Thermal AQFT, KMS States, and Nonequilibrium Structures

Thermal algebraic QFT replaces a finite-volume density matrix by a state on an algebra with a distinguished automorphism group. The KMS boundary identity is the equilibrium criterion; modular theory explains its representation-theoretic form; complete passivity gives its work-theoretic characterization; and phase decomposition shows why equilibrium need not be unique. None of these facts alone proves relaxation, transport, a local temperature, or horizon radiation.

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.

Enter this chapter

A $C^*$-dynamical system $(\mathcal A,\alpha)$ consists of a $C^*$-algebra and a strongly continuous one-parameter automorphism group. At inverse temperature $\beta>0$, a state $\omega$ is KMS when each suitable pair $A,B$ has an analytic strip function with boundary values

$$F_{A,B}(t)=\omega(A\alpha_t(B)), \qquad F_{A,B}(t+i\beta)=\omega(\alpha_t(B)A).$$

This formulation survives when no trace-class $e^{-\beta H}$ exists. In the GNS representation it is tied to modular flow, but equality between physical and modular parameters requires faithfulness, invariance, and the correct represented dynamics. Complete passivity characterizes KMS and ground states under the theorem’s regularity assumptions, while extremal KMS states identify pure thermodynamic phases rather than vector-pure states.

The dependency map below should be read as a sequence of licensed upgrades. KMS analyticity first fixes equilibrium relative to a chosen flow. Representation theory, passivity, phase structure, and relativistic analyticity branch from that input. Phase-space and Liouvillean estimates then address existence and relaxation. Scattering limits construct steady states; selected point fields define local thermality; geometric modular action supplies one horizon interface. No reverse arrow is implicit.

The selected automorphism group and KMS boundary identity are the common equilibrium input. Modular realization, complete passivity, factorial phase analysis, and relativistic tube analyticity require additional named hypotheses. Nuclearity can construct thermal states, while Liouvillean spectral information is needed for mixing; reservoir scattering constructs a NESS; a chosen set of local fields defines local thermal compatibility; and geometric modular action fixes the wedge normalization. The diagram is schematic and not to scale. Structured description and source data (JSON)

The infinite-system KMS framework was established in Haag, Hugenholtz, and Winnink 1967, §§2–5, pp. 217–232. The complete-passivity characterization is Pusz and Woronowicz 1978, Theorems 1.1–1.4 and 3.1, pp. 275–287.

The chapter sequence

Read the pages in this order.

1. C*-dynamical systems and the KMS condition defines the analytic strip identity and checks it in finite and free-field examples.
2. Modular dynamics and equilibrium representations identifies the represented physical flow with the modular group under explicit faithfulness hypotheses.
3. Passivity, complete passivity, and ground states separates single-copy work inequalities from their tensor-power strengthening.
4. Relativistic KMS analyticity and spectrum replaces a one-dimensional strip by a covariant thermal tube and records its spectral consequences.
5. Factorial KMS states, phases, and symmetry breaking relates extremal equilibrium states, centers, disjoint representations, and phase mixtures.
6. Thermal nuclearity, return to equilibrium, and mixing distinguishes construction of KMS states from Liouvillean decay and relaxation.
7. Nonequilibrium steady states and entropy production constructs stationary flux states from reservoir scattering and fixes the entropy-current sign.
8. Local thermal equilibrium and thermal observables compares nonstationary states with convex thermal references at finite observational resolution.
9. Horizon KMS, Unruh, and Hawking theorem interfaces separates wedge modular thermality, detector response, bifurcate-horizon state theorems, and outgoing radiation.

The order moves from equilibrium definition to representation and work, then to phase and dynamical questions. Local and horizon temperatures come last because both require a specified observable or geometric flow; neither is a substitute for the underlying state hypotheses.

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) preserves the caption, scoped headers, rows, and reading order.

These rows separate four questions often conflated in informal thermal reasoning: whether an equilibrium state exists, whether it is unique, whether other states approach it, and whether a chosen observable looks thermal. Each question has its own topology, representation, and limiting procedure.

The failure map organizes the fastest countertests. Read each lower box as the strongest statement that survives after one assumption is removed. In particular, stationarity survives without KMS analyticity, equilibrium states survive without mixing, time-averaged steady states survive without a scattering limit, and a local acceleration scale survives without a global Hawking state.

Each checkpoint removes one logically independent input. A stationary non-Gibbs state defeats the KMS converse; a passive non-Gibbs state can fail on tensor powers; multiple phases or a conserved Liouvillean mode block uniqueness and mixing; finite reservoirs recur and a Cesàro NESS need not be unique; one local thermal moment does not fix a temperature distribution; and local Rindler form does not choose a regular global black-hole state or produce an asymptotic flux. The diagram is schematic and not to scale. Structured description and source data (JSON)

Scope boundaries

This chapter establishes theorem hypotheses and algebraic constructions. Explicit many-body thermal spectra, kinetic coefficients, hydrodynamic constitutive relations, and detector switching calculations remain in their physical volumes. Finite-volume Gibbs matrices are useful test cases but never replace an infinite-volume state construction. Likewise, modular flow is representation-dependent algebraic structure; it becomes physical time only when a theorem identifies the two flows.

The nonequilibrium pages treat stationary reservoir limits and local thermal comparison, not a universal theory of thermalization. Horizon results are conditional on geometry and state regularity. The black-hole evaporation problem additionally requires an asymptotic state, scattering through the exterior, renormalized stress energy, and backreaction.

Review the chapter

For any proposed thermal theorem, ask in order:

1. What is the algebra, what is the automorphism group, and in which topology is it continuous?
2. Is the claim KMS, ground, passive, completely passive, stationary, mixing, or merely locally thermal?
3. Which representation is used, and is the state faithful on the relevant support?
4. Does the thermodynamic limit exist, and is local normality controlled?
5. Are phases factorial, disjoint, or centrally mixed?
6. Which Liouvillean spectral statement licenses decay, and in what folium and topology?
7. For a NESS, which large-time limit exists and which sign convention defines the currents?
8. For local thermality, which observable space and reference measures are allowed?
9. For a horizon temperature, which flow is normalized and which global state theorem is available?

Synthesis exercise

A state is invariant under time translations, passive for the tested one-copy cycles, and has a two-point function periodic at one imaginary-time separation. May one conclude that it is a unique mixing KMS phase?

Solution

No. Invariance is weaker than the full KMS strip identity for all analytic pairs. One-copy passivity is weaker than complete passivity. A single periodicity check does not establish the required analytic function or positivity. Even a proved KMS state can belong to a nontrivial phase simplex, and mixing requires separate Liouvillean spectral information. Each proposed upgrade therefore needs an independent hypothesis and theorem.

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.
• Jakšić, V., and Pillet, C.-A. (2002). “Mathematical theory of non-equilibrium quantum statistical mechanics.” Journal of Statistical Physics 108, 787–829. 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.