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I would like to understand a bit more the aspects of the gauge theory in statistical field theory. In particular, I would like to understand how the replacement $\tau \rightarrow it/\hbar$ is performed in a mathematically proper way, when $\tau$ is an inverse temperature. This replacement comes from the resemblance between the evolution operator $e^{-iHt/\hbar}$ in quantum field theory, and the statistical weight $e^{-H\tau}$ in statistical physics (one sees then that $\tau=\left(k_{B}T\right)^{-1}$ in case you wonder :-).

In principle it leads to a compact momentum space, when the frequencies become discrete, and called Matsubara frequencies $\omega_{n}=2\pi k_B T \left(n+1/2\right)$ and $\omega_{n}=2\pi n k_{B}T$ with $n$ an integer for fermions and bosons. I'm perfectly aware of the classic book

Methods of quantum field theory in statistical physics by Abrikosov, Gor'kov and Dzyalochinski - Dover Books on Physics

but I'm stuck on the gauge formalism. Can we do a gauge transformation in imaginary time-$\tau$ as we do with real time-$t$ ? Does the imaginary-time-covariant derivative $\partial_{\tau}+A_{\tau}$ make any sense? Are there some precautions to take?

Any comment, answer, indications or even good reference (or even just keywords) about this topic is warm welcome. I precise I'm a condensed matter physicist, so if you could adapt your parlance to me (for instance, please talk slowly and loudly), I would greatly appreciate :-)

EDIT: Clearly the keyword is thermal quantum field theory and there is an associated Wikipedia page with plenty good references. Anyway, any comment are still welcome, since I progress really slowly understanding this, especially what a gauge choice mean? Thanks in advance.

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Under a Wick rotation, which is what you do in order to go from Minkowski to Euclidean space, both the partial derivative $\partial_0$ with respect to time and the zero-component of the gauge field transform as

$$\partial_0\rightarrow i\partial_\tau$$

$$A_0\rightarrow iA_0.$$

This defines the covariant derivative for statistical field theory.


The general gauge transformation behaviour of the theory does not change, apart from the sign changes due to imaginary time.

In a theory that exhibits confinement, Wilson loops wrapped around the compact time direction, i.e. Polyakov loops, tell us something about the deconfinement phase transition of the theory.

For a pedagocial introduction to thermal/statistical field theory, you might consider these lecture notes.

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Thanks for your answer. The compactness of the reciprocal space (the fact that the Matsubara frequencies are discrete) is not a problem ? What is the gauge transform in that case ? Thanks in advance for more details. – FraSchelle Mar 11 '14 at 10:07
@FraSchelle I have edited my answer and added some points. – Frederic Brünner Mar 17 '14 at 23:15
@FraSchelle: There are subtleties with the Matsubara frequencies and Ward Identities. For instance, one has to be careful when formally deriving an expression with respect to $\omega_n$ in an expression. – Adam Mar 18 '14 at 13:54
@FredericBrünner Thanks a lot for the complement and the reference as well. I'll have a check as soon as possible. – FraSchelle Mar 19 '14 at 10:06
@FraSchelle: they are useful! Ward identities impose symmetries on the propagators and other correlation functions, that you can use to prove powerful results, for example. – Adam Mar 19 '14 at 13:27

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