Thermal expectation value ($⟨O{⟩}_{\beta }=\mathrm{Tr}(e^{-\beta H}O)/Z$)

The canonical ensemble assigns the density matrix $e^{-\beta H}/Z$, with $Z=\mathrm{Tr}{e}^{-\beta H}$. 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.