# Tag Info

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

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

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

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

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

The definition of dimension used here is that of a dimension of a manifold - essentially, how many coordinates (=real numbers) we need to describe the manifold (thought of as spacetime). Manifolds may carry a notion of length, and one of volume. They may also be compact or non-compact, roughly1 corresponding to finite and infinite. E.g. a sphere of radius ...

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: ...

5

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

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 ...

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

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" ...

4

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.

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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

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

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 ...

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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 ...

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

Let us quickly run through the standard KK compactification. We start with a $d+1$ dimensional theory $$S = \frac{1}{16\pi G_{d+1}} \int d^{d+1}x \sqrt{G} R_{d+1}$$ More general actions on the $d+1$ dimensional space can be considered, but this will suffice for our purposes. The metric $G_{MN}$ can be decomposed as $$ds^2 = G_{MN} dx^M dx^N = e^{2\Phi} ... 3 The eigenvalue of the generator t_a are integer multiples of g_{min} because t_a is a generator of a (cyclic) U(1) group and$$ \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)$$is a direct sum of simple Lie algebras. Concretely a Lie algebra element in L may be represented as block diagonal 6\times 6 matrix, where a 3\times 3 submatrix, 2\times 2 submatrix and a 1\times 1 submatrix carry the su(3) element, the su(2) element, and the u(1) element, ... 3 Disclaimer: while I have a good grasp of GR fundamentals, it is not my area of expertise. The gravitational mass is distinct from the inertial mass as follows. The gravitational mass defines how strongly the body curves space-time, qualitatively it answers the question "how strong is the gravitational force from this object?". The inertial mass defines how ... 3 The mass of a quantum of a field is defined from the second derivative of the potential term$$ m^2 = \left. \frac{\partial^2 V(\phi)}{\partial \phi^2} \right|_{\phi=0}  and similar for fields with spin (fields that are not scalar fields). The general form of the potential – or the whole Lagrangian – is always more complicated but only this leading term ...

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