KMS States, Imaginary Time, and Thermal Spectra

The Kubo–Martin–Schwinger (KMS) condition expresses thermal equilibrium directly in correlation functions. It connects imaginary-time periodicity, thermal spectra, and real-time response, including infinite systems where a global Gibbs density matrix need not exist.

Enter this chapter

Begin with Thermal Density Operators and the KMS Condition to derive the equilibrium condition from a Gibbs trace.

Helpful background. Thermal Density Operators and the KMS Condition and Retarded, Advanced, and Spectral Correlators develop the definitions used by the chapter’s opening pages.

Topics

1. Thermal Density Operators and the KMS Condition — Derives KMS from Gibbs cyclicity, tests a harmonic oscillator, and separates thermality from stationarity.
2. Infinite-Volume KMS States, Passivity, and Phase Multiplicity — Replaces the trace by a state on local observables and distinguishes extremal thermal phases from mixtures.
3. Imaginary Time and Matsubara Frequencies — Derives the thermal circle, mode frequencies, free propagators, zero modes, and sum contours.
4. Thermal Boundary Conditions and Graded Traces — Derives symmetry twists and separates thermal traces, supertraces, and refined indices.
5. Thermal Propagators and Spectral Representations — Builds the Wightman, causal, and Euclidean functions from one normalized spectral density.
6. Thermal Spectral Positivity and Sum Rules — States the positive-metric hypotheses, commutator moments, subtraction needs, and gauge or charged-channel exceptions.
7. Exact Euclidean–Real-Time Analytic Continuation — States the analytic and growth class that makes continuation unique and checks a free-field round trip.
8. Noisy Spectral Reconstruction as an Inverse Problem — Consumes a Volume 8 reconstruction and assigns a thermal limit on the conclusion from resolution, covariance, channel, limits, and constraints.

How the ideas fit together

Cyclicity of the thermal trace relates the two Wightman orderings by an imaginary-time shift. Together with the operator’s statistics, this yields thermal boundary conditions and Matsubara frequencies; the same relation fixes detailed balance in spectral representations Landsman and van Weert 1987, §§ 2–3, pp. 149–201.

These are exact relationships between functions with specified analyticity and growth. Recovering such a function from finitely many noisy Euclidean samples is a separate inverse problem. Positivity and sum rules restrict candidate spectra but cannot, by themselves, resolve a narrow peak or its width.

Review the chapter

Does a time-independent density operator necessarily satisfy KMS at some temperature?

Solution

No. Commuting with the Hamiltonian gives stationarity, but KMS additionally fixes relative occupations by the Boltzmann weights appropriate to the chosen dynamics. A stationary mixture with arbitrary energy populations generally violates thermal detailed balance.

Where to go next

Continue to Finite Density and Conserved Charges to identify which conserved quantities admit chemical potentials.

References

• Landsman, N. P., and Ch. G. van Weert. “Real- and Imaginary-Time Field Theory at Finite Temperature and Density.” Physics Reports 145, nos. 3–4 (1987): 141–249. doi:10.1016/0370-1573(87)90121-9.