What are some good resources for understanding Gauge transformation in QED?

Question

I have been trying to understand the gauge transformation in the context of Cavity QED and was wondering if anyone could suggest good field theory textbooks that would give me the basic understanding to work with Gauge transformations.

Answer 1

There are multiple layers to this: gauge transformations in electrodynamics in general and then in cavity QED specifically.

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For the general part, you will find useful information in almost any quantum optics textbook. ScullyZubairy has a treatment. For others see Book Recommendation: Quantum optics

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For cavity QED specifically, there is not much to add unless you really want to go into the details. Then there is much to add and you are partially looking at current research.

For some general and detailed insight into the underlying theory, I can recommend a review on macroscopic QED Scheel&Buhmann2008.

Recently, it has been realized that the gauge aspects of the theory become particularly relevant and non-trivial in the ultra-strong coupling regime. See for example DiStefano2019, DeBernardis2018 or Malekakhlagh2017, which can can also be found on arxiv (https://arxiv.org/abs/1809.08749 , https://arxiv.org/abs/1805.05339 , https://arxiv.org/abs/1701.07935).

Answer 2

Just want to add a note to @Wolpertinger's answer regarding gauge aspects of the ultra-strong coupling regime.

The perturbative QED's way of splitting the minimal coupling covariant derivative into free portion (free Dirac fermion) and interaction portion (fermion-photon vertex) is unfortunately distasteful, since the separate terms in isolation are NOT gauge invariant. As a remediation, we resort to Ward identities to safeguard the gauge invariance of the physically detectable calculation results.

Physics practitioners sometimes may get carried away in various approximation schemes in QED, and forget about the doctrine of consistently keeping in line with minimal coupling. And this is a sure recipe for disaster, as demonstrated in the ultra-strong coupling regime references pointed out by @Wolpertinger.