I recently became away of the Coleman-Mermin-Wagner theorem presented in [1802.07747] for higher-form symmetries and I have a question about how it might be applied to electromagnetism.

The theorem states: continuous $p$-form symmetries in $D$ spacetime dimensions are never broken if $p ≥ D − 2$.

In 3+1 spacetime dimensions, we consider the photon to be the Goldstone associated with a continuous $U(1)$ one-form symmetry. However, as stated in [1802.07747], in 2+1 dimensions, this interpretation as a Goldstone is no longer allowed.

Questions:

1. Suppose I have a dielectric material at finite temperature in 3+1 spacetime dimensions. Effectively, I now have a 3 dimensional system. Does this mean that the $U(1)$ one-form symmetry cannot be spontaneously broken?

2. If the answer to the above question is "yes," then the photon field can no longer be viewed as a Goldstone and thus need not satisfy Goldstone's theorem. As a result, I would expect the low-frequency dispersion relation $\omega \propto k$ should not hold (in the absence of fine-tuning). Are there any circumstances in which this dispersion relation holds exactly at arbitrarily low frequency (again in the absence of fine-tuning)?

3. If the answer to the above question in "no," then can we interpret the Coleman-Mermin-Wagner theorem as mandating that all finite-temperature materials have a finite skin-depth beyond which electromagnetic radiation gets exponentially damped?

I recently became aware of the Coleman-Mermin-Wagner theorem presented in [1802.07747] for higher-form symmetries and I have a question about how it might be applied to electromagnetism.

The theorem states: continuous $p$-form symmetries in $D$ spacetime dimensions are never broken if $p ≥ D − 2$.

In 3+1 spacetime dimensions, we consider the photon to be the Goldstone associated with a continuous $U(1)$ one-form symmetry. However, as stated in [1802.07747], in 2+1 dimensions, this interpretation as a Goldstone is no longer allowed.

Questions:

1. Suppose I have a dielectric material at finite temperature in 3+1 spacetime dimensions. Effectively, I now have a 3 dimensional system. Does this mean that the $U(1)$ one-form symmetry cannot be spontaneously broken?

2. If the answer to the above question is "yes," then the photon field can no longer be viewed as a Goldstone and thus need not satisfy Goldstone's theorem. As a result, I would expect the low-frequency dispersion relation $\omega \propto k$ should not hold (in the absence of fine-tuning). Are there any circumstances in which this dispersion relation holds exactly at arbitrarily low frequency (again in the absence of fine-tuning)?

3. If the answer to the above question in "no," then can we interpret the Coleman-Mermin-Wagner theorem as mandating that all finite-temperature materials have a finite skin-depth beyond which electromagnetic radiation gets exponentially damped?