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Unitary entropy and Gibbs equilibrium identities

Physical scope

Normal-state unitary model predictions and nonzero finite-dimensional canonical stationarity/KMS identity; real time is measured in seconds, and no irreversible relaxation, infinite-volume KMS theorem, or apparatus claim is made.

Statement

Assume the normal-state and canonical-preparation interpretations of Normal-state quantum measurement interpretation and Canonical quantum thermal preparation, and fix the positive entropy scale $k_{B} > 0$ in J/K. Model an isolated system on a separable complex Hilbert space $H$ by the unitary evolution $U (t) = e^{- i t H / ℏ}$ for real time $t$ measured in seconds, where the self-adjoint Hamiltonian $H = H^{*}$ has energy units J and $ℏ > 0$ has action units J s. For every trace-class normal density $ρ$ on $H$ , the evolved state $U (t) ρ U (t)^{*}$ is a normal density and has the same von Neumann entropy, including when that entropy is $+ \infty$ . For the finite canonical/KMS clause, require $d = \dim H \geq 1$ and inverse-energy parameter $β > 0$ in J $^{- 1}$ ; then the canonical density $σ_{β} = e^{- β H} / Z_{β}$ is stationary under this evolution and obeys the finite Gibbs KMS strip identity $F_{A, B} (t + i β ℏ) = Tr (σ_{β} α_{t} (B) A), F_{A, B} (z) = Tr (σ_{β} A α_{z} (B)), α_{z} (B) = e^{i z H / ℏ} B e^{- i z H / ℏ},$ for bounded matrices $A, B$ , with complexified time $z$ measured in seconds. These are deductions inside the specified model; they do not imply equilibration, entropy increase, or infinite-volume KMS existence.

Facts & Assumptions

Given: Full Choice as retained by the spectral suppliers; a separable Hilbert space $H$ , a trace-class density, a self-adjoint Hamiltonian $H$ with energy units, real time $t$ in seconds, positive action unit $ℏ$ in J s, $k_{B} > 0$ for the entropy clause; the adopted normal-state interpretation; and, for the KMS clause, finite dimension $d \geq 1$ with inverse-energy parameter $β > 0$ in J $^{- 1}$ .

[A1]

The Axiom of Choice: full Choice is retained by the unbounded spectral and trace-state supplier closure.

[F1]

Normal-state quantum measurement interpretation: prepared states are trace-class densities on a specified separable Hilbert space, with the stated probability interpretation.

[F2]

Canonical quantum thermal preparation: in the finite canonical model, the adopted preparation is $e^{- β H} / Tr (e^{- β H})$ .

[F3]

Trace-state cyclicity and spectral probability calculus: a trace-class density has a finite or countable positive spectral-weight list; its entropy is the corresponding nonnegative sum, possibly infinite.

[F4]

Finite unitary dynamics and Gibbs KMS identity: for finite Hermitian $H$ , the unitary is $e^{- i t H / ℏ}$ , the Gibbs state is stationary, and the stated analytic strip boundary identity holds for bounded matrices.

[F5]

Trace class is a two sided Banach operator ideal: multiplication of a trace-class operator on either side by a bounded operator preserves trace class.

[F6]

Spectral theorem for unbounded self-adjoint operators (PVM form), Integral of a measurable function against a projection-valued measure, and Pvm integral is a star homomorphism: the self-adjoint $H$ has a spectral PVM $E_{H}$ ; bounded Borel functions define bounded spectral integrals, and the integral preserves products, conjugation, and the constant function. Thus for $f_{t} (λ) = e^{- i t λ / ℏ}$ , $U (t) = f_{t} (E_{H})$ satisfies $U (t)^{*} U (t) = U (t) U (t)^{*} = I$ , since $\overline{f_{t}} f_{t} = 1$ .

[L1]

Nuclear bounded cyclicity for trace-class operators, parts (a)–(b): under Countable Choice, trace-class $T$ and bounded $B$ satisfy $Tr (B T) = Tr (T B)$ . In the proof take $T = U ρ$ and $B = U^{*}$ ; no partial-trace or unbounded cyclicity clause is invoked.

Proof

1.1givenA1F1F3F5F6L1algebra

By [F6], the bounded Borel function $f_{t} (λ) = e^{- i t λ / ℏ}$ defines a unitary $U (t) = f_{t} (E_{H})$ for each real $t$ . By [F3], write $ρ = \sum_{j} p_{j} ∣ u_{j} ⟩ ⟨ u_{j} ∣$ in trace norm, with $p_{j} > 0$ , orthonormal $u_{j}$ , and $\sum_{j} p_{j} = 1$ . Since $U (t)$ is bounded, [F5] preserves trace class under $U (t) ρ U (t)^{*}$ . Its quadratic forms are nonnegative because $⟨ U ρ U^{*} v, v ⟩ = ⟨ ρ U^{*} v, U^{*} v ⟩ \geq 0$ . The bounded cyclic trace identity [L1] gives $Tr (U ρ U^{*}) = Tr (ρ U^{*} U) = 1$ . The vectors $U u_{j}$ are orthonormal eigenvectors with the same eigenvalues $p_{j}$ ; the orthogonal complement of their span is the conjugated kernel. Therefore the entropy sum is termwise identical to that of $ρ$ . Since all summands are nonnegative, the equality also holds when the sum is $+ \infty$ . This mathematical statement is interpreted under [F1].

2.1givenstep 1.1F2F4∎

In finite dimension, apply [F4] to the canonical state from [F2]; it gives stationarity and the analytic strip identity under the exact bounded-matrix hypotheses. The entropy invariance in step 1.1 also applies to this finite Gibbs state. The conclusions are conditional predictions of this unitary model, with no claim that a system reaches the canonical state dynamically.

Unitary entropy and Gibbs equilibrium identities

Statement

Facts & Assumptions

Proof

Depends on

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Sources