Does Compactification Affect the Uncertainty of Field Localization on a Torus?

[thoughtgaze]

Integrating the lagrangian over spacetime in regular field theory (by regular i mean field theories with noncompact dimensions) gives the action. To do this, one integrates over all spacetime , minus infinity to plus infinity in each dimension. For field theories with compactified dimensions, does one still need to integrate over minus infinity to plus infinity? Or is it necessary to integrate only over the region 0 to 2piR where R is the radius of the compactified dimension?

And if anyone can point me to some useful discussions on this, it would be much appreciated.

 

[Orodruin]

Integrate over the base manifold. There is no need to cover it more than once for this purpose (which you will do if you integrate over more than 0 to 2pi if the base manifold is a circle of radius 1).

 

[ChrisVer]

There are many ways to do this... You can as well choose to go from -infinity to +infinity in your manifold. If the manifold is an $S_1$ (circle), you are just going to rotate around the circle again and again, which will give you a winding number similar to the integer you are getting by expanding your field in Fourier modes at the compactified dimension if you try to rotate it only once around the circle...
That is in general because the set of unimodular complex numbers is topologically equivalent to the unit circle.

(If I'm wrong someone could try to correct me)

 

[thoughtgaze]

Well the strange thing to me is that, let's say we are given some field configuration with the appropriate periodicity in the compactified domain. When we Fourier transform this function, it will in general be nonzero over the entire domain.

But if in the field theory we are only interested in the domain from 0 to 2piR, then why couldn't we also think of Fourier transforming a function that has the same configuration of interest within 0 to 2piR but is zero outside the region 0 to 2piR. You could then formulate it as an integral over the entire domain but the function itself would induce the cutoff because it is zero outside 0 to 2piR.

 

[ChrisVer]

Then what are you compactifying?

 

[Orodruin]

Depending on what you are doing, you can have boundaries on your compact space, in that case you need boundary conditions. If you are doing something with periodic boundary conditions I do not really see the benefit in considering it as part of the full real line.

 

[thoughtgaze]

Ultimately I am trying to understand it in the context of localized field configurations (let's say on a torus S¹×S¹) and the uncertainty associated with those field configurations.

Intuitively it seems there should be a maximum uncertainty on the localization of an event in this torus. The location of some event in this torus is maximally uncertain up to 2piR. However if I think of it in terms of integrating over the full real line, it is infinitely uncertain in a trivial sense because if a field is localized around 0 then it is also localized around n2piR for any integer n. Typically we Fourier transform the field with the given periodic boundary conditions, but this seems to miss the notion that the field is localized via compactification up to a maximal distance 2piR.

However if we say the function is well defined over some range with a distance 2piR, and zero elsewhere (equivalent to changing the integration region in the field theory), then we might retrieve the notion that it is maximally uncertain up to that distance.