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9

If your definition of a lens space means $S^3/\Gamma$, a quotient of the three-sphere, then the $U(1)$, $SU(2)$ couplings may (classically) be derived from the parent theory on the full three-sphere whose isometry is $SO(4) \approx SU(2) \times SU(2)$ (at the level of Lie algebras). The subgroup $\Gamma$ acts only inside one of the $SU(2)$ factors, let's say ...


9

When you write the five dimensional Kaluza-Klein metric tensor as $$ g_{mn} = \left( \begin{array}{cc} g_{\mu\nu} & g_{\mu 5} \\ g_{5\nu} & g_{55}\\ \end{array} \right) $$ where $g_{\mu\nu}$ corresponds to the ordinary four dimensional metric and $ g_{\mu 5}$ is the ordinary four dimensional vector potetial, $g_{55}$ appears as an ...


9

1. Kaluza-Klein theory. This is similar to General Relativity, but instead of three space dimensions plus time, there are four space dimensions plus time. The fourth dimension is cyclic, and satisfies some symmetry conditions. The electromagnetic potential appears as the components of the metric in the fourth space dimension. It is usually rejected on the ...


8

It is not so much as one single particle will be seen with different masses as it is that that one type of particle will be seen as having multiple different masses when it is detected multiple times. For example if the extra dimension is like a rolled up microscopic cylinder, the particle can have an infinite number of discrete masses starting from the ...


7

In addition to what dmckee said, another hint at ("large") extra dimensions would be the detection of Kaluza-Klein particles at the LHC for example. Kaluza-Klein particles are in principle nothing but the known standard model particles which can propagate into the extra dimensions if these are large enough. It can be shown that the angular momentum in these ...


6

Dear Mitchell, this is a very nice research project - at least judging by the fact that I have made a similar proposal. ;-) In this very form, however, it can't be right because any hypothetical 27-dimensional theory fails to be supersymmetric and the supersymmetry breaking can't be quite undone. However, brave souls have played with the transmutation of ...


6

A very sensible (and advanced) question. Yes, it is typical for 3-cycles to have the topology you mentioned. For example, this paper http://arxiv.org/abs/hep-th/0011190 studies only associative 3-cycles that are either $S^3$ or orbifolds of it, or the hyperbolic 3-space $H^3$ and orbifolds of that. And in some cases, the spherical cycle is rigid. The ...


6

As the very formulation of your question makes clear, we know what the actual algebra of local symmetries is. It is the five-dimensional diffeomorphism invariance assuming the $M^4\times S^1$ topology of the five-dimensional spacetime. The term "Kač-Moody generalization of an algebra" is nothing else than an alternative name for this algebra, especially for ...


5

Are they identifying inertial mass with the ADM mass? There is an old style way of writing $g_{00}$ of a stationary metric, far from the matter source, as $-\left(1 - \frac{1}{2}\phi({\vec x})\right)$, where $\phi$ is the potential function for the metric, and then you can monopole expand this and identify the numerator of the $\frac{1}{r}$ term of the ...


5

You missed a term in expanding the upper-indexed metric. The full version is below: \begin{align} \tilde{\Gamma}^\lambda_{\mu\nu} & = \frac{1}{2} \tilde{g}^{\lambda X} \left(\partial_\mu \tilde{g}_{\nu X} + \partial_\nu \tilde{g}_{\mu X} - \partial_X \tilde{g}_{\mu\nu}\right) \\ & =\frac{1}{2} \tilde{g}^{\lambda\sigma} \left(\partial_\mu ...


5

"Any transformation that changes one theory into another" (or the same) theory is not called T-duality. It is just a "duality". A condition is that the two theories seemingly look different - otherwise the equivalence would be vacuous - but it must be true that their spectrum and the strength of interactions between their states must be totally isomorphic: ...


4

The group manifold $U(1) \times SU(2)\times SU(3)$ is $1+3+8=12$-dimensional, not 7-dimensional. You probably meant the dimension of a manifold that may have this group as its isometry group. But one may show that no such low-dimensional manifold can be interpreted as the extra dimensions of string theory to produce a realistic model. The oldest ...


4

They are distinct. The mini-black holes are objects that are localized in the bulk and are approximate Schwarzschild thermal ensembles formed by high-energy collisions of incoming particles. The Kaluza Klein excitations are particles zooming around the extra dimensions, and are not thermal, but just as cold as any other single particle states. The only link ...


4

Here is a nice illustration of the Calabi Yau manifold. One can visualize at each point of our 3 dimensional space as a tiny manifold like that which encloses the extra dimensions. Alternatively: if our third dimension were curled up we would be living in Flatland , without knowledge of the third dimension. One can rotate a two dimensional figure into the ...


4

Annav gave the correct answer, but here's some help on visualisation. First thing: We cannot imagine more than 3 space dimensions. You can try, and get tantalizingly close, but it's extremely hard to wrap one's brain around it Due to this, I shall explain this in lower dimensions,and you can try to generalise it. Try. Alright. Let's imagine a thin hose. ...


4

Make some assumptions about the physics associated with the dimensions in questions (say electric field strength goes by $r^{-(n-1)}$ over distances in which $n$ dimensions are significant). Make predictions on that basis Compare to experiment Many predictions can be made and tested in the realms of high energy particle physics, but so far all are null.


3

The key is in the phrase following immediately after To us, the particle would not look like one particle, but a set of particles – all with different masses. which is The faster the particle moves along the extra dimension, the larger this apparent mass seems to be. (here is a link) I think that it means that the same particle in extra ...


3

I'm not aware of a standard notation. Associative 3-folds can be rigid, but need not be. Although it doesn't address your question in any kind of generality, associative 3-cycles can be understood quite easily for a special class of G2 manifolds of the form $(X \times S^1)/Z_2$ where $X$ is a Calabi-Yau 3-fold and the involution acts as $(\sigma_X,-1)$ where ...


3

If you're after an intuitive understanding then I think the original Kaluza-Klein theory is a good place to start. NB no-one believes the Kaluza-Klein theory to be a realistic description of the universe, but it's a good illustration of how extra dimensions may influence physics. See http://en.wikipedia.org/wiki/Kaluza%E2%80%93Klein_theory for a good article ...


3

Not my domaine of expertise, but I would say that Kaluza-Klein, by itself, can't properly unify GR and EM. If we stick to the 5-dimensional approach, the first studied by KK, we get a few drawbacks right on the starting points. The deviation of the photon, for instance, would be affected by the presence of a 5-dimensional parameter, we can calculate it for a ...


3

How's this: http://www.weylmann.com/kaluza.pdf ? Be careful though, because there are a couple very big errors that the author makes. For example in equation (6) there should not be a factor of 1/2 in front of the second term on the right-hand side. It should just be a factor of one. If you do the Christoffel symbol calculations and expand out the 5D ...


3

Not data, but... The string theorists and other TOE people would say that a single theory exhibiting all the required symmetries only has representations at higher dimensions. Then they deal "So why can't we see the other ones?" by giving them periodic boundary conditions on length scales smaller than we have been able to probe. This would be "extra" ...


3

If we consider the heat capacity of a mono-atomic gas, it will answer the question how many independent degrees of freedom exist in space. Experiment gives three space dimensions.


3

By the way what's the actual state of these searches for extra dimensions considering the data already available ? The LHC experiments are looking for extra dimensions in various channels (search for "extra dimensions LHC) and putting limits. At the moment the standard model reigns. They are trying for KK s too, though still at a planning stage. I do ...


3

Assuming you mean spatial dimensions, Nima Arkani-Hamed has some ideas and papers on the subject. Here is a popular article: http://www.nature.com/nature/journal/v433/n7021/full/433010a.html The idea of a fourth dimension was popularized by Charles Hinton in the late 1800s: http://en.wikipedia.org/wiki/Charles_Howard_Hinton and, of course, by Edwin Abbott ...



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