerType=='windows') var flag=!p.ShowControls;
	if (d.playerType=='realone') var flag=true; // TBD
	if (d.playerType=='quicktime') var flag=!p.getControllerVisible();
	if (d.playerType=='flash') var flag=true; // TBD
	this.showControls(id,flag);
}
config.macros.player.fullScreen=function(id) {
	var d=document.getElementById(id+"_div"); if (!d) return false;
	var p=document.getElementById(id); if (!p) return false;
	if (d.playerType=='windows') p.DisplaySize=3;
	if (d.playerType=='realone') p.SetFullScreen();
	if (d.playerType=='quicktime') { alert('full screen not available'); }
	if (d.playerType=='flash') { alert('full screen not available'); }
}
//}}}

// // Player HTML

//{{{
// placeholder (no player)
config.macros.player.html.none=' \
	<table id="%i%" width="%w%" height="%h%" style="background-color:#111;border:0;margin:0;padding:0;"> \
	<tr style="background-color:#111;border:0;margin:0;padding:0;"> \
	<td width="%w%" height="%h%" style="background-color:#111;color:#ccc;border:0;margin:0;padding:0;text-align:center;"> \
	&nbsp; \
	%u% \
	&nbsp; \
	</td></tr></table>';
//}}}

//{{{
// Windows Media Player
// v7.1 ID: classid=CLSID:6BF52A52-394A-11d3-B153-00C04F79FAA6
// v9	ID: classid=CLSID:22d6f312-b0f6-11d0-94ab-0080c74c7e95
config.macros.player.html.windows=' \
	<object id="%i%" width="%w%" height="%h%" style="margin:0;padding:0;" \
		classid="CLSID:22d6f312-b0f6-11d0-94ab-0080c74c7e95" \
		codebase="http://activex.microsoft.com/activex/controls/mplayer/en/nsmp2inf.cab#Version=6,4,5,715" \
		align="baseline" border="0" \
		standby="Loading Microsoft Windows Media Player components..." \
		type="application/x-oleobject"> \
		<param name="FileName" value="%u%"> <param name="ShowControls" value="%s%"> \
		<param name="ShowPositionControls" value="1"> <param name="ShowAudioControls" value="1"> \
		<param name="ShowTracker" value="1"> <param name="ShowDisplay" value="0"> \
		<param name="ShowStatusBar" value="1"> <param name="AutoSize" value="1"> \
		<param name="ShowGotoBar" value="0"> <param name="ShowCaptioning" value="0"> \
		<param name="AutoStart" value="0"> <param name="AnimationAtStart" value="1"> \
		<param name="TransparentAtStart" value="0"> <param name="AllowScan" value="1"> \
		<param name="EnableContextMenu" value="1"> <param name="ClickToPlay" value="1"> \
		<param name="InvokeURLs" value="1"> <param name="DefaultFrame" value="datawindow"> \
		%x% \
		<embed src="%u%" style="margin:0;padding:0;" \
			align="baseline" border="0" width="%w%" height="%h%" \
			type="application/x-mplayer2" \
			pluginspage="http://www.microsoft.com/windows/windowsmedia/download/default.asp" \
			name="%i%" showcontrols="%s%" showpositioncontrols="1" \
			showaudiocontrols="1" showtracker="1" showdisplay="0" \
			showstatusbar="%s%" autosize="1" showgotobar="0" showcaptioning="0" \
			autostart="0" autorewind="0" animationatstart="1" transparentatstart="0" \
			allowscan="1" enablecontextmenu="1" clicktoplay="0" invokeurls="1" \
			defaultframe="datawindow"> \
		</embed> \
	</object>';
//}}}

//{{{
// RealNetworks' RealOne Player
config.macros.player.html.realone=' \
	<table width="%w%" style="border:0;margin:0;padding:0;"><tr style="border:0;margin:0;padding:0;"><td style="border:0;margin:0;padding:0;"> \
	<object id="%i%" width="%w%" height="%h%" style="margin:0;padding:0;" \
		CLASSID="clsid:CFCDAA03-8BE4-11cf-B84B-0020AFBBCCFA"> \
		<PARAM NAME="CONSOLE" VALUE="player"> \
		<PARAM NAME="CONTROLS" VALUE="ImageWindow"> \
		<PARAM NAME="AUTOSTART" Value="false"> \
		<PARAM NAME="MAINTAINASPECT" Value="true"> \
		<PARAM NAME="NOLOGO" Value="true"> \
		<PARAM name="BACKGROUNDCOLOR" VALUE="#333333"> \
		<PARAM NAME="SRC" VALUE="%u%"> \
		%x% \
		<EMBED width="%w%" height="%h%" controls="ImageWindow" type="audio/x-pn-realaudio-plugin" style="margin:0;padding:0;" \
			name="%i%" \
			src="%u%" \
			console=player \
			maintainaspect=true \
			nologo=true \
			backgroundcolor=#333333 \
			autostart=false> \
		</OBJECT> \
	</td></tr><tr style="border:0;margin:0;padding:0;"><td style="border:0;margin:0;padding:0;"> \
	<object id="%i%_controls" width="%w%" height="60" style="margin:0;padding:0;display:%s%" \
		CLASSID="clsid:CFCDAA03-8BE4-11cf-B84B-0020AFBBCCFA"> \
		<PARAM NAME="CONSOLE" VALUE="player"> \
		<PARAM NAME="CONTROLS" VALUE="All"> \
		<PARAM NAME="NOJAVA" Value="true"> \
		<PARAM NAME="MAINTAINASPECT" Value="true"> \
		<PARAM NAME="NOLOGO" Value="true"> \
		<PARAM name="BACKGROUNDCOLOR" VALUE="#333333"> \
		<PARAM NAME="SRC" VALUE="%u%"> \
		%x% \
		<EMBED WIDTH="%w%" HEIGHT="60" NOJAVA="true" type="audio/x-pn-realaudio-plugin" style="margin:0;padding:0;display:%s%" \
			controls="All" \
			name="%i%_controls" \
			src="%u%" \
			console=player \
			maintainaspect=true \
			nologo=true \
			backgroundcolor=#333333> \
		</OBJECT> \
	</td></tr></table>';
//}}}

//{{{
// QuickTime Player
config.macros.player.html.quicktime=' \
	<OBJECT ID="%i%" WIDTH="%w%" HEIGHT="%h%" style="margin:0;padding:0;" \
		CLASSID="clsid:02BF25D5-8C17-4B23-BC80-D3488ABDDC6B" \
		CODEBASE="http://www.apple.com/qtactivex/qtplugin.cab"> \
		<PARAM name="SRC" VALUE="%u%"> \
		<PARAM name="AUTOPLAY" VALUE="true"> \
		<PARAM name="CONTROLLER" VALUE="%s%"> \
		<PARAM name="BGCOLOR" VALUE="#333333"> \
		<PARAM name="SCALE" VALUE="aspect"> \
		<PARAM name="SAVEEMBEDTAGS" VALUE="true"> \
		%x% \
		<EMBED name="%i%" WIDTH="%w%" HEIGHT="%h%" style="margin:0;padding:0;" \
			SRC="%u%" \
			AUTOPLAY="true" \
			SCALE="aspect" \
			CONTROLLER="%s%" \
			BGCOLOR="#333333" \
			EnableJavaSript="true" \
			PLUGINSPAGE="http://www.apple.com/quicktime/download/"> \
		</EMBED> \
	</OBJECT>';
//}}}

//{{{
// Flash Player
config.macros.player.html.flash='\
	<object id="%i%" width="%w%" height="%h%" style="margin:0;padding:0;" \
		classid="clsid:D27CDB6E-AE6D-11cf-96B8-444553540000" \
		codebase="http://download.macromedia.com/pub/shockwave/cabs/flash/swflash.cab#version=6,0,29,0"> \
		<param name="movie" value="%u%"> \
		<param name="quality" value="high"> \
		<param name="SCALE" value="exactfit"> \
		<param name="bgcolor" value="333333"> \
		%x% \
		<embed name="%i%" src="%u%" style="margin:0;padding:0;" \
			height="%h%" width="%w%" quality="high" \
			pluginspage="http://www.macromedia.com/go/getflashplayer" \
			type="application/x-shockwave-flash" scale="exactfit"> \
		</embed> \
	</object>';
//}}}

/***

''Inspired by [[TiddlyPom|http://www.warwick.ac.uk/~tuspam/tiddlypom.html]]''

|Name|SplashScreenPlugin|
|Created by|SaqImtiaz|
|Location|http://lewcid.googlepages.com/lewcid.html#SplashScreenPlugin|
|Version|0.21 |
|Requires|~TW2.08+|
!Description:
Provides a simple splash screen that is visible while the TW is loading.

!Installation
Copy the source text of this tiddler to your TW in a new tiddler, tag it with systemConfig and save and reload. The SplashScreen will now be installed and will be visible the next time you reload your TW.

!Customizing
Once the SplashScreen has been installed and you have reloaded your TW, the splash screen html will be present in the MarkupPreHead tiddler. You can edit it and customize to your needs.

!History
* 20-07-06 : version 0.21, modified to hide contentWrapper while SplashScreen is displayed.
* 26-06-06 : version 0.2, first release

!Code
***/
//{{{
var old_lewcid_splash_restart=restart;

restart = function()
{   if (document.getElementById("SplashScreen"))
        document.getElementById("SplashScreen").style.display = "none";
      if (document.getElementById("contentWrapper"))
        document.getElementById("contentWrapper").style.display = "block";

    old_lewcid_splash_restart();

    if (splashScreenInstall)
       {if(config.options.chkAutoSave)
			{saveChanges();}
        displayMessage("TW SplashScreen has been installed, please save and refresh your TW.");
        }
}


var oldText = store.getTiddlerText("MarkupPreHead");
if (oldText.indexOf("SplashScreen")==-1)
   {var siteTitle = store.getTiddlerText("SiteTitle");
   var splasher='\n\n<style type="text/css">#contentWrapper {display:none;}</style><div id="SplashScreen" style="border: 3px solid #ccc; display: block; text-align: center; width: 320px; margin: 100px auto; padding: 50px; color:#000; font-size: 28px; font-family:Tahoma; background-color:#eee;"><b>'+siteTitle +'</b> is loading<blink> ...</blink><br><br><span style="font-size: 14px; color:red;">Requires Javascript.</span></div>';
   if (! store.tiddlerExists("MarkupPreHead"))
       {var myTiddler = store.createTiddler("MarkupPreHead");}
   else
      {var myTiddler = store.getTiddler("MarkupPreHead");}
      myTiddler.set(myTiddler.title,oldText+splasher,config.options.txtUserName,null,null);
      store.setDirty(true);
      var splashScreenInstall = true;
}
//}}}

/***
|Name|Plugin: jsMath|
|Created by|BobMcElrath|
|Email|my first name at my last name dot org|
|Location|http://bob.mcelrath.org/tiddlyjsmath.html|
|Version|1.5.1|
|Requires|[[TiddlyWiki|http://www.tiddlywiki.com]] &ge; 2.0.3, [[jsMath|http://www.math.union.edu/~dpvc/jsMath/]] &ge; 3.0|
!Description
LaTeX is the world standard for specifying, typesetting, and communicating mathematics among scientists, engineers, and mathematicians.  For more information about LaTeX itself, visit the [[LaTeX Project|http://www.latex-project.org/]].  This plugin typesets math using [[jsMath|http://www.math.union.edu/~dpvc/jsMath/]], which is an implementation of the TeX math rules and typesetting in javascript, for your browser.  Notice the small button in the lower right corner which opens its control panel.
!Installation
In addition to this plugin, you must also [[install jsMath|http://www.math.union.edu/~dpvc/jsMath/download/jsMath.html]] on the same server as your TiddlyWiki html file.  If you're using TiddlyWiki without a web server, then the jsMath directory must be placed in the same location as the TiddlyWiki html file.

I also recommend modifying your StyleSheet use serif fonts that are slightly larger than normal, so that the math matches surrounding text, and \\small fonts are not unreadable (as in exponents and subscripts).
{{{
.viewer {
  line-height: 125%;
  font-family: serif;
  font-size: 12pt;
}
}}}

If you had used a previous version of [[Plugin: jsMath]], it is no longer necessary to edit the main tiddlywiki.html file to add the jsMath <script> tag.  [[Plugin: jsMath]] now uses ajax to load jsMath.
!History
* 11-Nov-05, version 1.0, Initial release
* 22-Jan-06, version 1.1, updated for ~TW2.0, tested with jsMath 3.1, editing tiddlywiki.html by hand is no longer necessary.
* 24-Jan-06, version 1.2, fixes for Safari, Konqueror
* 27-Jan-06, version 1.3, improved error handling, detect if ajax was already defined (used by ZiddlyWiki)
* 12-Jul-06, version 1.4, fixed problem with not finding image fonts
* 26-Feb-07, version 1.5, fixed problem with Mozilla "unterminated character class".
* 27-Feb-07, version 1.5.1, Runs compatibly with TW 2.1.0+, by Bram Chen
!Examples
|!Source|!Output|h
|{{{The variable $x$ is real.}}}|The variable $x$ is real.|
|{{{The variable \(y\) is complex.}}}|The variable \(y\) is complex.|
|{{{This \[\int_a^b x = \frac{1}{2}(b^2-a^2)\] is an easy integral.}}}|This \[\int_a^b x = \frac{1}{2}(b^2-a^2)\] is an easy integral.|
|{{{This $$\int_a^b \sin x = -(\cos b - \cos a)$$ is another easy integral.}}}|This $$\int_a^b \sin x = -(\cos b - \cos a)$$ is another easy integral.|
|{{{Block formatted equations may also use the 'equation' environment \begin{equation}  \int \tan x = -\ln \cos x \end{equation} }}}|Block formatted equations may also use the 'equation' environment \begin{equation}  \int \tan x = -\ln \cos x \end{equation}|
|{{{Equation arrays are also supported \begin{eqnarray} a &=& b \\ c &=& d \end{eqnarray} }}}|Equation arrays are also supported \begin{eqnarray} a &=& b \\ c &=& d \end{eqnarray} |
|{{{I spent \$7.38 on lunch.}}}|I spent \$7.38 on lunch.|
|{{{I had to insert a backslash (\\) into my document}}}|I had to insert a backslash (\\) into my document|
!Code
***/
//{{{

// AJAX code adapted from http://timmorgan.org/mini
// This is already loaded by ziddlywiki...
if(typeof(window["ajax"]) == "undefined") {
  ajax = {
      x: function(){try{return new ActiveXObject('Msxml2.XMLHTTP')}catch(e){try{return new ActiveXObject('Microsoft.XMLHTTP')}catch(e){return new XMLHttpRequest()}}},
      gets: function(url){var x=ajax.x();x.open('GET',url,false);x.send(null);return x.responseText}
  }
}

// Load jsMath
jsMath = {
  Setup: {inited: 1},          // don't run jsMath.Setup.Body() yet
  Autoload: {root: new String(document.location).replace(/[^\/]*$/,'jsMath/')}  // URL to jsMath directory, change if necessary
};
var jsMathstr;
try {
  jsMathstr = ajax.gets(jsMath.Autoload.root+"jsMath.js");
} catch(e) {
  alert("jsMath was not found: you must place the 'jsMath' directory in the same place as this file.  "
       +"The error was:\n"+e.name+": "+e.message);
  throw(e);  // abort eval
}
try {
  window.eval(jsMathstr);
} catch(e) {
  alert("jsMath failed to load.  The error was:\n"+e.name + ": " + e.message + " on line " + e.lineNumber);
}
jsMath.Setup.inited=0;  //  allow jsMath.Setup.Body() to run again

// Define wikifers for latex
config.formatterHelpers.mathFormatHelper = function(w) {
    var e = document.createElement(this.element);
    e.className = this.className;
    var endRegExp = new RegExp(this.terminator, "mg");
    endRegExp.lastIndex = w.matchStart+w.matchLength;
    var matched = endRegExp.exec(w.source);
    if(matched) {
        var txt = w.source.substr(w.matchStart+w.matchLength,
            matched.index-w.matchStart-w.matchLength);
        if(this.keepdelim) {
          txt = w.source.substr(w.matchStart, matched.index+matched[0].length-w.matchStart);
        }
        e.appendChild(document.createTextNode(txt));
        w.output.appendChild(e);
        w.nextMatch = endRegExp.lastIndex;
    }
}

config.formatters.push({
  name: "displayMath1",
  match: "\\\$\\\$",
  terminator: "\\\$\\\$\\n?", // 2.0 compatability
  termRegExp: "\\\$\\\$\\n?",
  element: "div",
  className: "math",
  handler: config.formatterHelpers.mathFormatHelper
});

config.formatters.push({
  name: "inlineMath1",
  match: "\\\$",
  terminator: "\\\$", // 2.0 compatability
  termRegExp: "\\\$",
  element: "span",
  className: "math",
  handler: config.formatterHelpers.mathFormatHelper
});

var backslashformatters = new Array(0);

backslashformatters.push({
  name: "inlineMath2",
  match: "\\\\\\\(",
  terminator: "\\\\\\\)", // 2.0 compatability
  termRegExp: "\\\\\\\)",
  element: "span",
  className: "math",
  handler: config.formatterHelpers.mathFormatHelper
});

backslashformatters.push({
  name: "displayMath2",
  match: "\\\\\\\[",
  terminator: "\\\\\\\]\\n?", // 2.0 compatability
  termRegExp: "\\\\\\\]\\n?",
  element: "div",
  className: "math",
  handler: config.formatterHelpers.mathFormatHelper
});

backslashformatters.push({
  name: "displayMath3",
  match: "\\\\begin\\{equation\\}",
  terminator: "\\\\end\\{equation\\}\\n?", // 2.0 compatability
  termRegExp: "\\\\end\\{equation\\}\\n?",
  element: "div",
  className: "math",
  handler: config.formatterHelpers.mathFormatHelper
});

// These can be nested.  e.g. \begin{equation} \begin{array}{ccc} \begin{array}{ccc} ...
backslashformatters.push({
  name: "displayMath4",
  match: "\\\\begin\\{eqnarray\\}",
  terminator: "\\\\end\\{eqnarray\\}\\n?", // 2.0 compatability
  termRegExp: "\\\\end\\{eqnarray\\}\\n?",
  element: "div",
  className: "math",
  keepdelim: true,
  handler: config.formatterHelpers.mathFormatHelper
});

// The escape must come between backslash formatters and regular ones.
// So any latex-like \commands must be added to the beginning of
// backslashformatters here.
backslashformatters.push({
    name: "escape",
    match: "\\\\.",
    handler: function(w) {
        w.output.appendChild(document.createTextNode(w.source.substr(w.matchStart+1,1)));
        w.nextMatch = w.matchStart+2;
    }
});

jsMath.Extension.Require("AMSmath");
jsMath.Extension.Require('underset-overset');
jsMath.Extension.Require("boldsymbol");
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>All of the recent theories of elementary particles have been shaped by the paper by Wigner, containing the classification of the irreducible representations of $ SL(2,\mathbb R) \rtimes \mathbb R^{1,3}$.... It is difficult to overestimate the "importance of this paper, which will certainly stand as one of the great intellectual achievements of our century.
> - Shlomo Sternberg - Group Theory and Physics

A ''Poincaré Transformation'' is an inhomogeneous [[Lorentz transformation|Lorentz Transformation]] given by
\[
x'^\mu={\Lambda^\mu}_\nu x^\nu+a^\mu
\]
The ''Poincaré Group''  $ISO(1,3)$ (or  ''Inhomogeneous Lorentz Group'') is the [[semi-direct product|Semi Direct Product]] of the [[Lorentz group|Lorentz Transformation]] and the translation group ($\mathbb R^{1,3} \rtimes O(1,3)$), such that the translations form an invariant subgroup but the the Lorentz-transformations do not. It is a $10$-parameter [[Lie group|Lie Group]] and the associated [[Lie algebra|Lie Algebra]] is spanned by the $4$ generators of translations $P_\mu$ and the $6$ Lorentz-generators $M_{\mu\nu}$ which satisfy
\[
\begin{align}  {}[P_\mu, P_\nu] &= 0 \\
{}[M_{\mu\nu}, P_\rho] &= \eta_{\mu\rho} P_\nu-\eta_{\nu\rho} P_\mu \\
{}[M_{\mu\nu}, M_{\rho\sigma}] &= \eta_{\mu\rho} M_{\nu\sigma}- \eta_{\mu\sigma} M_{\nu\rho}- \eta_{\nu\rho}  M_{\mu\sigma} + \eta_{\nu\sigma}  M_{\mu\rho} \end{align}
\]
The full Poincaré group which includes time reversal and space reflections is the [[isometry group|Isometry]] of Minkowski space. The conservation laws of [[energy-momentum|Stress Energy Tensor]] and angular-momentum in special relativity are connected with the Poincaré group. In fact, according to [[Noether’s theorem the invariance of a physical system under a spacetime translation leads to the conservation of the canonical energy-momentum tensor, whereas the invariance under a Lorentz transformation leads to the conservation of the canonical angular-momentum tensor. When passing to general relativity, these two tensors are modified by the presence of gravitation.

The Lie algebra of the Poincaré group has two basic invariants, interpreted physically as mass and spin.
The Poincaré group provides a correct description of masses and spin of elementary particles which is not the case for the mere Lorentz group.

<html><center><iframe name="content" src="http://www.markus-maute.de/trajectory/advert_60.html" width=51% height=86></iframe></center></html>Lectures:
* [[The Lorentz and Poincaré Groups and their Representations - R. M. D. Delgado|http://www.hep.manchester.ac.uk/u/rosa/Symmetries.pdf]]

The ''Polyvector Action Principle'' or ''Principle of least Polyvector Action'' is the conventional action principle applied to a [[polyvector action |Polyvector Action]] $S[\mb{X}]$:
\[
\delta S[\mb{X}] = 0
\]
The (minimum-) solutions of this equation are the [[Polyvector Euler-Lagrange equations|Polyvector Euler-Lagrange Equations]].

The minimum of the polyvector action represents the "shortest path" in polyvector space (polyvector geodesic). Contrary to a classical vector space this path is represented by a linear combination of points, curves, surfaces, volumes, etc.
This generalizes the concept of a [[geodesic|Geodesic Equation]] of a point particle in a curved spacetime background.
If one also considers second grades in the polyvector action (as is done in the Clifford algebra approach) one gets the description of a spinning particle in a curved background. Such particles obey an extended equivalence principle, the [[polyvector equivalence principle|Polyvector Equivalence Principle]]. Their trajectory can be interpreted as one of an extended object and might be seen as an alternative to string theory where the Polyakov and [[Nambu-Goto actions|Dirac-Nambu-Goto Action]] for example describe the trajectory of a string that sweeps out a worldsheet.
If one furthermore takes into account all relevant grades in $4$ spacetime dimensions, i.e. grades up to order $4$, one gets the geodesic of the whole universe including the "subgeodesics" of matter within it: ''The Trajectory of the Universe''. This opens up the possibility of a description of the dynamics of the whole universe which is somewhat reminiscent of the Hartle Hawking [[wavefunction of the universe|Wavefunction of the Universe]] description.

<html><center><img src="images/spin.gif" style="width: 340px; "/></center></html>
The ''Weak Polyvector Equivalence Principle'' or ''Generalized Weak Equivalence Principle'' states that all particles follow the same path in a gravitational field independent of their [[polyvector mass|Physical Polyvectors]].
This generalizes the conventional [[equivalence principle|Equivalence Principles]] of general relativity which only holds for point particles. If e.g. a spinning particle is included, which at least classically can be interpreted as an extended object, the particle is supposed to carry out a Zitterbewegung instead of following the more straight path of the point particle. The ''Weak Polyvector Equivalence Principle'' restores the equivalence in such situations in that it introduces additional [[polyvector mass terms|Physical Polyvectors]].

The ''Strong Polyvector Equivalence Principle'' states that, given fields, an [[autoparallel|Autoparallelity]] system in [[P-space|Polyvector Space]] can be found that can be regarded as a generalized inertial system, i.e. one in which the laws of a "flat" P-space are valid. In this system it is not only compensated for linear accelerations (as is the case for the strong equivalence principle of general relativity, but also for angular and higher order ones). In general relativity one gets to a local inertial system by carrying out a boost which transforms away the gravitational field. It is conjectured that a generalization of boosts to P-space allows for transforming away all other fields as well. A boost in polyvector space can be understood as a rotation in this space which generalizes a Lorentz boost, which is a rotation in the Minkowski subspace (grade 1 subspace) only. (One can therefore regard it as an "enhanced" or "extended" Lorentz transformation). The algebraic consequences are, that the grades of physical entities depend on the observer's polyvector frame.
The principle was suggested in the context of [[Clifford algebras|Clifford Algebra]] by [[W. M. Pezzaglia Jr.|http://www.clifford.org/~wpezzag/index.html]] under the name [[local automorphism invariance|Principle of Local Automorphism Invariance]]. As a Clifford space is a special P-space, a generalization to P-spaces is therefore suggestive.

>...gravity is that field which corresponds to a gauge invariance with respect to displacement transformations.
> - Richard Feynman -

{{center{//This is a [[Draft]] (and currently merely a sketch of ideas !)//}}}
Suppose we have a [[polyvector field|Polyvector Space]] $\mb P (\mb x)$, depending on the spacetime ($1$-vector-) coordinate $\mb x$. (A spinor field would be a special case for which $\mb P(\mb x)$ is even graded). 

Then the [[principle of local automorphism invariance|Principle of Local Automorphism Invariance]] applied to polyvector space states, that the equations governing $\mb P$ should be invariant under spacetime-translations and rotations, which means that physics should be invariant under these transformations. In other words, one has two kinds of gauge degrees of freedom, a translational and a rotational one.

This fact can be expressed in terms of a gauge invariant derivative $\mb D$, which acting upon $\mb P$ should look as follows: 
\begin{eqnarray}
\mb D \mb P (\mb x) &\equiv& \mb P' (\mb x+ \mb{dx}) - \mb P (\mb x) \\
& =& \left [ \mb P' (\mb x+ \mb{dx}) - \mb P (\mb x + \mb{dx}) \right ] + \left [ \mb P (\mb x+ \mb{dx}) - \mb P (\mb x) \right ] \\
& =& \text{rotation} + \text{translation}
\end{eqnarray}
Rotations, which take place keeping the point in spacetime fixed, correspond to classical [[Yang-Mills fields|Yang-Mills Theory]], represented by a Yang\-Mills connection. These are known as inner gauge transformations. 
Translations correspond to the gravitational (gauge) field, represented by a gravitational connection, which in the most simple case is a [[Christoffel connection|Christoffel Symbols]]. These are known as outer gauge transformations.
So, roughly speaking the covariant derivative should look like
\[
\mb D = \bs \partial + \mb A_{Y.M.} + \bs \Gamma_{grav.}
\]
Note, that a rotation is not necessarily restricted to be a spacetime-rotation, rather it can involve all grades of the polyvector. This suggests a mixing internal and external degrees of freedom which is reminiscent of what is seen in [[supersymmtry|Supersymmetry]]. 
Carrying out such a covariant derivative is therefore supposed to be more difficult and involved than is a classical (non-polyvectorial) one. 

{{center{[img(377px+, )[images/translationField.jpg]]}}}
As a spinor which is the even grade part of the polyvector is treated as a field, it seems natural to do the same with spacetime, which is treated as the grade $1$-part of the polyvector.
We therefore write the polyvector as
\begin{eqnarray}
\mb P (\mb x) = \mb{\tilde x}(\mb x)& \oplus& \bs \Psi( \mb {\tilde x} (\mb x)) \\
\text{(grade 1)}&\oplus& \text{(even grades)}
\end{eqnarray}
The "$\oplus$"-symbol indicates that we are adding different grades. 
We'll call $\mb {\tilde x} (\mb x)$ a translation field, as given a spacetime point $\mb x$, $\mb {\tilde x}$ describes a translated point in respect to $\mb x$. $\mb {\tilde x}$ parametrizes the spacetime manifold. We do not say that $\mb x$ is fixed, this would not make sense, as there is no (absolute) reference which allows one to do this (in fact this is the translational (gauge) invariance mentioned above). Hence only a relative description of points is considered. This is in the spirit of general relativity and anticipates the [[equivalence principle|Equivalence Principles]].
The covariant derivative of our polyvector then reads

\begin{eqnarray}
\frac{\mb{DP}}{\mb{Dx}}&=&\frac{\mb{D\tilde x}}{\mb{Dx}} \oplus \frac{\mb{D \bs \Psi (\tilde x)}}{\mb{D\tilde x}} \frac{\mb{D\tilde x}}{\mb{Dx}} \\
\end{eqnarray}
// ... TODO: I still have to take the pain and work that out to see if the whole concept is worth something ... //

As the product $\mb{DP}$ is a [[geometric product|Geometric Product]] there seems to be no a priori reason as to why to restrict $\mb D$ to be merely a grade-$1$ object. 

A final remark: 
The idea of a polyvector gauge theory of gravity was highly inspired by [[GTG]] and bears close resemblance to it, the main difference being that we do not assume the correct algebra to be a [[Clifford algebra|Clifford Algebra]] right away.

The ''Polyvector Klein\-Gordon Equation'' (''PKGE'') will be understood as the equation that results from the variation of the following [[polyvector action|Polyvector Action]]: 
\[
S[\Phi(\mb X), \partial_{X_\alpha} \Phi(\mb X)] \equiv M^2 \Phi(\mb X)^2 \pm \sum_\alpha (\partial_{X_\alpha} \Phi(\mb X))^2
\]
with $M$ a scalar constant and plus- and minus signs dependending on the algebra.
$\Phi(\mb X)$ is supposed to be a scalar polyvector function, i.e. $\Phi: P \rightarrow \mathbb R$ with $P$ designating polyvetor space. (One might also consider, $\Phi: P \rightarrow \mathbb C$, but we'll not do this for the moment).  

This action may be better understood as follows: 
Define the [[polyvector derivative|Polyvector Derivative]] as
\[
\bs \partial \equiv (M,\partial_x, \partial_y, \partial_z, \partial_t, \partial_{xy}, \ldots, \partial_{xyzt})
\]
then 
\[
\langle \bs \partial | \bs \partial \rangle = M^2 \pm \partial_x^2 \pm \ldots \pm \partial_{xyzt}^2 
\]
and thus 
\[
\langle \bs \partial | \bs \partial \rangle \Phi^2 = (M^2 \pm \partial_x^2 \pm \ldots \pm \partial_{xyzt}^2) \Phi^2 = M^2 \Phi^2 \pm (\partial_x \Phi)^2 \pm \ldots \pm (\partial_{xyzt} \Phi)^2
\]

We can therefore express the action in the more symmetrical form:
\[
S[\Phi(\mb X), \partial_{X_\alpha} \Phi(\mb X)] = \langle \bs \partial | \bs \partial \rangle \Phi^2 (\mb X) 
\]

Variation of $S$ leads to the [[polyvector Euler-Lagrange equations|Polyvector Euler-Lagrange Equations]]
\[
\partial_\Phi S[\Phi(\mb X), \partial_{X_\alpha} \Phi(\mb X)] - \partial_{X_\beta} \partial_{\partial_{X_\beta} \Phi(\mb X)} S[\Phi(\mb X), \partial_{X_\alpha} \Phi(\mb X)] = 0
\] 
which, using 
\[
\partial_\Phi S[\Phi(\mb X), \partial_{X_\alpha} \Phi(\mb X)] = 2 M^2 \Phi
\]
and
\[
\partial_{\partial_{X_\alpha} \Phi(\mb X)} \Phi S[\Phi(\mb X), \partial_{X_\alpha} \Phi(\mb X)]  =  \pm 2 \partial_{X_\alpha} \Phi  
\]
yields 
\[
M^2\Phi (\mb X) = \pm \sum_\alpha \partial_{X_\alpha}^2 \Phi (\mb X)   
\]
which is the polyvector Klein\-Gordon equation.

In the special case, when only the $1$-vectorial components of the partial polyvector derivative are non-zero, one recovers the classical [[Klein-Gordon equation|Klein-Gordon Equation]]. (Yet, under the assumption that the algebra is such that it "generates" the right signs). 
Contrary to the classical version which is of second order, the full PKGE is of $8^{th}$ order.
Yet, as the classical Dirac equation is the square root of the KG equation this gives us a hint as to how one might get a fourth-order equation from the PKGE. This is desirable, as [[fourth order theories|Fourth Order Theory]] are in principle [[renormalizable|Renormalization]].

As in classical mechanics and field theory we'll distinguish between a Lagrange function and a Lagrange density function in [[polyvector space|Polyvector Space]], called ''Polyvector Lagrange Function'' $L$ and ''Polyvector Lagrange Density'' $\mathcal L$ respectively. 
Both are related via 
\[
L(\bs \Psi (\mb X), \bs \partial_{\mb X} \bs \Psi (\mb X), \mb X) = \int_V \mathcal L(\bs \Psi (\mb X), \bs \partial_{\mb X} \bs \Psi (\mb X), \mb X) d V_{\mb X}
\]
where the integration is to be taken over all degrees of freedom of polyvector space. $\bs \partial_{\mb X}$ designates the [[polyvector derivative|Polyvector Derivative]]. 

If the polyvector Lagrangians only consist of $1$-vector parts, they coincide with the classical Lagrangians. 

!!!!Construction of a polyvector Lagrangian
We make the assertion: 
\[
L(\bs \Psi (\mb X), \bs \partial_{\mb X} \bs \Psi (\mb X)) \propto  \langle \bs \Psi (\mb X) | \bs \partial \bs \Psi (\mb X) \rangle 
\]
(As an aside: Notice the similarity between this expression and the fermionic part of the [[spectral action|Spectral Action]]).

The goal now is to determine the most realistic algebra underlying polyvector space and then to extract the rich physics this Lagrangian should contain. It certainly must reproduce all what is already known ([[GR|General Relativity]] and most critically, the [[standard model|Standard Model]]). 

[[Clifford algebras|Clifford Algebra]] have already shown some success in this respect but problems seem to persist (e.g. [[anomalies|Anomaly]]).

!!!!Some remarks
The polyvector field $\bs \Psi (\mb X)$ is an [[endomorphism|Homomorphism]]. In terms of physics this means that the spacetime manifold is mapped onto itself. As there is only one reality, i.e. one spacetime manifold, the polyvector field establishes a relationship among all relevant entities in spacetime coded in the polyvector. E.g. even grade polyvectors as a function of (one-)vectors describe spinor fields. Yet the polyvector field also includes the relationship between any two points (one-vectors) of the manifold. Therefore a polyvector can code nonlocality and the Lagrangian given above allows for a description of nonlocal theories.
These have a bad reputation, as they violate causality, i.e. interactions can propagate faster than the speed of light. Yet across the $20$ orders or so of "noman's land" between the shortest distances for which the the constancy of the speed of light has been tested yet and the Planck (or GUT) scale, there seems to be no reason why to believe that causality should still hold (this is at best a vague extrapolation, a bit in the spirit of the [[big desert hypothesis|Big Desert]]). See also "[[non-metricity tensor|Non-Metricity Tensor]]" for further remarks in this respect. 
As there is a clash of general relativity and quantum mechanics when it comes to very small scales, some radical modification seems to be required to allow for the unification of the two theories.
Probably the majority of physicists these days believe that general relativity is an effective theory and rather it than quantum mechanics needs an adaption. There are not too many possibilities (free parameters) to accomplish this and giving up the constancy of the speed of  light is one of them.
(Interestingly, the only extended objects that have so far been shown not to violate causality are those based on string field theory, [1], p. 262). 

A polyvector field possesses a certain symmetry: Given two "points" $\mb X$ and $\mb Y$ in polyvector space and a polyvector function $\mb Y = \mb F(\mb X)$ that relates them, and assuming that the map $\mb F$ is one-to-one, the polyvector field contains the inverse relationship as well, i.e. $\mb X = \mb F^{-1}(\mb Y) = \mb F^{-1}(\mb F(\mb X)))$.      
Concerning this fact, instead of using $\bs \Psi(\mb X)$ above it might have been better to use for example $\mb X'$, indicating the fact that $\mb X$ and $\mb X'$ belong to the same physical field. 
Yet the use of $\bs \Psi(\mb X)$ was intended to allude to the close relationship with the Dirac equation. 
We'll use $\bs \Psi$ and capital Latin letters interchangeably henceforward.

(There might be a relationship with the [[principle of event symmetry/permutation symmetry|Principle of Event Symmetry]] or with [[PT-symmetry|PT Symmetry]], although I do not really see it at the moment).

<html><center> <iframe name="content" src="http://www.markus-maute.de/trajectory/advert_60.html" width=51% height=86></iframe></center></html>
Google books: 
* [[[1] Introduction to Superstrings and M-theory - M. Kaku|http://books.google.com/books?id=9n-2O7wHnZ4C&dq=%22Introduction+to+Superstrings+and+M-Theory%22+download&printsec=frontcover&source=bn&hl=de&ei=VohrTIa9IaSTOJj76WE&sa=X&oi=book_result&ct=result&resnum=4&ved=0CCwQ6AEwAw#v=onepage&q&f=false]] {{t100Cite{[[bct. 349|http://scholar.google.de/scholar?cites=11923859208152693671&as_sdt=2005&sciodt=2000&hl=de]]}}}

In a polyvector manifold we define the ''Polyvector Path Length'' $L_{\mb A,\mb B}[\mb X]$ between the two fixed points $\mb A$ and $\mb B$ as the integral over the square root of the [[polyvector line element|Polyvector Line Element]]:
\[
L_{\mb A, \mb B}[\mb X] = \int^{\mb B}_{\mb{A}} |d\mb{\Sigma}| = \int^{\mb{B}}_{\mb A} \sqrt {G_{\alpha\beta} (\mb{X}){ {\frac {dX^\alpha}{d\sigma} \frac {dX^\beta}{d\sigma}}}} d\sigma =\int^{\mb B}_{\mb A} \sqrt {G_{\alpha\beta} (\mb X(\sigma)){P^\alpha (\sigma)P^\beta (\sigma)}} d\sigma
\]{{floatright { <html><span style="padding-left:1px"><iframe name="content" src="http://www.markus-maute.de/trajectory/advert_sky.html" width=180 height=620 scrolling="no"></iframe> </html> }}}with  $G_{\alpha\beta}(\mb{X}(\sigma))$ the [[polyvector metric tensor|Polyvector Metric Tensor]] and $P_\alpha$, $P_\beta$ components of the [[polyvector momentum|Physical Polyvectors]].
$L_{\mb A, \mb B}[\mb X]$ is a functional of the polyvector path $\mb X(\sigma)$.
All information about the "poly-curvature"' is contained in the [[polyvector metric|Polyvector Metric Tensor]]  $G_{\alpha\beta}(\sigma)$.

Variation leads to the ''Polyvector Geodesic Equation''
\[
\ddot X^\gamma - \Gamma^\gamma_{\alpha\beta}  \dot X^\alpha \dot X^\beta = 0
\]
The solution to this equation is the shortest path in polyvector space between $\mb A$ and $\mb B$.

!!!!Flat Polyvector Space
For a flat metric polyvector space, i.e. $G(\sigma) = N(\sigma)$, the polyvector path length is given by:
\[
L_{\mb{A,B}}[\mb{X}] = \int^{\mb B}_{\mb A} \sqrt {\langle d\mb  X(\sigma) \vert d\mb X(\sigma) \rangle}  = \int^{\mb B}_{\mb A}  \sqrt { \left\langle \frac {d\mb X}{d\sigma} \big | \frac {d\mb X}{d\sigma} \right\rangle }d\sigma  =  \int^{\mb B}_{\mb A} \sqrt {\langle \mb P(\sigma) \vert\mb P (\sigma) \rangle }d\sigma
\]
with $\mb P$ the [[polyvector momentum|Physical Polyvectors]].
Therefore the geodesic equation simplifies to:
\[
\dot P^\gamma  = F^\gamma = 0
\]
which means that the [[polyvector momentum|Physical Polyvectors]] is conserved and no resulting [[polyvector forces|Physical Polyvectors]] act.
The shortest path in polyvector space is a straight line. $L$ can be interpreted as the overall energy which is conserved in this case.

One can split up the polyvector momentum into its grades according to
\begin{eqnarray}
\mb P(\sigma) & =& \langle \mb P(\sigma) \rangle_0 +  \langle \mb P(\sigma) \rangle_1 +  \langle \mb P(\sigma) \rangle_2 +  \langle \mb P(\sigma) \rangle_3 +  \langle \mb P(\sigma) \rangle_4 \\
& =& \mb P(\sigma)_{rest} + \mb P(\sigma)_{kin} + \mb P(\sigma)_{rot} + ...
\end{eqnarray}
For the overall energy $E_{tot}$ one then gets (ignoring factors of the individual terms) 
\begin{eqnarray}
E_{tot} & =&  \int^{\mb B}_{\mb A} \sqrt {|\mb P_{rest}(\sigma)|^2 + |\mb P(\sigma)|^2_{kin} + |\mb P(\sigma)|^2_{rot} + \ldots} \;d\sigma  \\
& =& \int^{\mb B}_{\mb A} \sqrt {m_0^2 c^2 + p_\mu p^\mu +  S_{\mu\nu}S^{\mu\nu} + \ldots} \; d\sigma
\end{eqnarray}

Therefore the decomposition of the energy into the different degrees of freedom depends on the event $\mb \Sigma(\sigma)$ and the energy can be transformed from one degree of freedom to another one. (This can lead to the effect of a "Zitterbewegung" which is a deviation of the particle's trajectory from that of a straight line in $4$-dimensional spacetime).

The shortest path in polyvector space therefore can be interpreted as determined by the minimisation of the overall energy, that is the sum of the rest energy, the translational energy, the rotational energy, etc.

A ''Polyvector Worldline'' $\mb{X}(\sigma)$ describes a 1-dimensional curve in [[polyvector space|Polyvector Space]] which is parameterized by an affine parameter $\sigma$.

The ''Pomeransky\-Khriplovich Equations'' describe the motion of a spinning particle in an electromagnetic and gravitational field. Spin is considered as being linear and quadratic (pole-dipole approximation).

Papers:
* [[Spinning Relativistic Particles in External Fields - I.B. Khriplovich|http://arxiv.org/PS_cache/arxiv/pdf/0801/0801.1881v1.pdf]]
* [[Classical and Quantum Spins in Curved Spacetimes - A. J. Silenko|http://th-www.if.uj.edu.pl/acta/sup1/pdf/s1p0087.pdf]]   [[Transparencies|http://www.fuw.edu.pl/~krp/mathisson/wyklady/Silenko.ppt]]

The ''Poynting Theorem'' in electrodynamics states:
\begin{equation}
\vec E \cdot \vec  j + \vec \nabla \cdot \vec  S  -\frac{\partial \rho}{\partial t} = 0
\end{equation}
With $\rho$ the energy density of the electromagnetic field:
\begin{equation}
\rho = \frac{1}{2} ( \vec E  \cdot \vec D + \vec B \cdot \vec H)
\end{equation}
and $\vec S$ the Poynting vector
\begin{equation}
\vec S = \vec E \times \vec H
\end{equation}

Papers: 
* [[This Time – What a Strange Turn of Events! (2010)- P. E. Gibbs|http://prespacetime.com/index.php/pst/article/viewFile/12/9]] pct. 0

Links: 
* [[WIKIPEDIA - Event Symmetry|http://en.wikipedia.org/wiki/Event_symmetry]]

>The scene of the crime of an awful lot of stuff of speculative contemporary theoretical physics is that nobody knows how to do quantum mechanics correctly at the level of cosmology.
> - Lee Smolin Lecture at Perimeter Institure (2009): The quantization of unimodular gravity and the cosmological constant problem -

* Paradoxes (e.g. information)
* Ambiguities (e.g. Gribov)
* Crises (Spin)
* Anomalies
* Singularities (e.g. spacetime)
* Infinities
* Non\-Renormalizability
* Ghosts
* Tachyons
* Non-unitarity
* Obstruction

So there seem to be enough possiblities left to obtain a Nobel Price !
My favourite: [[Anomalies|Anomaly]].

Links:
* [[Open Issues in Fundamental Physics: The Millennium List|http://www.motionmountain.net/research/index.html#mill]]
* [[24 Questions for Elementary Physics - Sean Caroll|http://blogs.discovermagazine.com/cosmicvariance/2010/01/15/24-questions-for-elementary-physics/]]

> All is flux, nothing stays still.
> - Heraclit -

''Process Physics'' models reality as self-organising relational information and takes account of the limitations of logic, discovered by [[Gödel|Gödel's Theorems]] and extended by [[Chaitin|Gregory Chaitin]] by using the concept of self-referential noise (SNR). It provides a dynamical model where space and matter are seen to emerge from a fundamentally random but self-organising system. The system operates by forming a dissipative structure, driven by the SRN, and which is characterised by an emergent and expanding three-dimensional fractal space in which are embedded self-replicating fractal topological defects. This emergence is a non-algorithmic increase in complexity in the system. The emergent space is continually undergoing replacement of its components. The key behavioural mode for defects which are sufficiently large is that their existence, as identified by their topological properties, will survive the ongoing process of mutation, decay and regeneration; they are topologically self-replicating.
In process physics the collapse of the wavefunction finds its explanation in Gödel’s incompleteness theorem and its associated SRN within a process-system. Process physics predicts both fermionic and bosonic quantum modes, but identified as topologically encoded information and not with objects or ‘particles’. Unlike conventional quantum field theory the fermionic/bosonic modes are fractal in nature. At all levels the model exhibits evolved processes for self-replicating information.

Links:
* [[Website|http://www.scieng.flinders.edu.au/cpes/people/cahill_r/processphysics.html]]

Papers:
* [[Process Physics - R. T. Cahill|http://www.mountainman.com.au/process_physics/HPS13.pdf]] [[pct. 53|http://scholar.google.de/scholar?cites=11399556518260808129&hl=de]]
* [[Self-Referential Noise as a Fundamental Aspect of Reality - R. T. Cahill, C. M. Klinger|http://arxiv.org/PS_cache/gr-qc/pdf/9905/9905082v1.pdf]] [[pct. 30|http://scholar.google.de/scholar?cites=13250125711378012782&hl=de]]
* [[Process Physics: From Quantum Foam to General Relativity - R. T. Cahill|http://arxiv.org/PS_cache/gr-qc/pdf/0203/0203015v1.pdf]] [[pct. 23|http://scholar.google.de/scholar?cites=9301220453371540843&hl=de]]
* [[Process Physics: Modelling Reality as Self-Organising Information - R. T. Cahill, C. M. Klinger, K. Kitto|http://arxiv.org/PS_cache/gr-qc/pdf/0009/0009023v1.pdf]] [[pct. 19|http://scholar.google.de/scholar?cites=14094756012964523909&hl=de]]

Articles:
* [[Random Reality - M. Chown|http://www.newscientist.com/article/mg16522274.300-random-reality.html]]

Papers:
* [[Projective Relativity: Present Status and Outlook - B. Fauser|http://arxiv.org/PS_cache/gr-qc/pdf/0011/0011015v1.pdf]]

Papers:
* [[Quark Confinement without a Confining Force - P.S. Isaac, W.P. Joyce, J. Links|http://arxiv.org/PS_cache/hep-th/pdf/0307/0307047v1.pdf]]
* [[Transposition in Clifford Algebra: SU(3) from Reorientation Invariance - B. Schleikal|http://books.google.com/books?id=b6mbSCv_MHMC&pg=PA351&lpg=PA351&dq=%22SU(3)+from+reorientation%22&source=web&ots=KgA_AVmXgo&sig=e-CpIyd8scGsWmPPO05a03APHdM&hl=de&sa=X&oi=book_result&resnum=5&ct=result]] - SU(3) and the spacetime algebra.
*[[Gauge Fields with Unified Weak, Electromagnetic, and Strong Interactions - G. 't Hooft|http://www.phys.uu.nl/~thooft/gthpub/GaugeFields_75.pdf]]

!!!! Canonical Quantization
The canonical quantisation problem in physics consists of a commutative algebra of functions equipped with a Poisson bracket and the search for a [[noncommutative algebra|Noncommutative Geometry]] with [[commutators|Commutator]] reproducing this to lowest order in a deformation parameter.
It is well known that actually the converse problem is more well posed: Given a noncommutative algebra which is a flat deformation one may recover its semiclassical structure and Poisson bracket of which it is a quantisation.
Either way Poisson brackets are the semiclassical data for associative noncommutative algebras.
!!!! [[Path Integral Quantization|Feynman Path Integral]]
!!!! [[Stochastic Quantization|Stochastic Quantization]]

Books:
* [[Non-associative Algebra and its Applications (chap. 17) - L. V. Sabinin, L. Sbitneva, I. P. Shestakov|books/LevSabininLarissaSbitnevaIvanShestakov_Non-AssociativeAlgebraAndItsApplications.pdf]]

The idea of quantum computation is generally attributed to Feynman who proposed a computational scheme based on quantum mechanical laws.

Papers:
* [[Simulating Physics with Computers - R. P. Feynman|http://www.gdi.uni-bamberg.de/teaching/SS09/GdI-Seminar/Papers/feynman82.pdf]] {{t1000Cite{[[pct. 1755|http://scholar.google.de/scholar?cites=10169512770508744313&hl=de&as_sdt=2000]]}}}
* [[On the Power of Quantum Computation - D. R. Simon|http://www.hep.princeton.edu/~mcdonald/examples/QM/simon_siamjc_26_1474_97.pdf]]  {{t500Cite{[[pct. 570|http://scholar.google.de/scholar?cites=15888631324782233622&hl=de&as_sdt=2000]]}}}
* [[Looking at Nature as a Computer - N. Margolus|http://people.csail.mit.edu/nhm/looking-at-nature.pdf]] [[pct. 9|http://scholar.google.de/scholar?cites=11943823166538797011&hl=de]]

>It seems almost inevitable that in quantum gravity space will be modified or cut off at short distance. One can imagine many sorts of modification to the structure of space, but it is much harder to alter the nature of time in a consistent way.
> - J. Polchinski -

At present no perturbative [[renormalizable|Renormalization]] and [[unitary|Unitarity]] theory of quantum gravity seems to be known. All suggested metric models of gravity are nonrenormalizable or nonunitary. The known $N = 1$ [[supergravity|Supergravity]] models are finite up to two loops but may generate nonvanishing three-loop divergent counterterms. Models with extended [[supersymmetry|Supersymmetry]] (e.g. $N = 8$) or some other additional symmetry (e.g. local [[conformal symmetry|Conformal Group]]) have better renormalization features, but there is no proof of their complete finiteness by now.

Papers:
* [[How far are we from the Quantum Theory of Gravity? (2003) - L. Smolin|http://users.sch.gr/papostol/downloads/0303185.pdf]] {{t100Cite{[[pct. 143|http://scholar.google.de/scholar?cites=2019903338763155264&as_sdt=2005&sciodt=2000&hl=de]]}}}
* [[Quantum Gravity Phenomenology (2008) - G. Amelino-Camelia|http://arxiv.org/PS_cache/arxiv/pdf/0806/0806.0339v1.pdf]] [[pct. 90|http://scholar.google.de/scholar?cites=17561921306899221093&hl=de&as_sdt=2000]]
* [[The Phase Space Structure of Multi-particle Models in 2+1 Gravity - H.-J. Matschull|http://arxiv.org/PS_cache/gr-qc/pdf/0103/0103084v1.pdf]] [[pct. 17|http://scholar.google.de/scholar?cites=6556301508943061997&hl=de&as_sdt=2000]]
* [[Algebraic Approach to Quantum Gravity II: Noncommutative Spacetime - S. Majid|http://arxiv.org/PS_cache/hep-th/pdf/0604/0604130v1.pdf]] [[pct. 8|http://scholar.google.de/scholar?cites=10496328941943801231&as_sdt=2005&sciodt=2000&hl=de]]
* [[Is Quantum Gravity Necessary? - J. Mattingly|http://jmattingly.org/papers/IsGravityNecessarilyQuantized.proofs.pdf]] [[pct. 7|http://scholar.google.de/scholar?cites=14015950769707842957&hl=de&as_sdt=2000]]
* [[Algebraic Approach to Quantum Gravity III: Noncommmutative Riemannian Geometry - S. Majid|http://arxiv.org/PS_cache/hep-th/pdf/0604/0604132v1.pdf]]  [[pct. 3|http://scholar.google.de/scholar?cites=12492913097235504353&as_sdt=2005&sciodt=2000&hl=de]]
* [[Algebraic Approach to Quantum Gravity I: Relative Realism - S. Majid|http://philsci-archive.pitt.edu/archive/00003345/01/qg1.pdf]] [[pct. 2|http://scholar.google.de/scholar?cites=14435033245775806038&hl=de&as_sdt=2000]]
* [[Quantum Gravity and Jordan's Non-associative Algebras - H.-J. Treder|http://www.springerlink.com/content/l9577l8133n34146/]] pct. 0

Books: 
* [[The Structural Foundations of Quantum Gravity (2006) - D. Rickles, S. French, J. Saatsi|books/DeanRicklesStevenFrenchJuahSaatsi_TheStructuralFoundationsOfQuantumGravity.pdf]] [[bct.12|http://scholar.google.de/scholar?cites=12240841731844568339&as_sdt=2005&sciodt=2000&hl=de]]
* [[Approaches to Quantum Gravity (2008) - D. Oriti|books/DanieleOriti_ApproachesToQuantumGravity.pdf]] [[bct. 9|http://scholar.google.de/scholar?cites=6228387390848420882&as_sdt=2005&sciodt=2000&hl=de]]

See [[Hopf Algebra]].

According John von Neumann one has the following interpretations/correspondences: 
* [[Idempotents|Idempotency]] = properties,
*  algebraic elements = observables,
* spectral interval = smallest interval containing all possible values of the observable,
* commutativity = simultaneous observability.

In the axiomatic approach to quantum mechanics, the event structure of a physical system is identified with a quantum logic or an orthoalgebra, whereas in classical mechanics one is dealing with a [[Boolean algebra|Boolean Algebra]].

See also: [[Measurement algebra|Measurement Algebra]].

Google books:
* [[The Structure and Interpretation of Quantum Mechanics - R. I. G. Hughes|http://books.google.com/books?id=Z5_awfuGzC8C&pg=PA220&lpg=PA220&dq=orthoalgebra&source=bl&ots=CqRiTZwd9U&sig=piq39VTu1xQ8kA6otB-22MzcuyQ&hl=de&ei=UabeSszhM4GTsAamna2nDg&sa=X&oi=book_result&ct=result&resnum=2&ved=0CA4Q6AEwAQ#v=onepage&q=orthoalgebra&f=false]] {{t500Cite{[[bct. 254|http://scholar.google.de/scholar?cites=6010492491240308053&as_sdt=2005&sciodt=2000&hl=de]]}}}
* [[The Neumann Compendium - J. Von Neumann, F. Bródy, T. Vámos|http://books.google.de/books?hl=de&lr=&id=MY2_V2BfP5cC&oi=fnd&pg=PA103&ots=Z81miUS_EE&sig=-i-fmhYrcBSosjvPWLu_ofdQIvQ#v=onepage&q=&f=false]] [[bct. 2|http://scholar.google.de/scholar?cites=5161878676174486680&as_sdt=2005&sciodt=2000&hl=de]]

Videos: 
* [[Single electrons build up interference pattern|http://www.youtube.com/watch?v=ZUI3lhRje_0]]

''Quasicrystals'' are structures that fill all the space but lack translational symmetry. Classical theory of crystals allows only for $2-$, $3-$, $4-$, and $6-$ fold rotational symmetries whereas quasicrystals display symmetry other than these. They can be said to be in a state intermediate between crystal and glass.

The observation of quasicrystalline structures was first reported by Dan Shechtman et al. in 1984.

Mathmatically, quasicristals can be described by means of [[aperiodic tilings|Tiling]].

!!!!Examples:
Quasicrystals have been discovered in approximately 100 synthetic intermetallic compounds. Scores of binary and ternary metallic alloys are known to form quasicrystalline phases - mostly with icosahedral or decagonal point-group symmetry.

Verba volant, scripta manent.
----
A good part of science is distinguishing between useful crazy ideas and those that are just plain nutty. - Princeton University book advertisement -
----
An expert is a person who has made all the mistakes that can be made in a very narrow field. - Niels Bohr -
----
The ordinary man wonders at marvellous things; the wise man wonders at the commonplace. - Confucius -
----
I hear and I forget. I see and I remember. I do and I understand - Confucius -
----
One who learns but does not think, is lost. One who thinks but does not learn is in great danger. - Confucius -
----
Nothing in life is to be feared, it is only to be understood. Now is the time to understand more, so that we may fear less. - Marie Curie -
----
Any fool can know. The point is to understand. - Albert Einstein -
----
Not everything that can be counted counts, and not everything that counts can be counted. - Albert Einstein -
----
Everything should be made as simple as possible, but not simpler. - Albert Einstein -
----
Solange man jung ist, gehören alle Gedanken der Liebe - später gehört alle Liebe den Gedanken. - Albert Einstein -
----
Geniale Menschen sind selten ordentlich, ordentliche selten genial. - Albert Einstein -
----
Reading, after a certain age, diverts the mind too much from its creative pursuits. Any man who reads too much and uses his own brain too little falls into lazy habits of thinking. - Albert Einstein -
----
Only two things are infinite, the universe and human stupidity, and I'm not sure about the former. - Albert Einstein -
----
Strive not to be a success, but rather to be of value. - Albert Einstein -
----
Only those who will risk going too far can possibly find out how far one can go. - Thomas Stearns Eliot -
----
I wonder why I wonder why. I wonder why I wonder. I wonder why I wonder why I wonder why I wonder! - Richard Feynman -
----
I think I can safely say that nobody understands quantum mechanics. - Richard Feynman -
----
Even if there is only one possible unified theory, it is just a set of rules and equations. What is it that breathes fire into the equations and makes a universe for them to describe? - Stephen Hawking -
----
When I hear of Schrödinger's cat, I reach for my gun. - Stephen Hawking -
----
Der Horizont vieler Menschen ist ein Kreis mit Radius Null - und das nennen sie ihren Standpunkt. - David Hilbert -
----
Physics is much too hard for physicists. - David Hilbert -
----
The whole purpose of physics is to find a number, with decimal points, etc! Otherwise you haven't done anything. - Richard Feynman -
----
Every scientific statement must remain tentative for ever. - Karl Raimund Popper -
----
When all else fails, you can always tell the truth. - Abdus Salam -
----
Die Grenzen meiner Sprache bedeuten die Grenzen meiner Welt. - Ludwig Wittgenstein -
----
Nicht wie die Welt ist, ist das Mystische, sondern dass sie ist. - Ludwig Wittgenstein -
----
Wovon man nicht sprechen kann, darüber muss man schweigen. - Ludwig Wittgenstein -
----
The effort to understand the universe is one of the very few things that lifts human life a little above the level of farce, and gives it some of the grace of tragedy. - Steven Weinberg -
----
Weʹre all lying in the gutter, but some of us are gazing at the stars. - Oscar Wilde -
----
The whole purpose of science is to find meaningful simplicity in the midst of disorderly complexity - Herbert Simon -
----
The unreasonable effectiveness of mathematics in the natural sciences - Eugene Wigner -
----
Science is a wonderful thing if one does not have to earn one's living at it - Albert Einstein -
----
As far as I see, all a priori statements in physics have their origin in symmetry. - Hermann Weyl -
----
Number rules the universe. - Pythagoras -
----
Use examples; that such as thou teachest may understand thee the better! - Pythagoras -
----
A fool is known by his Speech; and a wise man by Silence. - Pythagoras -
----
Time is the soul of this world. - Pythagoras -
----
The oldest, shortest words - "yes" and "no" - are those which require the most thought. - Pythagoras -
----
You cannot teach a man anything; you can only help him discover it in himself. - Galileo Galilei -
----
The great book of nature can be read only by those who know the language in which it was written. And this language is mathematics. - Galileo Galilei -
----
All is flux, nothing stays still. - Heraclit -
----
Nothing endures but change. - Heraklit -
----
Enthusiasm is followed by disappointment and even depression, and then by renewed enthusiasm. - Murray Gell-Mann -
----
Available energy is the main object at stake in the struggle for existence and the evolution of the world. - Ludwig Boltzmann -
----
What then is time? If no one asks me, I know what it is. If I wish to explain it to him who asks, I do not know. - Saint Augustine -
----
Omnibus ex nihil decendis sufficit unum. (One suffices to derive all out of nothing.) - Gottfried Leibniz
----
Never make a calculation until you know the answer. - John Archibald Wheeler
----
The most powerful method of advance is to perfect and generalize the mathematical formalism that forms the existing basis of theoretical physics. - Paul Adrien Maurice Dirac -
----
The notion of existence is one of the primitive concepts with which we must begin as given. It is the clearest concept we have. - Kurt Gödel
----
Numquam ponenda est pluralitas sine necessitate. - William of Ockham
----
If A equals success, then the formula is A = X + Y + Z. X is work. Y is play. Z is keep your mouth shut. - Albert Einstein -
----
Ich behaupte aber, dass in jeder besonderen Naturlehre nur so viel eigentliche Wissenschaft angetroffen werden könne, als darin Mathematik anzutreffen ist. - Immanuel Kant -
----
Mathemata mathematicis scribuntur (mathematics is written for mathematicians) - Nicolaus Copernicus -
----
The art of doing mathematics consists in finding that special case which contains all the germs of generality. - David Hilbert -
----
Fundamental concepts are rare. - Shiing Shen Chern -
----
If I could explain it to the average person, I wouldn't have been worth the Nobel Prize. - Richard Feynman -
----
If you haven't found something strange during the day, it hasn't been much of a day. - Archibald Wheeler -
----
Publish or perish. - J. C. Polkinghorne -
----
Shut up and calculate! - David Mermin -
----
The most incomprehensible thing about the world is that it is comprehensible. - Albert Einstein -

/***
|Name|QuoteOfTheDayPlugin|
|Source|http://www.TiddlyTools.com/#QuoteOfTheDayPlugin|
|Documentation|http://www.TiddlyTools.com/#QuoteOfTheDayPluginInfo|
|Version|1.4.1|
|Author|Eric Shulman|
|License|http://www.TiddlyTools.com/#LegalStatements <br>and [[Creative Commons Attribution-ShareAlike 2.5 License|http://creativecommons.org/licenses/by-sa/2.5/]]|
|~CoreVersion|2.1|
|Type|plugin|
|Requires||
|Overrides||
|Description|Display a randomly selected "quote of the day" from a list defined in a separate tiddler|

!!!!!Documentation
>see [[QuoteOfTheDayPluginInfo]]
!!!!!Revisions
<<<
2008.03.21 [1.4.1] in showNextItem(), corrected handling for random selection so that //initial// index value will randomized correctly instead of always showing first item, even when randomizing.  Thanks to Riccardo Gherardi for finding this.
| Please see [[QuoteOfTheDayPluginInfo]] for previous revision details |
2005.10.21 [1.0.0] Initial Release.  Based on a suggestion by M.Russula
<<<
!!!!!Code
***/
//{{{
version.extensions.QuoteOfTheDayPlugin= {major: 1, minor: 4, revision: 1, date: new Date(2008,3,21)};
config.macros.QOTD = {
	clickTooltip: "click to view another item",
	timerTooltip: "auto-timer stopped...  'mouseout' to restart timer",
	timerClickTooltip: "auto-timer stopped...  click to view another item, or 'mouseout' to restart timer",
	handler:
	function(place,macroName,params) {
		var tid=params.shift(); // source tiddler containing HR-separated quotes
		var p=params.shift();
		var click=true; // allow click for next item
		var inline=false; // wrap in slider for animation effect
		var random=true; // pick an item at random (default for "quote of the day" usage)
		var folder=false; // use local filesystem folder list
		var cookie=""; // default to no cookie
		var next=0; // default to first item (or random item)
		while (p) {
			if (p.toLowerCase()=="noclick") var click=false;
			if (p.toLowerCase()=="inline") var inline=true;
			if (p.toLowerCase()=="norandom") var random=false;
			if (p.toLowerCase().substr(0,7)=="cookie:") var cookie=p.substr(8);
			if (!isNaN(p)) var delay=p;
			p=params.shift();
		}
		if ((click||delay) && !inline) {
			var panel = createTiddlyElement(null,"div",null,"sliderPanel");
			panel.style.display="none";
			place.appendChild(panel);
			var here=createTiddlyElement(panel,click?"a":"span",null,"QOTD");
		}
		else
			var here=createTiddlyElement(place,click?"a":"span",null,"QOTD");
		here.id=(new Date()).convertToYYYYMMDDHHMMSSMMM()+Math.random().toString(); // unique ID
		// get items from tiddler or file list
		var list=store.getTiddlerText(tid,"");
		if (!list||!list.length) { // not a tiddler... maybe an image directory?
			var list=this.getImageFileList(tid);
			if (!list.length) { // maybe relative path... fixup and try again
				var h=document.location.href;
				var p=getLocalPath(decodeURIComponent(h.substr(0,h.lastIndexOf("/")+1)));
				var list=this.getImageFileList(p+tid);
			}
		}
		if (!list||!list.length) return false; // no contents... nothing to display!
		here.setAttribute("list",list);
		if (delay) here.setAttribute("delay",delay);
		here.setAttribute("random",random);
		here.setAttribute("cookie",cookie);
		if (click) {
			here.title=this.clickTooltip
			if (!inline) here.style.display="block";
			here.setAttribute("href","javascript:;");
			here.onclick=function(event)
				{ config.macros.QOTD.showNextItem(this); }
		}
		if (config.options["txtQOTD_"+cookie]!=undefined) next=parseInt(config.options["txtQOTD_"+cookie]);
		here.setAttribute("nextItem",next);
		config.macros.QOTD.showNextItem(here);
		if (delay) {
			here.title=click?this.timerClickTooltip:this.timerTooltip
			here.onmouseover=function(event)
				{ clearTimeout(this.ticker); };
			here.onmouseout=function(event)
				{ this.ticker=setTimeout("config.macros.QOTD.tick('"+this.id+"')",this.getAttribute("delay")); };
			here.ticker=setTimeout("config.macros.QOTD.tick('"+here.id+"')",delay);
		}
	},
	tick: function(id) {
		var here=document.getElementById(id); if (!here) return;
		config.macros.QOTD.showNextItem(here);
		here.ticker=setTimeout("config.macros.QOTD.tick('"+id+"')",here.getAttribute("delay"));
	},
	showNextItem:
	function (here) {
		// hide containing slider panel (if any)
		var p=here.parentNode;
		if (p.className=="sliderPanel") p.style.display = "none"
		// get a new quote
		var index=here.getAttribute("nextItem");
		var items=here.getAttribute("list").split("\n----\n");
		if (index<0||index>=items.length) index=0;
		if (here.getAttribute("random")=="true") index=Math.floor(Math.random()*items.length);
		var txt=items[index];
		// re-render quote display element, and advance index counter
		removeChildren(here); wikify(txt,here);
		index++; here.setAttribute("nextItem",index);
		var cookie=here.getAttribute("cookie");
		if (cookie.length) {
			config.options["txtQOTD_"+cookie]=index.toString();
			saveOptionCookie("txtQOTD_"+cookie);
		}
		// redisplay slider panel (if any)
		if (p.className=="sliderPanel") {
			if(anim && config.options.chkAnimate)
				anim.startAnimating(new Slider(p,true,false,"none"));
			else p.style.display="block";
		}
	},
	getImageFileList: function(cwd) { // returns HR-separated list of image files
		function isImage(fn) {
			var ext=fn.substr(fn.length-3,3).toLowerCase();
			return ext=="jpg"||ext=="gif"||ext=="png";
		}
		var files=[];
		if (config.browser.isIE) {
			cwd=cwd.replace(/\//g,"\\");
			// IE uses ActiveX to read filesystem info
			var fso = new ActiveXObject("Scripting.FileSystemObject");
			if(!fso.FolderExists(cwd)) return [];
			var dir=fso.GetFolder(cwd);
			for(var f=new Enumerator(dir.Files); !f.atEnd(); f.moveNext())
				if (isImage(f.item().path)) files.push("[img[%0]]".format(["file:///"+f.item().path.replace(/\\/g,"/")]));
		} else {
			// FireFox (mozilla) uses "components" to read filesystem info
			// get security access
			if(!window.Components) return;
			try { netscape.security.PrivilegeManager.enablePrivilege("UniversalXPConnect"); }
			catch(e) { alert(e.description?e.description:e.toString()); return []; }
			// open/validate directory
			var file=Components.classes["@mozilla.org/file/local;1"].createInstance(Components.interfaces.nsILocalFile);
			try { file.initWithPath(cwd); } catch(e) { return []; }
			if (!file.exists() || !file.isDirectory()) { return []; }
			var folder=file.directoryEntries;
			while (folder.hasMoreElements()) {
				var f=folder.getNext().QueryInterface(Components.interfaces.nsILocalFile);
				if (f instanceof Components.interfaces.nsILocalFile)
					if (isImage(f.path)) files.push("[img[%0]]".format(["file:///"+f.path.replace(/\\/g,"/")]));
			}
		}
		return files.join("\n----\n");
	}
}
//}}}

The ''Randall\-Sundrum Model'' (also called ''5-dimensional Warped Geometry Theory'') is a [[brane world scenario|Brane World Scenario]] based on the metric
\begin{equation}
ds^2=e^{-2kR\phi}g_{\mu\nu}dx^\mu dx^\nu-R^2 d\phi^2,
\end{equation}
where $k$ is a scale on the order of the Planck scale, $g_{\mu\nu}$ is the [[metric tensor|Metric Tensor]] of the $4$-dimensional subspace of this $5$-dimensional warped spacetime, and $\phi\in[0,\pi]$ is the coordinate for an extra dimension of size $R$.

There are two popular models. The first, called \RS1, has a finite size for the extra dimension with two branes, one at each end. The second, \RS2, is similar to the first, but one brane has been placed infinitely far away, so that there is only one brane left in the model.

The Randall\-Sundrum scenario can also be obtained from a [[spectral action|Spectral Action]] in [[noncommutative geometry|Noncommutative Geometry]].
This is achieved by making the mass scale appearing in the Dirac operator dynamical by replacing it with a [[dilaton field|Dilaton]]. In this case the spectral action becomes almost scale invariant and gives the same low energy limit as the Randall\-Sundrum model (and provides a model for extended [[inflation|Inflation]]). 

Links:
* [[WIKIPEDIA - Randall-Sundrum-Modell|http://de.wikipedia.org/wiki/Randall-Sundrum-Modell]]

Links:
* [[WIKIPEDIA - Raychaudhuri Equation|http://en.wikipedia.org/wiki/Raychaudhuri_equation]]

/***
|Name|RecentChangesPlugin|
|Source|http://www.TiddlyTools.com/#RecentChangesPlugin|
|Version|2.1.0|
|Author|Eric Shulman - ELS Design Studios|
|License|http://www.TiddlyTools.com/#LegalStatements <br>and [[Creative Commons Attribution-ShareAlike 2.5 License|http://creativecommons.org/licenses/by-sa/2.5/]]|
|~CoreVersion|2.1|
|Type|plugin|
|Requires||
|Overrides||
|Description|display droplist of recently changed tiddlers with goto, edit, and preview buttons|
!!!!!Usage
<<<
The {{{<<recentChanges>>}}} macro displays a droplist of all tiddlers that have been changed within the last N days (default=10 days).
<<<
!!!!!Examples
<<<
{{smallform{
{{{<<recentChanges>>}}}
><<recentChanges>>
or
{{{<<recentChanges #ofdays summary>>}}}
>where:
>* #ofdays specifies the time limit for list changed tiddlers.  Use 0 (zero) to list all tiddlers in the document
>* "summary" is a keyword that outputs only the summary text (without the droplist or buttons)
>{{{<<recentChanges 14 summary>>}}}
><<recentChanges 14 summary>>
or
{{{<<recentChanges #ofdays previewheight previewclass>>}}}
>where:
>* #ofdays specifies the time limit for list changed tiddlers.  Use 0 (zero) to list all tiddlers in the document
>* previewheight is a CSS height measurement and sets the FIXED height of the tiddler preview area (default is 15em)
>* previewclass is any CSS classname, and can be used to apply custom styles to the preview area (default is to use the standard 'viewer' class)
>{{{<<recentChanges 14 10em groupbox>>}}}
><<recentChanges 14 10em groupbox>>
}}}
<<<
!!!!!Revisions
<<<
2008.07.01 [2.1.0] added optional "summary" keyword for simply text output
2008.05.01 [2.0.1] fixup for titles with double-quotes
2007.07.26 [2.0.0] re-written as plugin
2006.10.02 [1.0.0] initial release (as inline script ShowRecentChanges)
<<<
!!!!!Code
***/
//{{{
version.extensions.RecentChangesPlugin= {major: 2, minor: 1, revision: 0, date: new Date(2008,7,1)};

config.shadowTiddlers.RecentChanges="<<recentChanges>>";

config.macros.recentChanges = {
	layout: '<form><!--\
		--><select size=1 name="list" style="width:69.5%" \
			onchange=" \
				this.form.goto.disabled=this.form.edit.disabled=this.form.preview.disabled=!this.value.length; \
				var target=this.parentNode.parentNode.nextSibling; removeChildren(target); \
				if (!this.value.length) \
					{ target.style.display=\'none\'; this.form.preview.value=\'preview\'; } \
				else if (target.style.display==\'block\') { \
					wikify(\'<\'+\'<tiddler [[\'+this.value+\']]>\'+\'>\',target); \
					target.style.display=\'block\'; \
					this.form.preview.value=\'done\'; \
				} \
			"><!--\
		-->%options%<!--\
		--></select><!--\
		--><input type="button" name="goto" value="goto" disabled title="view selected tiddler" style="width:10%" \
			onclick="var target=this.parentNode.parentNode.nextSibling; removeChildren(target); \
				target.style.display=\'none\'; this.form.preview.value=\'preview\'; \
				story.displayTiddler(story.findContainingTiddler(this),this.form.list.value); \
			"><!--\
		--><input type="button" name="edit" value="edit" disabled title="edit selected tiddler" style="width:10%" \
			onclick="var target=this.parentNode.parentNode.nextSibling; removeChildren(target); \
				target.style.display=\'none\'; this.form.preview.value=\'preview\'; \
				story.displayTiddler(story.findContainingTiddler(this),this.form.list.value,DEFAULT_EDIT_TEMPLATE); \
			"><!--\
		--><input type="button" name="preview" value="preview" disabled title="show/hide tiddler preview" style="width:10%" \
			onclick="var target=this.parentNode.parentNode.nextSibling; \
				if (this.value==\'preview\') { \
					removeChildren(target); \
					wikify(\'<\'+\'<tiddler [[\'+this.form.list.value+\']]>\'+\'>\',target); \
					target.style.display=this.form.list.value.length?\'block\':\'none\'; this.value=\'done\'; \
				} else { \
					removeChildren(target); \
					target.style.display=\'none\'; this.value=\'preview\'; \
				} \
			"><!--\
		--></form>',
	handler: function(place,macroName,params,wikifier,paramString,tiddler) {
		var days=10; if (!isNaN(params[0])) days=parseInt(params[0]); // time limit in days (use 0 for all tiddlers)
		var summary=params[1]&&params[1].toLowerCase()=="summary"; if (summary) params.shift();
		var height='15em'; if (params[1]) height=params[1]; // preview area fixed height
		var previewclass='viewer'; if (params[2]) previewclass=params[2]; // preview area CSS class
		var tiddlers=store.getTiddlers('modified','excludeLists').reverse();
		var count=tiddlers.length;
		if (days) {
			var timelimit=(new Date()).getTime()-86400000*days;
			for (var count=0; count<tiddlers.length && tiddlers[count].modified>timelimit; count++);
		}
		var s=count+' tiddlers have changed since ';
		s+=new Date(timelimit).formatString("DDD, MMM DDth YYYY 0hh:0mm");
		s+=' ('+days+' days ago)';
		if (summary)
			{ wikify(s,place); return; }
		var opts='<option value="">'+s+'</option>';
		for (var i=0; i<count; i++) { var t=tiddlers[i];
			opts+='<option value="'+t.title.replace(/"/g,"&#x22;")+'">';
			opts+=t.modified.formatString('YYYY.0MM.0DD 0hh:0mm')+' - '+t.title;
			opts+='</option>';
		}
		createTiddlyElement(place,"div").innerHTML=this.layout.replace(/%options%/,opts);
		var preview=createTiddlyElement(place,"div",null,previewclass);
		preview.style.display='none';
		preview.style.whiteSpace='normal';
		preview.style.overflow='auto';
		preview.style.height=height;
	}
}
//}}}

''Renormalization'' originated not from abstract theory but rather from the struggle to overcome a severe technical problem: If one supposes that spacetime is a continuum, then in any finite volume of space there is an infinite number of degrees of freedom, and in summing their contributions to physical processes one often finds divergent, and hence meaningless results. Renormalization originated as a technical trick to absorb these divergences into redefinitions of the couplings: it relates so called "bare" couplings, which appear in the fundamental Lagrangian and have no direct physical significance, to "renormalized" couplings, which correspond to what one actually measures in the laboratory.

Later on, thanks largely to the work of Wilson, renormalization came to be understood in more general terms: 
Given a system consisting of a large number of oscillators with different frequencies ω i. When one deals with a problem which is characterized by some energy scale $E$, one cannot directly excite the oscillators whose energy levels are higher than $E$. Nevertheless, the presence of those degrees of freedom affects low energy physics: through vacuum polarization effects, they change the effective values of the charges. In a functional integral, one can compute the effective charges as coefficients in an ''Effective Action'' which is obtained by “integrating out” all the degrees of freedom with energies larger than $E$. Consequently, the observed (renormalized) strength of the interaction between two particles will depend on the energy of the interacting particles.
Although the formal definition of the effective action as the result of a functional integration inevitably involves the regularization of divergent quantities, the difference between two Wilsonian effective actions associated to two energy scales is finite, because it involves only a finite range of momenta.
One could take the attitude that since only renormalized quantities can be measured, it is never necessary to talk about the bare action, nor about a UV regulator.

Papers:
* [[Gravity from a Particle Physicists’ Perspective| Roberto Percacci|http://pos.sissa.it/archive/conferences/081/011/ISFTG_011.pdf]] pct. 0

> Of all the physicists I have known, Feynman impressed me most with the simplicity and elegance with which he articulates and solves complicated problems.
> - Hagen Kleinert -

> I hope you understand the simple examples, because if you do, you'll understand all of the generalities right away. At least that's the way I understand things. 
> - Richard Feynman [1] - 

Videos: 
* [[[1] Elementary Particles and the Laws of Physics|http://video.google.com/videoplay?docid=7108406426776765294&hl=de&emb=1#docid=-8958142021831702044]]
* [[Richard Feynman - The Relation of Mathematics & Physics. Part 1|http://www.youtube.com/watch?v=1SrHzSGn-I8]]
* [[Feynman: Take the World from Another Point of View|http://www.youtube.com/watch?v=PsgBtOVzHKI&feature=related]]
* [[Richard Feynman on Quantum Mechanics - Part 1 - Photons: Corpuscles of Light|http://video.google.com/videoplay?docid=-6203969264735574264&amp;q=marcolli&total=9&start=0&num=10&so=0&type=search&amp;plindex=6#docid=1501838765715417418]]

In the following robotic observatories are listed that offer access to the general public.


!!![[SLOOH|http://www.slooh.com/]] {{floatright { <html><iframe name="content" src="http://www.markus-maute.de/trajectory/advert_sky.html" width=180 height=620 scrolling="no"></iframe> </html> }}} 

{{center{[img(370px+, )[http://www.slooh.com/webcams/obs_teide.jpg]]}}}

Examples:
* [[SLOOH Australia, Macedon Ranges Observatory, 15. 01. 2010|http://www.waa.at/bericht/2010/01/20100115api14.html]]
* [[Deep Sky und Komet Lulin Observatorio del Teide, Teneriffa, 22. 02. 2009|http://www.waa.at/bericht/2009/02/20090222api21.html]]
* [[MySlooh.com - Dave M.|http://www.myslooh.com/DaveM]]
* [[David Mihalic's Astronomy Site|http://www.astro.lightarts.net/]]

!!![[Bradford Robotic Telescope|http://www.telescope.org/]]

{{center{[img(370px+, )[http://www.telescope.org/webcam-tn-observatorycam-i.php]][img(370px+, )[http://www.telescope.org/webcam-tn-ogscam-i.php]]}}}

!!![[Micro-Observatory|http://mo-www.cfa.harvard.edu/MicroObservatory/]]

!!![[Global Rent-a-Scope|http://www.global-rent-a-scope.com/]]
* [[Life Images|http://gis.global-rent-a-scope.com/imagePreview.aspx]]
* [[Global Rent a Scope Demonstration|http://www.youtube.com/watch?v=Um_YKprUKgE]]

!!![[LightBuckets|http://www.lightbuckets.com/index.php]]

Links:
* [[Window to the Universe|http://www.markus-maute.de/universe/WindowToTheUniverse.html]] - My own snap-shots with robotic telescopes.
{{center{[img(77px+, )[http://www.markus-maute.de/universe/pictures/M20.jpg]][img(79px+, )[http://www.markus-maute.de/universe/pictures/M13.jpg]][img(98px+, )[http://www.markus-maute.de/universe/pictures/M17.jpg]]}}}
* [[The Night Sky Live|http://nightskylive.net/index.php]]
* [[Four Online Telescopes Serve the Stars to Interstellar Paparazzi - P. di Just|http://www.wired.com/science/discoveries/magazine/16-03/st_telescopes]]
* [[Internet Observatorien: Astrophotographie unter besten Bedingungen - K.-O. Detken|http://www.detken.net/papers/HIPO_0209_Astrophotografie.pdf]]
* [[STELLA - Links|http://www.aip.de/stella/index.php?id=links]]

> I do not believe that a real understanding of the nature of elementary particles can ever be achieved without a simultaneous deeper understanding of the nature of spacetime itself.
> - [[Roger Penrose|http://pipl.com/search/?FirstName=roger&LastName=penrose&City=&State=&Country=&CategoryID=2&Interface=2]] -

Videos:
* [[Are We Due for a New Revolution in Fundamental Physics? (2004)|http://www.perimeterinstitute.ca/index.php?option=com_content&task=view&id=318&Itemid=81&lecture_id=3292]]
* [[Princeton Lectures - Fashion, Faith and Fantasy in the New Physics of the Universe (3 lectures, 2003) |http://www.princeton.edu/WebMedia/lectures/]]

Books:
* [[The Road to Reality - A Complete Guide to the Laws of the Universe|books/RogerPenrose_TheRoadToRealityACompleteGuideToTheLawsOfTheUniverse.pdf]]  {{t100Cite{[[bct. 214|http://scholar.google.de/scholar?cites=2893587437361748042&hl=de]]}}}

The ''Scattering Matrix'' (or ''S-matrix'') relates the initial state and the final state of a physical system undergoing a scattering process.
More formally, the S-matrix $S$ is defined as the [[unitary|Unitarity]] matrix (i.e. $S^ \dagger S = 1$) connecting asymptotic particle states $\Psi$ in the [[Hilbert space|Hilbert Space]] of physical states (scattering channels), i.e. $\Psi(\infty) = S \Psi(-\infty)$.

While the S-matrix may be defined for any background (spacetime) that is asymptotically solvable and has no horizons, it has a simple form in the case of the Minkowski space where the Hilbert space is a space of irreducible unitary representations of the [[inhomogeneous Lorentz group|Poincaré Transformation]]; the S-matrix is the evolution operator between time equal to minus infinity, and time equal to plus infinity. It is defined only in the limit of zero energy density (or infinite particle separation distance). It can be shown that if a quantum field theory in Minkowski space has a [[mass gap|Mass Gap]], the state in the asymptotic past and in the asymptotic future are both described by Fock spaces.

!!!!Limitations 
The S-matrix formulation is possible only if the assumption of the existence of non-interacting asymptotic states or fields before and after the scattering process is justified.
This is not the case in quantum electrodynamics and in [[quantum field theories|Quantum Field Theory]] in a curved spacetime background, where one encounters the co called "infrared problem". Here one is seeking for alternatives to the S-matrix description. 
In all these cases no formulation of a Fock space for asymptotic states for very early and very late times is possible. 

In [[conformal|Conformal Transformation]] quantum field theories the sheer definition of a S-matrix is not possible. Asymptotic states and fields are ill defined as far removed points can be mapped to close points through a dilation transformation.

''STEP'' (the ''S''atellite ''T''est of the ''E''quivalence ''P''rinciple) will advance experimental limits on violations of [[Einstein’s equivalence principle|Equivalence Principles]] from their present sensitivity of $2$ parts in $10^{13}$ to $1$ part in $10^{18}$ through multiple comparison of the motions of four pairs of test masses of different compositions in an earth-orbiting drag-free satellite.

Papers:
* [[The Science Case for STEP - J. Overduin, F. Everitt, J. Mester, P. Worden|http://arxiv.org/PS_cache/arxiv/pdf/0902/0902.2247v2.pdf]] [[pct. 1|http://scholar.google.de/scholar?cites=10364899760222302819&hl=de]]

''[[Sage|http://www.sagemath.org]]'' is a coherent consolidation platform of over 64 of the best open source scientific packages in pure and applied mathematics, working under a command line and a browser interface. Its internal scripting language is the Python.

*<<LaunchApplicationButton "Launch SUN Virtual Box" "System" "file:///C:\Program Files\Sun\VirtualBox\VirtualBox.exe">>
* [[Launch Browser Worksheet|http://192.168.56.101]]
* [[SAGE Online|http://www.sagenb.org/]]

SAGE manuals:
* [[Sage Reference Manual|http://www.sagemath.org/doc/reference/index.html]]

Systems included:
* [[GAP]]
* [[PARI GP]]
* Singular
* [[MAXIMA|MAXIMA]]

(Some) optional systems:
* [[Mathematica]]
* [[MAGMA]]
* [[Maple|Maple]]
* [[MATLAB|MATLAB]]

Examples:
* [[Short Sage-Combinatorics Demo|http://alpha.sagenb.org/home/pub/8/]]

Google books:
* [[Computational Group Theory and the Theory of Groups: AMS Special Session on Computational Group Theory, March 3-4, 2007, Davidson College, Davidson, North Carolina - L.-C. Kappe, A. Magidin, R. F. Morse|http://books.google.com/books?id=3i-XzHwgHGIC&pg=PA133&lpg=PA133&dq=gap+orthogonal+group+finite+fields&source=bl&ots=8cL1FgTdDR&sig=PdaUTJ7s7uTasW4_fsqI9JkUq0U&hl=de&ei=xxEtS_yrDYej_Ab--ISNCQ&sa=X&oi=book_result&ct=result&resnum=7&ved=0CDQQ6AEwBg#v=onepage&q=gap%20orthogonal%20group%20finite%20fields&f=false]] pct. 0

Videos:
* [[Sage Days: Algebraic Geometry - MSRI (Mathematical Sciences Institute)|http://www.msri.org/calendar/workshops/WorkshopInfo/502/show_workshop]]

The following search keywords are used:

"pct." - paper citations (Google scholar)
"jct." - journal citations (Google scholar)
"bct." - book citations (Google scholar)
"tct." - theses citations (Google scholar)

"prl." - paper relevance, scale 1,...,10, with 10 the best rating (a personal estimate)
"jrl." -  journal relevance, scale 1,...,10, with 10 the best rating (a personal estimate)
"trl." - presentation/talk relevance, scale 1,...,10, with 10 the best rating (a personal estimate)
"brl." - book relevance, scale 1,...,10, with 10 the best rating (a personal estimate)
"lrl." - (hyper-)link relevance, scale 1,...,10, with 10 the best rating (a personal estimate)
"vrl." - video relevance, scale 1,...,10, with 10 the best rating (a personal estimate)

TRD - "To Read" (optionally followed by a number from 1 to 10, 10 signifying highest relevance).

/***
|Name|SearchOptionsPlugin|
|Source|http://www.TiddlyTools.com/#SearchOptionsPlugin|
|Documentation|http://www.TiddlyTools.com/#SearchOptionsPluginInfo|
|Version|3.0.6|
|Author|Eric Shulman|
|License|http://www.TiddlyTools.com/#LegalStatements|
|~CoreVersion|2.1|
|Type|plugin|
|Description|extend core search function with additional user-configurable options|
Adds extra options to core search function including selecting which data items to search, enabling/disabling incremental key-by-key searches, and generating a ''list of matching tiddlers'' instead of immediately displaying all matches.  This plugin also adds syntax for rendering 'search links' within tiddler content to embed one-click searches using pre-defined 'hard-coded' search terms.
!!!!!Documentation
>see [[SearchOptionsPluginInfo]]
!!!!!Configuration
<<<
Search in:
<<option chkSearchTitles>> titles <<option chkSearchText>> text <<option chkSearchTags>> tags <<option chkSearchFields>> fields <<option chkSearchShadows>> shadows
<<option chkSearchHighlight>> Highlight matching text in displayed tiddlers
<<option chkSearchList>> Show list of matches
<<option chkSearchListTiddler>> Write list to [[SearchResults]] tiddler
<<option chkSearchTitlesFirst>> Show title matches first
<<option chkSearchByDate>> Sort matching tiddlers by modification date (most recent first)
<<option chkIncrementalSearch>> Incremental key-by-key search: {{twochar{<<option txtIncrementalSearchMin>>}}} or more characters,  {{threechar{<<option txtIncrementalSearchDelay>>}}} msec delay
<<option chkSearchOpenTiddlers>> Search only in tiddlers that are currently displayed
<<option chkSearchExcludeTags>> Exclude tiddlers tagged with: <<option txtSearchExcludeTags>>
<<<
!!!!!Revisions
<<<
2009.09.22 [3.0.6] in TiddlyWiki.prototype.search, added 'match' param for core compatibility
2009.01.16 [3.0.5] added chkSearchOpenTiddlers option to limit searches to displayed tiddlers only
|please see [[SearchOptionsPluginInfo]] for additional revision details|
2005.10.18 [1.0.0] Initial Release
<<<
!!!!!Code
***/
//{{{
version.extensions.SearchOptionsPlugin= {major: 3, minor: 0, revision: 6, date: new Date(2009,9,22)};

var defaults={
	chkSearchTitles:	true,
	chkSearchText:		true,
	chkSearchTags:		true,
	chkSearchFields:	true,
	chkSearchTitlesFirst:	true,
	chkSearchList:		true,
	chkSearchHighlight:	true,
	chkSearchListTiddler:	false,
	chkSearchByDate:	false,
	chkIncrementalSearch:	true,
	chkSearchShadows:	true,
	chkSearchOpenTiddlers:	false,
	chkSearchExcludeTags:	true,
	txtSearchExcludeTags:	'excludeSearch',
	txtIncrementalSearchDelay:	500,
	txtIncrementalSearchMin:	3
}; for (var id in defaults) if (config.options[id]===undefined)
	config.options[id]=defaults[id];

if (config.macros.search.reportTitle==undefined)
	config.macros.search.reportTitle="SearchResults"; // note: not a cookie!
config.macros.search.label+="\xa0"; // a little bit of space just because it looks better
//}}}
// // searchLink: {{{[search[text to find]] OR [search[text to display|text to find]]}}}
//{{{
config.formatters.push( {
	name: "searchLink",
	match: "\\[search\\[",
	lookaheadRegExp: /\[search\[(.*?)(?:\|(.*?))?\]\]/mg,
	prompt: "search for: '%0'",
	handler: function(w)
	{
		this.lookaheadRegExp.lastIndex = w.matchStart;
		var lookaheadMatch = this.lookaheadRegExp.exec(w.source);
		if(lookaheadMatch && lookaheadMatch.index == w.matchStart) {
			var label=lookaheadMatch[1];
			var text=lookaheadMatch[2]||label;
			var prompt=this.prompt.format([text]);
			var btn=createTiddlyButton(w.output,label,prompt,
				function(){story.search(this.getAttribute("searchText"))},"searchLink");
			btn.setAttribute("searchText",text);
			w.nextMatch = this.lookaheadRegExp.lastIndex;
		}
	}
});
//}}}
// // incremental search uses option settings instead of hard-coded delay and minimum input values
//{{{
var fn=config.macros.search.onKeyPress;
fn=fn.toString().replace(/500/g, "config.options.txtIncrementalSearchDelay||500");
fn=fn.toString().replace(/> 2/g, ">=(config.options.txtIncrementalSearchMin||3)");
eval("config.macros.search.onKeyPress="+fn);
//}}}
// // REPLACE story.search() for option to "show search results in a list"
//{{{
Story.prototype.search = function(text,useCaseSensitive,useRegExp)
{
	var co=config.options; // abbrev
	var re=new RegExp(useRegExp ? text : text.escapeRegExp(),useCaseSensitive ? "mg" : "img");
	if (config.options.chkSearchHighlight) highlightHack=re;
	var matches = store.search(re,co.chkSearchByDate?"modified":"title","");
	if (co.chkSearchByDate) matches=matches.reverse(); // most recent first
	var q = useRegExp ? "/" : "'";
	clearMessage();
	if (!matches.length) {
		if (co.chkSearchListTiddler) discardSearchResults();
		displayMessage(config.macros.search.failureMsg.format([q+text+q]));
	} else {
		if (co.chkSearchList||co.chkSearchListTiddler)
			reportSearchResults(text,matches);
		else {
			var titles = []; for(var t=0; t<matches.length; t++) titles.push(matches[t].title);
			this.closeAllTiddlers(); story.displayTiddlers(null,titles);
			displayMessage(config.macros.search.successMsg.format([matches.length, q+text+q]));
		}
	}
	highlightHack = null;
}
//}}}
// // REPLACE store.search() for enhanced searching/sorting options
//{{{
TiddlyWiki.prototype.search = function(searchRegExp,sortField,excludeTag,match)
{
	var co=config.options; // abbrev
	var tids = this.reverseLookup("tags",excludeTag,!!match,sortField);
	var opened=[]; story.forEachTiddler(function(tid,elem){opened.push(tid);});

	// eliminate tiddlers tagged with excluded tags
	if (co.chkSearchExcludeTags&&co.txtSearchExcludeTags.length) {
		var ex=co.txtSearchExcludeTags.readBracketedList();
		var temp=[]; for(var t=tids.length-1; t>=0; t--)
			if (!tids[t].tags.containsAny(ex)) temp.push(tids[t]);
		tids=temp;
	}

	// scan for matching titles first...
	var results = [];
	if (co.chkSearchTitles) {
		for(var t=0; t<tids.length; t++) {
			if (co.chkSearchOpenTiddlers && !opened.contains(tids[t].title)) continue;
			if(tids[t].title.search(searchRegExp)!=-1) results.push(tids[t]);
		}
		if (co.chkSearchShadows)
			for (var t in config.shadowTiddlers) {
				if (co.chkSearchOpenTiddlers && !opened.contains(t)) continue;
				if ((t.search(searchRegExp)!=-1) && !store.tiddlerExists(t))
					results.push((new Tiddler()).assign(t,config.shadowTiddlers[t]));
			}
	}
	// then scan for matching text, tags, or field data
	for(var t=0; t<tids.length; t++) {
		if (co.chkSearchOpenTiddlers && !opened.contains(tids[t].title)) continue;
		if (co.chkSearchText && tids[t].text.search(searchRegExp)!=-1)
			results.pushUnique(tids[t]);
		if (co.chkSearchTags && tids[t].tags.join(" ").search(searchRegExp)!=-1)
			results.pushUnique(tids[t]);
		if (co.chkSearchFields && store.forEachField!=undefined)
			store.forEachField(tids[t],
				function(tid,field,val) {
					if (val.search(searchRegExp)!=-1) results.pushUnique(tids[t]);
				},
				true); // extended fields only
	}
	// then check for matching text in shadows
	if (co.chkSearchShadows)
		for (var t in config.shadowTiddlers) {
			if (co.chkSearchOpenTiddlers && !opened.contains(t)) continue;
			if ((config.shadowTiddlers[t].search(searchRegExp)!=-1) && !store.tiddlerExists(t))
				results.pushUnique((new Tiddler()).assign(t,config.shadowTiddlers[t]));
		}

	// if not 'titles first', or sorting by modification date,
	// re-sort results to so titles, text, tag and field matches are mixed together
	if(!sortField) sortField = "title";
	var bySortField=function(a,b){
		if(a[sortField]==b[sortField])return(0);else return(a[sortField]<b[sortField])?-1:+1;
	}
	if (!co.chkSearchTitlesFirst || co.chkSearchByDate) results.sort(bySortField);

	return results;
}
//}}}
// // HIJACK core {{{<<search>>}}} macro to add "report" and "simple inline" output
//{{{
config.macros.search.SOP_handler=config.macros.search.handler;
config.macros.search.handler = function(place,macroName,params)
{
	// if "report", use SearchOptionsPlugin report generator for inline output
	if (params[1]&&params[1].substr(0,6)=="report") {
		var keyword=params[0];
		var options=params[1].split("=")[1]; // split "report=option+option+..."
		var heading=params[2]?params[2].unescapeLineBreaks():"";
		var matches=store.search(new RegExp(keyword.escapeRegExp(),"img"),"title","excludeSearch");
		if (matches.length) wikify(heading+window.formatSearchResults(keyword,matches,options),place);
	} else if (params[1]) {
		var keyword=params[0];
		var heading=params[1]?params[1].unescapeLineBreaks():"";
		var seperator=params[2]?params[2].unescapeLineBreaks():", ";
		var matches=store.search(new RegExp(keyword.escapeRegExp(),"img"),"title","excludeSearch");
		if (matches.length) {
			var out=[];
			for (var m=0; m<matches.length; m++) out.push("[["+matches[m].title+"]]");
			wikify(heading+out.join(seperator),place);
		}
	} else
		config.macros.search.SOP_handler.apply(this,arguments);
};
//}}}
// // SearchResults panel handling
//{{{
setStylesheet(".searchResults { padding:1em 1em 0 1em; }","searchResults"); // matches std tiddler padding

config.macros.search.createPanel=function(text,matches,body) {

	function getByClass(e,c) { var d=e.getElementsByTagName("div");
		for (var i=0;i<d.length;i++) if (hasClass(d[i],c)) return d[i]; }
	var panel=createTiddlyElement(null,"div","searchPanel","searchPanel");
	this.renderPanel(panel,text,matches,body);
	var oldpanel=document.getElementById("searchPanel");
	if (!oldpanel) { // insert new panel just above tiddlers
		var da=document.getElementById("displayArea");
		da.insertBefore(panel,da.firstChild);
	} else { // if panel exists
		var oldwrap=getByClass(oldpanel,"searchResults");
		var newwrap=getByClass(panel,"searchResults");
		// if no prior content, just insert new content
		if (!oldwrap) oldpanel.insertBefore(newwrap,null);
		else {	// swap search results content but leave containing panel intact
			oldwrap.style.display='block'; // unfold wrapper if needed
			var i=oldwrap.getElementsByTagName("input")[0]; // get input field
			if (i) { var pos=this.getCursorPos(i); i.onblur=null; } // get cursor pos, ignore blur
			oldpanel.replaceChild(newwrap,oldwrap);
			panel=oldpanel; // use existing panel
		}
	}
	this.showPanel(true,pos);
	return panel;
}

config.macros.search.renderPanel=function(panel,text,matches,body) {

	var wrap=createTiddlyElement(panel,"div",null,"searchResults");
	wrap.onmouseover = function(e){ addClass(this,"selected"); }
	wrap.onmouseout = function(e){ removeClass(this,"selected"); }
	// create toolbar: "open all", "fold/unfold", "close"
	var tb=createTiddlyElement(wrap,"div",null,"toolbar");
	var b=createTiddlyButton(tb, "open all", "open all matching tiddlers", function() {
		story.displayTiddlers(null,this.getAttribute("list").readBracketedList()); return false; },"button");
	var list=""; for(var t=0;t<matches.length;t++) list+='[['+matches[t].title+']] ';
	b.setAttribute("list",list);
	var b=createTiddlyButton(tb, "fold", "toggle display of search results", function() {
		config.macros.search.foldPanel(this); return false; },"button");
	var b=createTiddlyButton(tb, "close", "dismiss search results",	function() {
		config.macros.search.showPanel(false); return false; },"button");
	createTiddlyText(createTiddlyElement(wrap,"div",null,"title"),"Search for: "+text); // title
	wikify(body,createTiddlyElement(wrap,"div",null,"viewer")); // report
	return panel;
}

config.macros.search.showPanel=function(show,pos) {
	var panel=document.getElementById("searchPanel");
	var i=panel.getElementsByTagName("input")[0];
	i.onfocus=show?function(){config.macros.search.stayFocused(true);}:null;
	i.onblur=show?function(){config.macros.search.stayFocused(false);}:null;
	if (show && panel.style.display=="block") { // if shown, grab focus, restore cursor
		if (i&&this.stayFocused()) { i.focus(); this.setCursorPos(i,pos); }
		return;
	}
	if(!config.options.chkAnimate) {
		panel.style.display=show?"block":"none";
		if (!show) { removeChildren(panel); config.macros.search.stayFocused(false); }
	} else {
		var s=new Slider(panel,show,false,show?"none":"children");
		s.callback=function(e,p){e.style.overflow="visible";}
		anim.startAnimating(s);
	}
	return panel;
}

config.macros.search.foldPanel=function(button) {
	var d=document.getElementById("searchPanel").getElementsByTagName("div");
	for (var i=0;i<d.length;i++) if (hasClass(d[i],"viewer")) var v=d[i]; if (!v) return;
	var show=v.style.display=="none";
	if(!config.options.chkAnimate)
		v.style.display=show?"block":"none";
	else {
		var s=new Slider(v,show,false,"none");
		s.callback=function(e,p){e.style.overflow="visible";}
		anim.startAnimating(s);
	}
	button.innerHTML=show?"fold":"unfold";
	return false;
}

config.macros.search.stayFocused=function(keep) { // TRUE/FALSE=set value, no args=get value
	if (keep===undefined) return this.keepReportInFocus;
	this.keepReportInFocus=keep;
	return keep
}

config.macros.search.getCursorPos=function(i) {
	var s=0; var e=0; if (!i) return { start:s, end:e };
	try {
		if (i.setSelectionRange) // FF
			{ s=i.selectionStart; e=i.selectionEnd; }
		if (document.selection && document.selection.createRange) { // IE
			var r=document.selection.createRange().duplicate();
			var len=r.text.length; s=0-r.moveStart('character',-100000); e=s+len;
		}
	}catch(e){};
	return { start:s, end:e };
}
config.macros.search.setCursorPos=function(i,pos) {
	if (!i||!pos) return; var s=pos.start; var e=pos.end;
	if (i.setSelectionRange) //FF
		i.setSelectionRange(s,e);
	if (i.createTextRange) // IE
		{ var r=i.createTextRange(); r.collapse(true); r.moveStart("character",s); r.select(); }
}
//}}}
// // SearchResults report generation
// note: these functions are defined globally, so they can be more easily redefined to customize report formats//
//{{{
if (!window.reportSearchResults) window.reportSearchResults=function(text,matches)
{
	var cms=config.macros.search; // abbrev
	var body=window.formatSearchResults(text,matches);
	if (!config.options.chkSearchListTiddler) // show #searchResults panel
		window.scrollTo(0,ensureVisible(cms.createPanel(text,matches,body)));
	else { // write [[SearchResults]] tiddler
		var title=cms.reportTitle;
		var who=config.options.txtUserName;
		var when=new Date();
		var tags="excludeLists excludeSearch temporary";
		var tid=store.getTiddler(title); if (!tid) tid=new Tiddler();
		tid.set(title,body,who,when,tags);
		store.addTiddler(tid);
		story.closeTiddler(title);
		story.displayTiddler(null,title);
	}
}

if (!window.formatSearchResults) window.formatSearchResults=function(text,matches,opt)
{
	var body='';
	var title=config.macros.search.reportTitle
	var q = config.options.chkRegExpSearch ? "/" : "'";
	if (!opt) var opt="all";
	var parts=opt.split("+");
	for (var i=0; i<parts.length; i++) { var p=parts[i].toLowerCase();
		if (p=="again"||p=="all")   body+=window.formatSearchResults_again(text,matches);
		if (p=="summary"||p=="all") body+=window.formatSearchResults_summary(text,matches);
		if (p=="list"||p=="all")    body+=window.formatSearchResults_list(text,matches);
		if (p=="buttons"||p=="all") body+=window.formatSearchResults_buttons(text,matches);
	}
	return body;
}

if (!window.formatSearchResults_again) window.formatSearchResults_again=function(text,matches)
{
	var title=config.macros.search.reportTitle
	var body='';
	// search again
	body+='{{span{<<search "'+text.replace(/"/g,'&#x22;')+'">> /%\n';
	body+='%/<html><input type="button" value="search again"';
	body+=' onclick="var t=this.parentNode.parentNode.getElementsByTagName(\'input\')[0];';
	body+=' config.macros.search.doSearch(t); return false;">';
	body+=' <a href="javascript:;" onclick="';
	body+=' var e=this.parentNode.nextSibling;';
	body+=' var show=e.style.display!=\'block\';';
	body+=' if(!config.options.chkAnimate) e.style.display=show?\'block\':\'none\';';
	body+=' else anim.startAnimating(new Slider(e,show,false,\'none\'));';
	body+=' return false;">options...</a>';
	body+='</html>@@display:none;border-left:1px dotted;margin-left:1em;padding:0;padding-left:.5em;font-size:90%;/%\n';
	body+='	%/<<option chkSearchTitles>>titles /%\n';
	body+='	%/<<option chkSearchText>>text /%\n';
	body+='	%/<<option chkSearchTags>>tags /%\n';
	body+='	%/<<option chkSearchFields>>fields /%\n';
	body+='	%/<<option chkSearchShadows>>shadows\n';
	body+='	<<option chkCaseSensitiveSearch>>case-sensitive /%\n';
	body+='	%/<<option chkRegExpSearch>>text patterns /%\n';
	body+='	%/<<option chkSearchByDate>>sorted by date\n';
	body+='	<<option chkSearchHighlight>> highlight matching text in displayed tiddlers\n';
	body+='	<<option chkIncrementalSearch>>incremental key-by-key search: /%\n';
	body+='	%/{{twochar{<<option txtIncrementalSearchMin>>}}} or more characters, /%\n';
	body+='	%/{{threechar{<<option txtIncrementalSearchDelay>>}}} msec delay\n';
	body+='	<<option chkSearchOpenTiddlers>> search only in tiddlers that are currently displayed\n';
	body+='	<<option chkSearchExcludeTags>>exclude tiddlers tagged with:\n';
	body+='	{{editor{<<option txtSearchExcludeTags>>}}}/%\n';
	body+='%/@@}}}\n\n';
	return body;
}

if (!window.formatSearchResults_summary) window.formatSearchResults_summary=function(text,matches)
{
	// summary: nn tiddlers found matching '...', options used
	var body='';
	var co=config.options; // abbrev
	var title=config.macros.search.reportTitle
	var q = co.chkRegExpSearch ? "/" : "'";
	body+="''"+config.macros.search.successMsg.format([matches.length,q+"{{{"+text+"}}}"+q])+"''\n";
	var opts=[];
	if (co.chkSearchTitles) opts.push("titles");
	if (co.chkSearchText) opts.push("text");
	if (co.chkSearchTags) opts.push("tags");
	if (co.chkSearchFields) opts.push("fields");
	if (co.chkSearchShadows) opts.push("shadows");
	if (co.chkSearchOpenTiddlers) body+="^^//search limited to displayed tiddlers only//^^\n";
	body+="~~&nbsp; searched in "+opts.join(" + ")+"~~\n";
	body+=(co.chkCaseSensitiveSearch||co.chkRegExpSearch?"^^&nbsp; using ":"")
		+(co.chkCaseSensitiveSearch?"case-sensitive ":"")
		+(co.chkRegExpSearch?"pattern ":"")
		+(co.chkCaseSensitiveSearch||co.chkRegExpSearch?"matching^^\n":"");
	return body;
}

if (!window.formatSearchResults_list) window.formatSearchResults_list=function(text,matches)
{
	// bullet list of links to matching tiddlers
	var body='';
	var pattern=co.chkRegExpSearch?text:text.escapeRegExp();
	var sensitive=co.chkCaseSensitiveSearch?"mg":"img";
	var link='{{tiddlyLinkExisting{<html><nowiki><a href="javascript:;" onclick="'
		+'if(config.options.chkSearchHighlight)'
		+'	highlightHack=new RegExp(\x27'+pattern+'\x27.escapeRegExp(),\x27'+sensitive+'\x27);'
		+'story.displayTiddler(null,\x27%0\x27);'
		+'highlightHack = null; return false;'
		+'" title="%2">%1</a></html>}}}';
	for(var t=0;t<matches.length;t++) {
		body+="* ";
		if (config.options.chkSearchByDate)
			body+=matches[t].modified.formatString('YYYY.0MM.0DD 0hh:0mm')+" ";
		var title=matches[t].title;
		var fixup=title.replace(/'/g,"\\x27").replace(/"/g,"\\x22");
		var tid=store.getTiddler(title);
		var tip=tid?tid.getSubtitle():''; tip=tip.replace(/"/g,"&quot;");
		body+=link.format([fixup,title,tip])+'\n';
	}
	return body;
}

if (!window.formatSearchResults_buttons) window.formatSearchResults_buttons=function(text,matches)
{
	// embed buttons only if writing SearchResults to tiddler
	if (!config.options.chkSearchListTiddler) return "";
	// "open all" button
	var title=config.macros.search.reportTitle;
	var body="";
	body+="@@diplay:block;<html><input type=\"button\" href=\"javascript:;\" "
		+"onclick=\"story.displayTiddlers(null,[";
	for(var t=0;t<matches.length;t++)
		body+="'"+matches[t].title.replace(/\'/mg,"\\'")+"'"+((t<matches.length-1)?", ":"");
	body+="],1);\" accesskey=\"O\" value=\"open all matching tiddlers\"></html> ";
	// "discard SearchResults" button
	body+="<html><input type=\"button\" href=\"javascript:;\" "
		+"onclick=\"discardSearchResults()\" value=\"discard "+title+"\"></html>";
	body+="@@\n";
	return body;
}

if (!window.discardSearchResults) window.discardSearchResults=function()
{
	// remove the tiddler
	story.closeTiddler(config.macros.search.reportTitle);
	store.deleteTiddler(config.macros.search.reportTitle);
	store.notify(config.macros.search.reportTitle,true);
}
//}}}

''Self Organized Criticality'' (= ''SOC'').

Links:
* [[WIKIPEDIA - Self-organized Criticality|http://en.wikipedia.org/wiki/Self-organized_criticality]]

Papers:
* [[Phase Transitions and Complex Systems - R. V. Solé, S. C. Manrubia, B. Luque, J. Delgado|http://complex.upf.es/~ricard/COMPLEXITY-96.pdf]] [[pct. 66|http://scholar.google.de/scholar?cites=15790333541130288044&hl=de]]
* [[Cellular Automata and Self-organized Criticality - M. Creutz|http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.50.5288&rep=rep1&type=pdf]] [[pct. 8|http://scholar.google.com/scholar?cites=9131077436801048589&as_sdt=2005&sciodt=2000&hl=de]] TRD
* [[Symmetropy and Self-Organized Criticality - K. Nanjo, H. Nagahama, E. Yodogawa|http://www.scipress.org/journals/forma/pdf/1603/16030213.pdf]] [[pct. 4|http://scholar.google.de/scholar?cites=2013446146386239124&hl=de]]
* [[Y-Bias and Angularity: The Dynamics of Self-Organizing Criticality From the Zero Point to Infinity - D. G. Yurth, D. Ayres|http://www.pureenergysystems.com/academy/papers/Y-Bias_and_Angularity/Y-Bias_Monograph_Preview_Edition_25Apr06c.doc]] pct. 0

The concept of ''Shannon Entropy'' (a.k.a. ''Information Entropy'') was introduced by Claude E. Shannon in his 1948 landmark paper "A Mathematical Theory of Communication" [1].

For a stream of characters (taken from a given alphabet) Shannon entropy $S$ quantifies the predictability of each character’s occurrence in terms of those previously transmitted and is defined as
\[
 S \equiv  - \sum_i  p_i \ln (p_i)
\]
where $p_i$ is the probability (inverse of the frequency) of the occurrence of the $i^{th}$ character of the alphabet. 

The function $S$ is
*  positive definite, 
*  vanishing if the stream is constant (i.e. if it only consists of one character),
*  maximal for completely random sequences.

So generally speaking, the higher the Shannon entropy, the less predictable is the outcome of the next event. Thus Shannon entropy is a measure of how chaotic, random or disordered a system is.


<html><center><iframe name="content" src="http://www.markus-maute.de/trajectory/advert_60.html" width=51% height=86></iframe></center></html>Links:
* [[WIKIPEDIA - Entropy (Information Theory)|http://en.wikipedia.org/wiki/Information_entropy]]
* [[WIKIPEDIA - Entropy in Thermodynamics and Information Theory|http://en.wikipedia.org/wiki/Entropy_in_thermodynamics_and_information_theory]]

Papers:
* [[[1] A Mathematical Theory of Communication - C. E. Shannon|http://faculty.kfupm.edu.sa/RI/abdallah/041/ee406/shannon.pdf]] {{t1000Cite{[[pct. 32565|http://scholar.google.de/scholar?cites=14249175757432453165&hl=de]]}}}

/***
|Name|SinglePageModePlugin|
|Source|http://www.TiddlyTools.com/#SinglePageModePlugin|
|Documentation|http://www.TiddlyTools.com/#SinglePageModePluginInfo|
|Version|2.9.6|
|Author|Eric Shulman - ELS Design Studios|
|License|http://www.TiddlyTools.com/#LegalStatements <br>and [[Creative Commons Attribution-ShareAlike 2.5 License|http://creativecommons.org/licenses/by-sa/2.5/]]|
|~CoreVersion|2.1|
|Type|plugin|
|Requires||
|Overrides|Story.prototype.displayTiddler(), Story.prototype.displayTiddlers()|
|Options|##Configuration|
|Description|Show tiddlers one at a time with automatic permalink, or always open tiddlers at top/bottom of page.|
This plugin allows you to configure TiddlyWiki to navigate more like a traditional multipage web site with only one tiddler displayed at a time.
!!!!!Documentation
>see [[SinglePageModePluginInfo]]
!!!!!Configuration
<<<
<<option chkSinglePageMode>> Display one tiddler at a time
><<option chkSinglePagePermalink>> Automatically permalink current tiddler
><<option chkSinglePageKeepFoldedTiddlers>> Don't close tiddlers that are folded
><<option chkSinglePageKeepEditedTiddlers>> Don't close tiddlers that are being edited
<<option chkTopOfPageMode>> Open tiddlers at the top of the page
<<option chkBottomOfPageMode>> Open tiddlers at the bottom of the page
<<option chkSinglePageAutoScroll>> Automatically scroll tiddler into view (if needed)

Notes:
* The "display one tiddler at a time" option can also be //temporarily// set/reset by including a 'paramifier' in the document URL: {{{#SPM:true}}} or {{{#SPM:false}}}.
* If more than one display mode is selected, 'one at a time' display takes precedence over both 'top' and 'bottom' settings, and if 'one at a time' setting is not used, 'top of page' takes precedence over 'bottom of page'.
* When using Apple's Safari browser, automatically setting the permalink causes an error and is disabled.
<<<
!!!!!Revisions
<<<
2008.10.17 [2.9.6] changed chkSinglePageAutoScroll default to false
| Please see [[SinglePageModePluginInfo]] for previous revision details |
2005.08.15 [1.0.0] Initial Release.  Support for BACK/FORWARD buttons adapted from code developed by Clint Checketts.
<<<
!!!!!Code
***/
//{{{
version.extensions.SinglePageModePlugin= {major: 2, minor: 9, revision: 6, date: new Date(2008,10,17)};
//}}}
//{{{
config.paramifiers.SPM = { onstart: function(v) {
	config.options.chkSinglePageMode=eval(v);
	if (config.options.chkSinglePageMode && config.options.chkSinglePagePermalink && !config.browser.isSafari) {
		config.lastURL = window.location.hash;
		if (!config.SPMTimer) config.SPMTimer=window.setInterval(function() {checkLastURL();},1000);
	}
} };
//}}}
//{{{
if (config.options.chkSinglePageMode==undefined)
	config.options.chkSinglePageMode=true;
if (config.options.chkSinglePagePermalink==undefined)
	config.options.chkSinglePagePermalink=true;
if (config.options.chkSinglePageKeepFoldedTiddlers==undefined)
	config.options.chkSinglePageKeepFoldedTiddlers=false;
if (config.options.chkSinglePageKeepEditedTiddlers==undefined)
	config.options.chkSinglePageKeepEditedTiddlers=false;
if (config.options.chkTopOfPageMode==undefined)
	config.options.chkTopOfPageMode=false;
if (config.options.chkBottomOfPageMode==undefined)
	config.options.chkBottomOfPageMode=false;
if (config.options.chkSinglePageAutoScroll==undefined)
	config.options.chkSinglePageAutoScroll=false;
//}}}
//{{{
config.SPMTimer = 0;
config.lastURL = window.location.hash;
function checkLastURL()
{
	if (!config.options.chkSinglePageMode)
		{ window.clearInterval(config.SPMTimer); config.SPMTimer=0; return; }
	if (config.lastURL == window.location.hash) return; // no change in hash
	var tids=decodeURIComponent(window.location.hash.substr(1)).readBracketedList();
	if (tids.length==1) // permalink (single tiddler in URL)
		story.displayTiddler(null,tids[0]);
	else { // restore permaview or default view
		config.lastURL = window.location.hash;
		if (!tids.length) tids=store.getTiddlerText("DefaultTiddlers").readBracketedList();
		story.closeAllTiddlers();
		story.displayTiddlers(null,tids);
	}
}


if (Story.prototype.SPM_coreDisplayTiddler==undefined)
	Story.prototype.SPM_coreDisplayTiddler=Story.prototype.displayTiddler;
Story.prototype.displayTiddler = function(srcElement,tiddler,template,animate,slowly)
{
	var title=(tiddler instanceof Tiddler)?tiddler.title:tiddler;
	var tiddlerElem=document.getElementById(story.idPrefix+title); // ==null unless tiddler is already displayed
	var opt=config.options;
	var single=opt.chkSinglePageMode && !startingUp;
	var top=opt.chkTopOfPageMode && !startingUp;
	var bottom=opt.chkBottomOfPageMode && !startingUp;
	if (single) {
		story.forEachTiddler(function(tid,elem) {
			// skip current tiddler and, optionally, tiddlers that are folded.
			if (	tid==title
				|| (opt.chkSinglePageKeepFoldedTiddlers && elem.getAttribute("folded")=="true"))
				return;
			// if a tiddler is being edited, ask before closing
			if (elem.getAttribute("dirty")=="true") {
				if (opt.chkSinglePageKeepEditedTiddlers) return;
				// if tiddler to be displayed is already shown, then leave active tiddler editor as is
				// (occurs when switching between view and edit modes)
				if (tiddlerElem) return;
				// otherwise, ask for permission
				var msg="'"+tid+"' is currently being edited.\n\n";
				msg+="Press OK to save and close this tiddler\nor press Cancel to leave it opened";
				if (!confirm(msg)) return; else story.saveTiddler(tid);
			}
			story.closeTiddler(tid);
		});
	}
	else if (top)
		arguments[0]=null;
	else if (bottom)
		arguments[0]="bottom";
	if (single && opt.chkSinglePagePermalink && !config.browser.isSafari) {
		window.location.hash = encodeURIComponent(String.encodeTiddlyLink(title));
		config.lastURL = window.location.hash;
		document.title = wikifyPlain("SiteTitle") + " - " + title;
		if (!config.SPMTimer) config.SPMTimer=window.setInterval(function() {checkLastURL();},1000);
	}
	if (tiddlerElem && tiddlerElem.getAttribute("dirty")=="true") { // editing... move tiddler without re-rendering
		var isTopTiddler=(tiddlerElem.previousSibling==null);
		if (!isTopTiddler && (single || top))
			tiddlerElem.parentNode.inser