Coulomb interaction and non-trivial topology

[tom.stoer]

I have a question regarding the Coulomb interaction in spaces with non-trivial topology.

Suppose we have D large spatial dimensions (D>2). Then the Coulomb potential is V_C(r) ~ 1/r^D-2. Usually one shows in three dimensions that the Coulomb potential V_C(r) is nothing else but the Fourier transform of G(k) ~ 1/k² which is a Greens function for a massless particle; with a mass-term there would be an additional factor exp^-mr.

Now suppose that we do not have flat space but that space is compactified. The simplest example is a 3-torus with size L. Then one charge located at r=0 "feels" another charge located at r as if it were located at r, r + Le_i, r + 2Le_i, ... this is equivalent to say that the potential must be periodic on the 3-torus, i.e. it must respect the condition

V(r) = V(r + Le_i)

One can discretize the theory by using Fourier modes on the 3-torus T³ = S¹ * S¹ * S¹. Then the Coulomb potential V(r) can be derived from G(k) via a discrete Fourier series where one sums over 1/k² where k respects the periodicity of the 3-torus.

So far so good.

What happens if
1) the topology of space (spacetime) becomes more complex? There are e.g. speculations that our universe could have the topology of a dodecahedral space which could explain suppressions of CMB multipole moments.
2) the geometry of space becomes dynamic? In GR the geometry of space is not fixed; typically space(time) will expand.

I have no idea how a mode decomposition in a dodecahedral space would look like. I have no idea how this could affect the Coulomb interaction at early times (for a small universe). I have no idea how the topology of an expanding universe would restrict the mode decomposition.

Are there any hints how all this could affect such a simple law as the Coulomb interaction?

 

[qsa]

Tom
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[qsa]

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[Physics Monkey]

Before even getting into the second half of your question, I think you have to a be a little careful about even something as innocent as the torus. For example, you might try to compute the potential due to a charge on a torus using the method of images, this is just what you said. But this sum doesn't converge. You can try to argue by symmetry, but it matters how you regulate the problem. We meet these kinds of divergences often in solid state physics where one wants to sum the potential due to an infinite lattice of charge. Boundary conditions and other subtleties usually matter. For example, on a closed space like a torus you can't have a single isolated charge.

 

[tom.stoer]

In 1+1 dim QED on a circle S¹ everything is well-defined. The interaction term is sum over q_nq_n-k/n² where q_n is (troughly speaking) the Fourier mode of the charge distribution q(x). This can be derived via the standard canonical formalism with gauge A°(x)=0 and implementation of the Gauss law G(x)=0. G(x) generates residual gauge symmetries respecting A°(x)=0 and induces charge neutrality Q=0. Infinities can be regulated (even in QFT) quite easily; the simplest way to do this is point splitting / normal ordering. The divergence is trivial as 1+1 QED is super-renormalizable.

So the theory is well understood and everything can be derived rather rigorously. The same mechanism applies for 3+1 QED on the 3-torus, even if it is more involved. Again the Coulomb potential has the characteristic 1/n² term. So I think the problem is well understood, provided that Fourier modes can be used.

And this is exactly my problem: if the geometry is not the n-torus or if geometry is not fixed as in an expanding universe the Fourier decomposition is no longer applicable.

 

[Physics Monkey]

Well, if we forget about the expanding geometry for the minute, any fixed metric geometry has a laplacian with a well defined discrete spectrum. Also, the eigenfunctions of this laplacian form a complete basis analogous to the Fourier modes. So classically, the Poisson equation can still be solved so long as the net charge density remains zero. The eigenfunctions may be complicated, but there is no objection in principle. The issue of divergences is also not so bad, because these arise from the ultraviolet behavior of the theory and hence shouldn't care too much what the space is doing globally. This can be codified, for example, in the heat kernel expansion used in zeta function regularization. Divergences are related to local geometric features of the space and can removed with local counterterms.

Handing an expanding universe or trying to take into account propagation time effects is an extra layer of complication, but I'm not sure what really bothers you about this situation?

 

[tom.stoer]

... any fixed metric geometry has a laplacian with a well defined discrete spectrum. Also, the eigenfunctions of this laplacian form a complete basis analogous to the Fourier modes. So classically, the Poisson equation can still be solved so long as the net charge density remains zero. The eigenfunctions may be complicated, but there is no objection in principle.

So you say "fix the geometry", "solve the Laplacian" and finally "calculate the Coulomb potential". I agree that this is the correct way. Unfortunately in the case of dodecahedral spacetime already the first step is rather complicated.

Does anybody know about a paper where one can find the eigentfunctions? Is Mathematica able to do the job?

 

[qsa]

So you say "fix the geometry", "solve the Laplacian" and finally "calculate the Coulomb potential". I agree that this is the correct way. Unfortunately in the case of dodecahedral spacetime already the first step is rather complicated.

Does anybody know about a paper where one can find the eigentfunctions? Is Mathematica able to do the job?

This message has the attachment I could not send you through PM, it superimposes the simulation on a typical chart for hydrogen.

 

[Physics Monkey]

So you say "fix the geometry", "solve the Laplacian" and finally "calculate the Coulomb potential". I agree that this is the correct way. Unfortunately in the case of dodecahedral spacetime already the first step is rather complicated.

Does anybody know about a paper where one can find the eigentfunctions? Is Mathematica able to do the job?

I don't even know what the "dodecahedral spacetime" is, so I'm afraid I can't help on this point. Mathematica, MATLAB, etc can do some of these problems numerically, but it gets harder in 3 or more dimensions and you usually need a finite domain. Another approach you can use in two dimensions is conformal transformations. For example, Poisson's equation on a bumpy manifold with the topology of a torus is equivalent to Poisson's equation on the torus with a modified charge density. I'm not sure if this what you had in mind?

 

[tom.stoer]

Have a look at
http://en.wikipedia.org/wiki/Dodecahedron
and
http://en.wikipedia.org/wiki/Poincaré_homology_sphere#The_Poincar.C3.A9_homology_sphere_in_cosmology

You get dodecahedral spacetime in a similar way as you get the 2-torus out of a rectangle. Take two opposite edges of the rectangle and glue them together; the results is a cylinder. Then take the two circles of the cylinder and glue them together; the result is a 2-torus.

For dodecahedral spacetime it's a little bit more complicated: take the dodecahedron, identify two opposite faces and glue them together; there is one further condition regarding the twist (0°, 72°, 144°, ...). You get a multiple connected 3-manifold called the dodecahedral space.

 

[Physics Monkey]

Neat. Just out of curiosity, why are you interested in this space?

 

[tom.stoer]

Neat. Just out of curiosity, why are you interested in this space?

Because it predicts (I don't know how - that's why I am asking these silly questions :-) a kind of suppression of fluctuations of cosmic background radiation (which has been observed) and therefore is a good candidate for the topology of our universe!

 

[haael]

2) the geometry of space becomes dynamic? In GR the geometry of space is not fixed; typically space(time) will expand.

My intuition is, if some compactified dimension changes size, the Coulomb force changes strength. You can say that the fine structure constant changes or that the charge of an electron changes.
When you have D big dimensions plus some small compactified ones, the Coulomb force on large distances is proportional to $$r^{-D+2}$$. But the exact factor depends also on number and size of small dimensions. For only one compactified extra dimension, the strength is inversely proportional to the size of the dimension. If it gets 2 times bigger, all charges drop times 2.

For more small dimensions, the factor is proportional to the (hyper)volume of the compactified subspace.

1) the topology of space (spacetime) becomes more complex? There are e.g. speculations that our universe could have the topology of a dodecahedral space which could explain suppressions of CMB multipole moments.

If our universe's dimensions are compactified on such large scale, I would expect changes in momentum spectrum, just like only some momenta are allowed for electrons on atom orbits.

I also suspect, that the Coulomb force would be somewhat weaker on large distances. Suppose only one big circular compactified dimension. If two charges were placed "halfway" in the universe, anyone of them would feel the presence of the other from two opposite directions, thus the forces would cancel.

However, the most cool thing for me are small compactified dimensions. Maybe weak and strong forces are just the electromagnetic force, manifesting unusual behavior on scales smaller than the size of additional dimensions.

 

[tom.stoer]

When you have D big dimensions plus some small compactified ones ...

The question is how the large-scale structure (topology) affects the CMB.

If our universe's dimensions are compactified on such large scale, I would expect changes in momentum spectrum, just like only some momenta are allowed for electrons on atom orbits.

I also suspect, that the Coulomb force would be somewhat weaker on large distances. Suppose only one big circular compactified dimension. If two charges were placed "halfway" in the universe, anyone of them would feel the presence of the other from two opposite directions, thus the forces would cancel.

I fully agree. I can calculate all these effects for not too complicated topologies: T^3, S³, ... But for an expanding dodecahedral spacetime I have no idea: space is expanding, topology is far from trivial ...