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The fluctuation-dissipation theorem relates equilibrium fluctuations to dissipative linear response. The ordering and normalization of a quantum correlation function matter. For a Hermitian operator $A$ in thermal equilibrium, defineFor nonzero frequency, with spectral function $\rho (\omega )=-2\mathrm{Im}{G}_R(\omega )$ and $\beta =1/(k_{B}T)$,Here $k_B$ is the Boltzmann constant and $\hslash$ is the reduced Planck constant. In units $k_B=\hslash =1$, this reduces to $S^{>}(\omega )=\rho (\omega )/(1-e^{-\omega /T})$.

To see the origin, the Kubo--Martin--Schwinger condition gives $S^{<}(\omega )=e^{-\beta \hslash \omega }{S}^{>}(\omega )$ for the reversed ordering. The commutator spectral function obeys $\hslash \rho =S^{>}-S^{<}$, which proves the displayed relation. The symmetrized spectrum instead satisfiesAt $|\hslash \omega |\ll k_{B}T$, both forms approach $k_{B}T\rho (\omega )/\omega$. This connects a measured noise spectrum with the dissipative part of the corresponding response, after matching the experimental spectrum's ordering and Fourier conventions. ^[ref]

The Kubo--Martin--Schwinger condition uses cyclicity of the thermal trace to relate correlation functions separated by imaginary time $\beta$.

Let $\beta =1/T$ and $A(\tau )=e^{H\tau }A{e}^{-H\tau }$. For $0<\tau <\beta$, cyclicity of the trace givesThis is the bosonic Kubo--Martin--Schwinger condition. It identifies the two time orderings across the end of the thermal interval, so

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.