Thermal expectation value
= Thermal expectation value
{title2=$\langle O\rangle_\beta=\operatorname{Tr}(e^{-\beta H}O)/Z$}
The <canonical ensemble> assigns the <density matrix> $e^{-\beta H}/Z$, with $Z=\operatorname{Tr}e^{-\beta H}$. Its <expectation value> of an observable is the displayed normalized trace. In imaginary time, cyclicity of this trace gives periodic bosonic <correlation functions>; it does not make the imaginary-time Heisenberg operator itself periodic.