Thermal AQFT, KMS States, and Nonequilibrium Structures

The KMS condition defines equilibrium through an algebra, a state, and a chosen time evolution. This chapter separates that equilibrium statement from phase uniqueness, relaxation, local thermality, and nonequilibrium steady flow.

Enter this chapter

Begin with the KMS strip identity and its representation before considering work extraction or dynamical limits. Helpful background. Thermal density operators and the KMS condition supplies the finite-system calculation; infinite-volume KMS states, passivity, and phase multiplicity supplies the physical thermodynamic-limit setting; passivity, work, and information in QFT supplies the operational work language.

Topics

1. C*-Dynamical Systems and the KMS Condition — Defines equilibrium without requiring a trace-class Gibbs operator.
2. Modular Dynamics and Equilibrium Representations — Relates physical and modular dynamics under representation hypotheses.
3. Passivity, Complete Passivity, and Ground States — Distinguishes one-copy passivity from equilibrium under arbitrary tensor powers.
4. Relativistic KMS Analyticity and Spectrum — States the stronger spacetime analyticity available in relativistic settings.
5. Factorial KMS States, Phases, and Symmetry Breaking — Explains extremal thermal phases and their possible mixtures.
6. Thermal Nuclearity, Return to Equilibrium, and Mixing — Separates existence of equilibrium from decay toward equilibrium.
7. Nonequilibrium Steady States and Entropy Production — Constructs stationary reservoir-flow limits and entropy currents.
8. Local Thermal Equilibrium and Thermal Observables — Compares selected local observables with thermal reference states.
9. Horizon KMS, Unruh, and Hawking Theorem Interfaces — Distinguishes geometric modular thermality from detector and radiation claims.

How the ideas fit together

KMS analyticity survives the thermodynamic limit even when a Gibbs density matrix does not Haag, Hugenholtz, and Winnink 1967, §§ 2–5. Complete passivity supplies a work-based characterization of KMS and ground states under the theorem’s hypotheses Pusz and Woronowicz 1978, Theorems 1.1–1.4 and 3.1. Neither result supplies mixing. The table therefore keeps equilibrium, representation, phase structure, and large-time limits as separate questions.

Hypotheses and licensed conclusions

Domains, assumptions, directional conclusions, and failure tests for equilibrium and nonequilibrium claims

Object and domain	Required hypotheses	Licensed conclusion	Excluded converse or upgrade	Adversarial check
State on $(\mathcal A,\alpha)$	Strongly continuous automorphisms; analytic elements; positive normalized functional; KMS boundary identity at fixed $\beta$	Equilibrium relative to the chosen time flow, including infinite systems without a density matrix	Stationarity alone is not KMS, and a finite-volume Gibbs formula is not an infinite-volume construction	Test a stationary diagonal state whose level populations are not Gibbs weights
Cyclic process and tensor powers	Well-defined generator and differentiable cyclic perturbations; passivity for every finite tensor power	Under the Pusz–Woronowicz hypotheses, a completely passive state is KMS or a ground state	Single-copy passivity does not imply complete passivity	Use a passive non-Gibbs three-level state and activate work extraction on several copies
KMS representation and phase decomposition	GNS representation, normal extension, faithful support reduction; extremality taken inside the KMS simplex	Modular implementation of equilibrium and factorial characterization of a pure thermodynamic phase	Factorial does not mean type I or vector-pure, and a convex mixture is not one phase	Decompose a central mixture of two disjoint factorial KMS states
Thermal construction and relaxation	For existence, thermal/phase-space nuclearity and controlled volume limits; for mixing, Liouvillean spectral and perturbative estimates	Existence of locally normal KMS states, or return to equilibrium in the stated folium and topology	Nuclearity does not imply uniqueness or mixing; KMS analyticity does not remove nonzero Liouvillean resonances	Add a conserved quantity or phase multiplicity and test whether correlations decay
Reservoir scattering state	Thermodynamic reservoirs, Møller or Cesàro limit, current domains, relative-entropy balance, and steady conservation	A stationary NESS and nonnegative weighted entropy production with a fixed current convention	A finite recurrent system does not establish a unique NESS; zero production need not imply equal temperatures	Set the transmission to zero between reservoirs at unequal temperatures
Local or horizon temperature claim	For LTE, a selected local-observable space and positive reference measure; for horizons, geometric modular action or a regular invariant state plus normalized Killing flow	Thermal compatibility at stated resolution, or a KMS period for the specified boost/Killing parameter	One thermometer does not determine a universal temperature; a local Rindler argument does not prove Hawking flux	Match the Wick-square moment with two distinct mixtures, or change the global state while keeping the same local horizon geometry

Structured table data (JSON).

Review the chapter

A state is stationary and passive for the tested single-copy cycles. Must it be a unique mixing KMS phase?

Solution

No. Stationarity does not imply the analytic KMS boundary identity, and single-copy passivity does not imply complete passivity. Even a KMS state may coexist with other phases; mixing additionally requires dynamical spectral information.

Where to go next

Continue to noncommutative information and measurement for state comparison and operations, or curved-spacetime AQFT for geometric applications.

References

• Haag, R., Hugenholtz, N. M., and Winnink, M. (1967). “On the equilibrium states in quantum statistical mechanics.” Communications in Mathematical Physics 5, 215–236. DOI.
• Pusz, W., and Woronowicz, S. L. (1978). “Passive states and KMS states for general quantum systems.” Communications in Mathematical Physics 58, 273–290. DOI.