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

For a canonical ensemble with finite canonical partition function $Z=\mathrm{Tr}{e}^{-\beta H}$, the thermal density operator is ${\rho }_{\beta }=e^{-\beta H}/Z$. An observable $O$ haswhen this trace exists.

In finite dimension, define imaginary-time evolution by $A(\tau )=e^{\tau H}A{e}^{-\tau H}$, using $\hslash =1$. Cyclicity of the trace gives the thermal boundary identityThis Kubo--Martin--Schwinger condition swaps the operator order. It yields periodicity for bosonic imaginary-time-ordered correlation functions, rather than asserting that every unordered correlator is periodic.

Nor is the Heisenberg operator itself generally periodic. For $H|g⟩=0$ and $H|e⟩=\Delta |e⟩$, the raising operator $A=|e⟩⟨g|$ evolves as $A(\tau )=e^{\tau \Delta }A$, which is not periodic for real nonzero $\Delta$. The thermal periodicity belongs to the appropriately ordered expectation values.