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Integrability-qualified quantum variance Green–Kubo limit

Statement

Assume Countable Choice and the finite-matrix C $^{*}$ -identity hypothesis used by Bounded C-star Dyson response for finite-range spin dynamics. Use its finite-range spin algebra $A$ , strongly continuous automorphism group $α_{t}$ , and invariant state $ϕ$ obtained from consistent local densities. Let $V = V^{*} \in A$ satisfy $ϕ (V) = 0$ , and define $X_{T} = \int_{0}^{T} α_{t} (V) d t, G (t) = Re ϕ (V α_{t} (V)) .$ If $G \in L^{1} ([0, \infty))$ , then $\lim_{T \to \infty} \frac{ϕ (X_{T}^{2})}{2 T} = D : = \int_{0}^{\infty} G (t) d t \geq 0.$ For a finite list of centered self-adjoint observables $V_{1}, \dots, V_{m}$ , if each symmetrized covariance $G_{a b} (t) = \frac{1}{2} (Re ϕ (V_{a} α_{t} (V_{b})) + Re ϕ (V_{b} α_{t} (V_{a})))$ is absolutely integrable on $[0, \infty)$ , then $D_{a b} = \int_{0}^{\infty} G_{a b} (t) d t$ is a real positive semidefinite matrix. For a velocity observable measured in m s $^{- 1}$ , $D$ has units m $^{2}$ s $^{- 1}$ ; for general $V$ , its units are $[V]^{2}$ s. The result is a variance of a time-integrated observable, not a joint classical trajectory or a universal transport law.

Facts & Assumptions

Given: Countable Choice and the finite-matrix C $^{*}$ -identity hypothesis of the finite-range group construction; the invariant state and group of the algebraic response theorem; centered self-adjoint observables; and absolute integrability of the stated covariance functions.

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): Countable Choice is the exact choice assumption for the local-algebra completion; no full Axiom of Choice is used.

[F1]

Finite matrix C-star algebra and density-state duality: every finite local matrix algebra has its induced-norm C $^{*}$ -identity.

[F2]

Bounded C-star Dyson response for finite-range spin dynamics: under [A1], [F1], and the stated finite-range interaction assumptions, $α_{t}$ is a strongly continuous automorphism group on the quasi-local algebra and $ϕ$ is an invariant bounded state.

[F3]

Stationary-state covariance identity for time-integrated observables: for centered self-adjoint observables, the time-integrated symmetrized covariance equals the finite-time symmetrized variance and its matrix is positive semidefinite.

[F4]

Dominated convergence: pointwise convergence under an integrable majorant permits passage of limits through Lebesgue integrals.

Proof

technique · apply the stationary covariance identity and pass to the infinite-time limit by dominated convergence

Given: [A1], [F1]–[F4], the centered observables, and the integrability hypotheses.

1.1givenA1F1F2F3F4algebra

The algebraic group/state hypotheses follow from [A1], [F1], and the exact interaction and state assumptions carried by [F2]; applying the stationary covariance identity [F3] to the scalar observable gives $\frac{ϕ (X_{T}^{2})}{2 T} = \int_{0}^{T} (1 - \frac{t}{T}) G (t) d t .$ Extend the integrand by zero for $t > T$ . For each fixed $t \geq 0$ it converges to $G (t)$ as $T \to \infty$ , and its absolute value is at most $∣ G (t) ∣$ , which is integrable by hypothesis. Applying [F4] on $[0, \infty)$ yields the limit $\int_{0}^{\infty} G (t) d t$ .

2.1givenF2F3step 1.1algebra

For finite $T$ , $X_{T} = X_{T}^{*}$ by [F3], so $ϕ (X_{T}^{2}) = ϕ (X_{T}^{*} X_{T}) \geq 0$ . The limit in step 1.1 is therefore nonnegative.

3.1F2F3F4step 1.1algebra∎

For $m$ observables, [F3] gives the finite-time real symmetric covariance matrix with entries $D_{a b} (T) = \int_{0}^{T} (1 - t / T) G_{a b} (t) d t$ ; for every real vector $c$ , $c^{T} D (T) c = ϕ ((\sum_{a} c_{a} X_{a, T})^{2}) / (2 T) \geq 0$ . Each entry converges to $D_{a b} = \int_{0}^{\infty} G_{a b} (t) d t$ by the same dominated-convergence argument as step 1.1. For any $c$ , $c^{T} D c = \lim_{T} c^{T} D (T) c \geq 0$ , so $D$ is positive semidefinite.

Integrability-qualified quantum variance Green–Kubo limit

Statement

Facts & Assumptions

Proof

Depends on

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