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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 ...


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 ...


7

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 ...


7

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 ...


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 ...


5

I'll start with your last question. " Could one also assign a length (in meters) to time in this way? " Yes! In the (Minkowki) spacetime infinitesimal interval, $$\mbox{d}s^2=\left(ic_0\mbox{d}t\right)^2+\left(\mbox{d}x\right)^2+\left(\mbox{d}y\right)^2+\left(\mbox{d}z\right)^2$$ $\left(ic_0\mbox{d}t\right)$ has units of ?? Meters! Exactly! You could ...


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 ...


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.


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

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

"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

On Unification I presume you're asking whether just classical gravity & classical EM can be unified. They sure can! Classical General Relativity and Classical Electromagnetism are unified in Kaluza-Klein-Theory, which proves that 5-dimensional general relativity is equivalent to 4-dimensional general relativity plus 4-dimensional maxwell ...


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

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 ...


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

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

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 ...


3

The eigenvalue of the generator $t_a$ are integer multiples of $g_{min}$ because $t_a$ is a generator of a (cyclic) $U(1)$ group and $$ \exp(2\pi i t_a/ g_{min}) = 1 $$ holds as an operator equation. This equation says that the exponentiation of the generator with some imaginary coefficient that I parameterized as $2\pi i / g_{min}$ is equal to the identity. ...


3

The semisimple Lie algebra $$L=su(3)\oplus su(2)\oplus u(1)$$ is a direct sum of simple Lie algebras. Concretely a Lie algebra element in $L$ may be represented as block diagonal $6\times 6$ matrix, where a $3\times 3$ submatrix, $2\times 2$ submatrix and a $1\times 1$ submatrix carry the $su(3)$ element, the $su(2)$ element, and the $u(1)$ element, ...


2

Since you were here yesterday, you're certainly here at the moment and we hope you're here tomorrow, there's some pretty compelling evidence for an additional dimension! To be less flippant, here is a more vivid example. Suppose an infinitely thin disc (like a coin with no discernible thickness) came into existence on your desk Tuesday midday and remained ...


2

This is the problem of radion stabilization for KK theories. The radion is the effective 4D field which measures the size of the compactified dimension. For Calabi-Yau compactifications with unbroken SUSY, the radion turns out to be a moduli, which is experimentally unsatisfactory. There are many different radion stabilization mechanisms out there, e.g. ...


2

These researches are interesting. This connects with the projective Cayley plane, which is important in the Jordan algebra. Some of the structure discussed in the paper Hisham Sati http://arxiv.org/PS_cache/arxiv/pdf/0807/0807.4899v3.pdf makes contact with the Jordan algebra. A constraint on the diagonal scalars of the Jordan matrix, which is similar to ...


2

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" ...



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