Why is the periodicity of fields in finite temperature QCD consequence of Trace in the action? 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?
 A: A derivation is given for the case of a scalar field in these lecture notes. The same arguments apply for gauge fields in QCD.
The idea is that when calculating the partition function, which is equivalent to the euclidean action of the quantum field theory, we take the trace of the exponential of the hamiltonian $\hat{H}$ and possible other terms involving a chemical potential $\mu$ and the number operator $\hat{N}$:
$$Z=\text{Tr}\;\text{exp}[-\beta(\hat{H}-\mu \hat{N})].$$
The trace is given explicitely in terms of the fields $\phi$ by 
$$Z=\int d\phi\langle\phi|\text{exp}[-\beta(\hat{H}-\mu \hat{N})]|\phi\rangle.$$
As described in the lecture notes, further manipulation of this expression reveals a delta function, which ensures the periodicity of the fields with respect to time. 
A: 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.
