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Conditional local KMS clock temperature

Physical scope

Orbitwise KMS-time rescaling for each actually constructed finite cell in the selected stationary model; no spatially varying global time rescaling, generic curved-spacetime KMS existence, equilibration law, or empirical temperature measurement.

Statement

Fix one cell $i$ of the finite preparation in Actually constructed Gibbs/KMS preparation, at its assigned event $p_{i} \in O$ , with $N_{i} = N_{ξ} (p_{i})$ , $T_{i} = T (p_{i})$ , and $μ_{i} = μ (p_{i})$ . Set $β_{K} : = (k_{B} T_{K})^{- 1} > 0$ in joules $^{- 1}$ . Let $A_{i} = B (H_{i})$ , let $G_{i} : = H_{i} - μ_{i} Q_{i}$ be its proper grand generator in joules, let $G_{K, i} : = N_{i} G_{i}$ be its Killing-time grand generator, and let $ω_{i} (X) : = Tr (ρ_{i} X)$ . Use length coordinates $x^{0} = c t$ , with $c > 0$ the speed of light in metres per second and the reference-normalized $ξ$ dimensionless in these coordinates. Choose an open interval $I_{K, i} \subseteq R$ containing $0$ on which the integral curve through $p_{i}$ remains in $O$ . Define the unique smooth curve $γ_{i} : I_{K, i} \to O$ by $γ_{i} (0) = p_{i}, \frac{d γ_{i}}{d t_{K}} (t_{K}) = c ξ_{γ_{i} (t_{K})},$ where $t_{K}$ is physical Killing time in seconds; the equation states its parameterization by the reference-normalized Killing field. This curve is future-directed timelike. The stationary normalization used in KMS orbit clock rescaling gives $ξ (N_{ξ}) = 0$ , so $N_{ξ} (γ_{i} (t_{K})) = N_{i}$ throughout this interval. Its proper-time parametrization is $Γ_{i} : N_{i} I_{K, i} \to O, Γ_{i} (τ) : = γ_{i} (τ / N_{i}),$ which satisfies $g (d Γ_{i} / d τ, d Γ_{i} / d τ) = - c^{2}$ . Thus $τ$ is proper time in seconds and $τ = N_{i} t_{K}$ on the geometric orbit. No completeness of $ξ$ is assumed: the geometric curve is defined on its flow interval, while the finite-matrix dynamics below are defined for all real parameters by their generators and use the same constant rescaling.

Define

$α_{t_{K}}^{K, i} (X) : = e^{i t_{K} G_{K, i} / ℏ} X e^{- i t_{K} G_{K, i} / ℏ}, α_{τ}^{l o c, i} (X) : = e^{i τ G_{i} / ℏ} X e^{- i τ G_{i} / ℏ},$

where both $t_{K}$ and $τ$ are measured in seconds, and $N_{i} > 0$ is dimensionless. Write $A_{i}^{0} : = {X \in A_{i} : [X, Q_{i}] = 0}$ for the neutral observable algebra. Then $ω_{i}$ is $β_{K}$ -KMS for $α^{K, i}$ and is $β_{l o c, i}$ -KMS for $α^{l o c, i}$ , where

$β_{l o c, i} = N_{i} β_{K} = \frac{1}{k_{B} T_{i}}, ℏ β_{l o c, i} is the strip width in seconds .$

On $A_{i}^{0}$ , the same proper-time automorphism is generated by the physical Hamiltonian $H_{i}$ , because its chemical-potential term commutes with every observable in that subalgebra. Thus the local temperature is $T_{i} = T_{K} / N_{i}$ for this constructed state and selected orbit. The conclusion is only orbitwise; it does not make a position-dependent lapse into one global time-rescaled dynamics.

Facts & Assumptions

Given: The selected stationary bath and one actual finite-cell Gibbs/KMS preparation, with $T_{K} > 0$ , $N_{i} > 0$ , and $ℏ > 0$ in joule-seconds.

[A1]

Selected stationary thermal equilibrium adopts one connected stationary bath with smooth future timelike Killing field $ξ$ , lapse $N_{ξ}$ , observer $u = ξ / N_{ξ}$ , positive proper-temperature field $T$ , and Killing inverse-temperature vector $β = u / (k_{B} T)$ . It does not assert a KMS state.

[F1]

Actually constructed Gibbs/KMS preparation adopts the finite density $ρ = ⨂_{j} ρ_{j}$ and its Killing-time grand-generator dynamics; the mathematical construction is conditional on ${A C}_{ω}$ through the cited finite-cell supplier. That postulate makes no generic curved-QFT state claim.

[F2]

Physical-time KMS strip equilibrium condition: a $β$ -KMS state has, for every observable pair, a bounded function continuous on the closed strip of width $β ℏ$ , holomorphic inside, and with boundary values $ω (A α_{t} (B))$ and $ω (α_{t} (B) A)$ .

[F3]

KMS orbit clock rescaling: if $ω$ is $β_{K}$ -KMS for $α$ , then for fixed dimensionless $λ > 0$ , $ω$ is $λ β_{K}$ -KMS for $α_{τ}^{λ} = α_{τ / λ}$ . On one stationary orbit with $τ = N t_{K}$ , it gives $β_{l o c} = N β_{K}$ and $T_{l o c} = T_{K} / N$ ; a spatially varying lapse is not one global rescaling.

[F4]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): ${A C}_{ω}$ means every countable family of nonempty sets has a choice function. It is carried from the finite-cell construction; no stronger choice principle is used.

Proof

1.1givenA1F1F2algebra

By [F1], the full product state is $β_{K}$ -KMS for the grand-generator dynamics. The embedded cell algebra $A_{i}$ is invariant under that dynamics: all lifted cell generators commute, and conjugation by the other tensor factors cancels, leaving $α_{t_{K}}^{K, i}$ . Restricting the full-state strip functions to observables in this invariant subalgebra gives the KMS boundary values for $ω_{i}$ , since the product density restricts to $ρ_{i}$ . Thus $ω_{i}$ is $β_{K}$ -KMS for $α^{K, i}$ .

2.1F1F2F3F4step 1.1algebra∎

Since $G_{K, i} = N_{i} G_{i}$ and $N_{i}$ is constant along the selected orbit, $α_{τ}^{l o c, i} = α_{τ / N_{i}}^{K, i}$ . Apply [F3] with $λ = N_{i}$ to obtain the $N_{i} β_{K}$ -KMS strip and width $ℏ N_{i} β_{K}$ . The stationary normalization in [F3] gives $T_{i} = T_{K} / N_{i}$ , so $N_{i} β_{K} = N_{i} / (k_{B} T_{K}) = 1 / (k_{B} T_{i})$ . If $X \in A_{i}^{0}$ , then conjugation by $e^{i τ μ_{i} Q_{i} / ℏ}$ fixes $X$ , so the local grand and physical $H_{i}$ automorphisms agree there.

Conditional local KMS clock temperature

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources