Identify thermal equilibrium through correlations

Use $\hbar=k_B=c=1$, metric $(+,-,-,-)$, and Heisenberg evolution $A(t)=e^{iHt}Ae^{-iHt}$. Temperature therefore has energy units, and $\beta=1/T$ has inverse-energy units. Our real-time Fourier transform uses $e^{i\omega t-i\mathbf k\cdot\mathbf x}$, with inverse factors of two pi. For a neutral Hermitian bosonic operator A, define raw unordered correlations without factors of i. These conventions will also fix the retarded signs later in the module. Chemical potentials are set to zero throughout these elementary relations.

The Gibbs state commutes with H, giving time-translation invariance. Cyclicity of the trace, combined with the Boltzmann weight, relates an operator moved around that trace to evolution by an imaginary time interval. The resulting KMS identity holds in the appropriate analytic strip for well-defined correlations. It is stronger than stationarity and does not assume free fields. With nonzero conserved-charge chemical potentials, charged operators acquire additional charge-dependent factors; substituting a chemical potential by intuition would not preserve the present formulas.

Fourier transforming the imaginary-time shift yields $C^>(\omega)=e^{\beta\omega}C^<(\omega)$ under our positive-sign forward transform. A spectral decomposition makes the factor transparent: swapping the initial and final energy eigenstates changes their Gibbs probabilities by the exponential of the exchanged energy. For a Hermitian A, the lesser spectrum at positive frequency equals the greater spectrum at negative frequency. Positive-frequency fluctuations can therefore outweigh negative-frequency fluctuations without violating stationarity. The ratio is a thermal population factor, not an independently adjustable emission parameter.

Fermionic signs require separating raw products from signed Green functions. For an annihilation operator, the raw greater product and raw lesser product still satisfy a positive Boltzmann ratio from the ordinary trace. Standard fermionic conventions instead define the greater function with minus i and the lesser with plus i. Their KMS relation consequently contains a minus sign. Fermionic imaginary-time antiperiodicity additionally follows from fermionic time ordering. Attributing every minus sign directly to cyclicity would conflate three different operations and spoil the occupation factors.

A finite system diagonal in its energy basis is stationary for any chosen populations. Only special population ratios correspond to a Gibbs temperature. Checking correlations at several transition frequencies can distinguish those cases, whereas fitting one transition can always produce a misleading effective temperature. In a field theory, operator choice and accessible frequency range similarly determine how demanding the diagnostic is. An equilibrium claim should be supported by compatible spectral ratios and conserved quantities rather than a single approximately constant occupation number.

KMS does not say that a thermal state has no fluctuations or no dissipation. It constrains how fluctuations and response are related. Conversely, a driven stationary state may have constant macroscopic properties while violating this thermal relation. Nonequilibrium contour methods can describe such states without imposing KMS. Reserve the thermal identity for an actual equilibrium ensemble or a controlled local-equilibrium approximation, and record which time and length scales justify using the latter. A transient fit to a Boltzmann-looking slope is not a proof of global equilibrium.