B Can 5D Kaluza-Klein Theory Be Adapted for 2D Space-Time Models?

[Spinnor]

Should one be able to take the results of 5D Kaluza-Klein theory and turn it into a theory that works with two less space dimensions?

Thanks!

 

[Orodruin]

It is unclear what you are asking. The entire point of KK theory is to have more than four space-time dimensions.

 

[Spinnor]

It is unclear what you are asking. The entire point of KK theory is to have more than four space-time dimensions.

Could you take the results from the 5D theory and simplify it to work in a made up space of one time, one space, and one compact dimension.

Thanks!

 

[Orodruin]

You will have to be more precise. FWhat do you mean by "work"?

 

[haushofer]

You mean to compactify three spatial dimensions instead of one? I think that should be possible, e.g. compactifying 5D GR to 2D by taking your compactification space being spheres. Normally, 2D GR is not defined in terms of a metric because the Einstein eqns are identically satisfied, so I have to think about that or do the compactification explicitly :P

 

[Spinnor]

I guess GR is maybe boring with one space and one time dimension? Here I'm thinking we have one large space dimension and one compact space dimension in addition to time. I was hoping that KK theory could be simplified so that I might better understand the geometrical aspect of electrodynamics in lower dimensions.

Thanks!

 

[Spinnor]

You will have to be more precise. FWhat do you mean by "work"?

It should reproduce classical electrodynamics of one space and one time dimension by proper interpretation of an additional compact dimension.

 

[Spinnor]

So for a charged particle with no external forces the particles motion is a helix of a fixed pitch around a cylinder defined by the one large space dimension and the one small compact dimension? The greater the pitch of the helix the greater the momentum in the large dimension, assume momentum is constant in the compact dimension? If there is an external field which causes a force the pitch of the helix is no longer constant in time? The path the particle takes is the shortest by some measure?

Thanks!

 

[haushofer]

I guess GR is maybe boring with one space and one time dimension? Here I'm thinking we have one large space dimension and one compact space dimension in addition to time. I was hoping that KK theory could be simplified so that I might better understand the geometrical aspect of electrodynamics in lower dimensions.
Thanks!

Well, it's not that GR is 'boring' in 2D, it is simply not defined because the Einstein eqns are identities. So what people try to do (at least, what I know of), is to describe gravity by a scalar field (dilatonic gravity). The reason is that if you fix your 2 diffeomorphisms in two dimensions, a metric has 1/2*2*(2+1) - 2= 1 degree of freedom. But if I were you, I would carry out the compactification explicitly and see what you get :) There are several interestic geometric identities in 2 and 3 dimensions, and I'm not sure how these mix up with compactification.

 

[haushofer]

It should reproduce classical electrodynamics of one space and one time dimension by proper interpretation of an additional compact dimension.

Actually, that's an interesting question, because GR in D=3 doesn't have gravitational waves with the Einstein eqn's; the Riemann tensor is completely fixed by the Ricci tensor and the metric. So the question is how that translates itself into the electrodynamics.

 

[haushofer]

So to answer your question in the OP which I now understand: I think you have to be very careful in just copy-pasting the 5D results to the 3D results. You have to do the compactification explicitly to find out because of the reasons stated earlier.

 

[Spinnor]

Thanks for the replies! Been a bit under the weather. I guess to try and make things a bit more simple I wonder if one can start with 2D Minkowski space-time and then just add a single compact dimension as Kaluza did and formulate classical electrodynamics in this new space? I am really at this point only interested in the geometrical aspects of electrodynamics in this space. I was told in a thread that QED in 2D space-time is simple but not trivial.

6th post in thread below,

https://www.physicsforums.com/threads/qed-in-3d-qed-in-1d-what-changes.403293/#post-2721506

Thanks again for the help!

 

[haael]

You can construct a model with as many dimensions as you wish. It may be 1 spatial, 1 temporal and 1 compactified if you wish.

As whether this theory would be applicable to the real world: no. You will not reconstruct the Coulomb's law from it, for instance.

It might be useful however for such setups as quantum dots.