= Thermal trace as a closed Euclidean path integral
{title2=$Z=\int_{x(\beta\hbar)=x(0)}\mathcal D x\,e^{-S_E[x]/\hbar}$}
Taking the <trace> of the position-space imaginary-time kernel identifies its endpoints and integrates over the common <position>. The resulting closed paths have imaginary-time <length> $\beta\hbar$. Bosonic coherent fields are periodic, while fermionic coherent fields are antiperiodic. This temporal condition follows from the <trace>, independently of the spatial <boundary conditions>.
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