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In finite temperature QCD, the gauge fields must be periodic in temporal direction. They say this is the consequence of trace in the action for gauge fields. How does trace imply that the fields must be periodic?
This is related to the KMS condition. https://en.wikipedia.org/wiki/KMS_condition
I guess you mean the EUCLIDEAN temporal direction, and the period is $1/T$. This is part of the so called KMS condition as Arnold Neumaier already suggested. Actually, periodicity of the fields (or periodicity of the Euclidean temporal direction of the manifold where filds are defined) arises as soon as one tries to translate the KMS condition into the language of finite-temperature path integral.
let's say the trace is the expectation value. the action will be invariant so by calculating the expectation value of the action one would expect a minima on the path taken by a particle. This would be independent of time, the same physics will describe the dynamics tomorrow. hence, periodicity in the temporal direction is a way of saying that if something happens right now that something is equally likely to happen tomorrow, next week and so on.
It seems to me that you are making confusion between temporal invariance (that however holds since we are dealing with thermal equilibrium states) and Euclidean temporal periodicty.
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