KMS States, Imaginary Time, and Thermal Spectra

This chapter decides whether an equilibrium claim is a state identity, an imaginary-time rule, a spectral constraint, an exact analytic continuation, or a resolution-limited inference. KMS is the common origin: it replaces a finite-volume Gibbs trace in infinite systems, fixes thermal boundary conditions and detailed balance, and connects Euclidean correlators to causal real-time functions without erasing the difference between exact and noisy data.

The relation among KMS equilibrium, real-time functions, and imaginary-time thermal field theory is reviewed in Landsman and van Weert 1987, §§ 2–3, pp. 149–201.

Enter this chapter

There is no hard prerequisite for the overview, but a prepared reader can distinguish a density operator from a pure state and a retarded correlator from a time-ordered one. Review Thermal Density Operators and the KMS Condition for the former and Retarded, Advanced, and Spectral Correlators for the latter.

Use these diagnostics.

• Can you derive an imaginary-time shift by moving $e^{-\beta H}$ through a trace? Enter at KMS. If you can do so only in finite volume, continue immediately to infinite-volume KMS.
• Can you explain why a thermal fermion is antiperiodic but a fermion in a supertrace is periodic? Enter at boundary conditions and graded traces.
• Given $\rho(\omega)$, can you reconstruct both $G_R$ and $G_E$ with the correct sign? Enter at thermal propagators.
• Given twelve noisy Euclidean times, would you call $i\omega_n\to\omega+i0$ exact? If not, compare exact continuation with spectral inference.

Choose a route

Goal	Route	Result you should be able to demonstrate
Graduate thermal core	KMS → imaginary time → spectral representations	Derive periodicity, Matsubara modes, detailed balance, and the Euclidean/retarded dictionary
Infinite-system equilibrium	KMS → infinite-volume states → Chapter 1 limits	State equilibrium without a nonexistent global Gibbs trace and distinguish extremal phases from mixtures
Supersymmetric or twisted trace	imaginary time → graded traces	Derive the twist and refuse to identify an index with thermal pressure
Spectral consistency	propagators → positivity and sum rules	Qualify a channel and test normalization, signs, moments, and ultraviolet subtractions
Euclidean-to-real-time inference	exact continuation → Volume 8 inverse methods → thermal claim ceiling	Distinguish mathematical uniqueness from numerical resolution and report only identifiable features

One correlator dictionary

For a neutral bosonic Hermitian operator in the chapter’s convention,

$$\rho=G^>-G^<, \qquad G^>=e^{\beta\omega}G^<, \qquad \mathcal G(z)=\int\frac{\mathrm d\omega'}{2\pi} \frac{\rho(\omega')}{z-\omega'}.$$

The retarded and advanced correlators are the upper and lower boundary values of $\mathcal G$. For nonzero Matsubara frequencies the Euclidean correlator uses the same $\rho$ and obeys $G_E(i\omega_n)=-\mathcal G(i\omega_n)$ in the displayed convention; $n>0$ continues to the retarded boundary, whereas $n<0$ continues to the advanced boundary. This minus sign, the $i0$ side, the zero-mode limit, the operator adjoint, and any charge twist must travel together. Positivity applies only after the Hilbert-space metric and channel are qualified.

The central fork is evidential:

• a complete exact correlator in a controlled analytic class has a unique continuation;
• a finite noisy data vector determines only resolution-smeared spectral information;
• positivity and sum rules narrow the admissible family but do not guarantee a unique peak or transport coefficient.

Guide to the eight pages

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 claim ceiling from resolution, covariance, channel, limits, and constraints.

Conventions and recurrent checks

The site uses the mostly-minus Lorentzian metric and forward transform $e^{+ip\cdot x}$. This chapter defines $G_R=-i\theta(t)\langle[\mathcal O(t),\mathcal O(0)]\rangle$, so $\rho=-2\operatorname{Im}G_R$ on the upper boundary. Euclidean time has $0\le\tau<\beta$ and explicitly declared periodicity. A chemical potential, gauge holonomy, fermionic grading, or alternate retarded sign receives a local translation.

The harmonic oscillator and free scalar provide recurring round trips: Gibbs ratios imply KMS; KMS implies Bose weights; Matsubara sums recover equal-time occupation; one spectral delta pair reconstructs both Euclidean and retarded propagators; the first moment recovers the canonical commutator. These examples test normalization and signs but do not model broad interacting spectra or inverse-problem resolution.

Misconceptions to reject

• A stationary state need not be KMS.
• A global Gibbs density matrix need not exist in infinite volume.
• Periodic fermions compute a graded trace, not an ordinary thermal trace.
• Positive spectral weight is not guaranteed for gauge-variant or off-diagonal channels.
• Exact analytic uniqueness does not make finite noisy reconstruction stable.
• A fit-compatible transport peak is not uniquely identified by Euclidean closure.

Review the chapter

Derivation. Move $e^{-\beta H}$ through a trace and recover the direction of the KMS shift. Verify it with $a(t)$ for a harmonic oscillator.

Representation change. Starting from a free scalar $\rho$, calculate $G_R$, $G_E(i\omega_n)$, and $G_E(\tau)$. A correct result passes the commutator moment and thermal periodicity checks.

Comparison. Contrast $\operatorname{Tr}e^{-\beta H}$ with $\operatorname{Tr}[(-1)^Fe^{-\beta H}]$. A correct answer identifies boundary conditions, positivity, cancellations, and the different physical question.

Failure diagnosis. A gauge-fixed propagator reconstruction violates $\rho(\omega>0)\ge0$. Explain why this need not invalidate the data. A correct answer checks whether the operator belongs to a positive physical Hilbert-space channel before applying positivity.

Inference check. A narrow peak is stable under one regularization scan but narrower than every reported resolution kernel. A correct answer reports a smeared integral or model-conditional feature, not a resolved width.

Continue

Finite Density and Conserved Charges adds chemical twists, charge susceptibilities, thresholds, and access barriers. Thermal Perturbation Theory and Renormalization evaluates the sum-integrals introduced here. Real-Time Contours, Keldysh Bases, and Response removes equilibrium as an assumption while preserving causal and normalization checks.

References

• Baym, Gordon, and N. David Mermin. “Determination of Thermodynamic Green’s Functions.” Journal of Mathematical Physics 2, no. 2 (1961): 232–234. doi:10.1063/1.1703704.
• Haag, Rudolf, Nico M. Hugenholtz, and Marius Winnink. “On the Equilibrium States in Quantum Statistical Mechanics.” Communications in Mathematical Physics 5 (1967): 215–236. doi:10.1007/BF01646342.
• 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.