= Thermal expectation value
{title2=$\langle O\rangle_\beta=\operatorname{Tr}(e^{-\beta H}O)/Z$}
For a <canonical ensemble> with finite <canonical partition function> $Z=\operatorname{Tr}e^{-\beta H}$, the thermal <density operator> is $\rho_\beta=e^{-\beta H}/Z$. An <observable> $O$ has
$$
\langle O\rangle_\beta=\frac{\operatorname{Tr}(e^{-\beta H}O)}{Z}
$$
when this <trace> exists.
In finite dimension, define imaginary-time evolution by $A(\tau)=e^{\tau H}Ae^{-\tau H}$, using $\hbar=1$. Cyclicity of the <trace> gives the thermal boundary identity
$$
\langle A(\tau+\beta)B\rangle_\beta=\langle BA(\tau)\rangle_\beta.
$$
This <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 $H|g\rangle=0$ and $H|e\rangle=\Delta|e\rangle$, the raising operator $A=|e\rangle\langle g|$ 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>.
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