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Conserved spectral probability laws for commuting observables and stationary eigenrays

Statement

Assume the Axiom of Choice and the Axiom of Countable Choice (The Axiom of Choice, The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $H$ be a nonzero complex Hilbert space (Hilbert space), let $K$ be a self-adjoint operator on $H$ with spectral projection-valued measure $E = E_{K}$ on the Borel subsets of $R$ (Spectral theorem for unbounded self-adjoint operators (PVM form), Projection valued measure), and put $U (t) : = e^{- i t K / ℏ} = \int_{R} e^{- i t λ / ℏ} d E (λ)$ for $ℏ > 0$ , the strongly continuous unitary group generated by $- i K / ℏ$ (A self-adjoint operator generates a strongly continuous unitary group, Integral of a measurable function against a projection-valued measure).

(a) If a bounded operator $B \in B (H)$ satisfies $B E (C) = E (C) B$ for every Borel set $C \subseteq R$ , then $B U (t) = U (t) B$ for every $t \in R$ .

(a') If a bounded self-adjoint operator $S$ satisfies $S K \subseteq K S$ — that is $S (K x) = K (S x)$ for every $x \in D (K)$ — then $S U (t) = U (t) S$ for every $t$ .

(b) Let $F$ be a projection-valued measure on a measurable space $(Y, Σ)$ with $F (S) E (C) = E (C) F (S)$ for all $S \in Σ$ and all Borel $C \subseteq R$ . Then:

(i) $F (S) U (t) = U (t) F (S)$ for all $t$ and $S$ ;

(ii) for every density operator $ρ$ on $H$ — a self-adjoint positive trace-class operator with $tr ρ = 1$ (Trace class operator, Density operators and scalar spectral probability clause (a)) — the numbers $tr (ρ F (S))$ are independent of $t$ : $tr (U (t) ρ U (t)^{*} F (S)) = tr (ρ F (S)) (t \in R, S \in Σ);$

(iii) for every unit vector $ψ \in H$ and every $S$ , $∥ F (S) U (t) ψ ∥^{2} = ∥ F (S) ψ ∥^{2}$ for all $t$ ;

(c) applying (b) with $F = E$ : for every density operator $ρ$ the energy law $tr (ρ E (C))$ is independent of $t$ , for every unit $ψ$ the law $∥ E (C) U (t) ψ ∥^{2} = ∥ E (C) ψ ∥^{2}$ is independent of $t$ , and if $ψ \in D (K)$ then both existing scalar moments are conserved: $\int_{R} λ^{k} d E_{K, U (t) ψ} (λ) = \int_{R} λ^{k} d E_{K, ψ} (λ), k = 1, 2,$ with the integrals finite at $t = 0$ and equal to $⟨ K ψ, ψ ⟩$ and $∥ K ψ ∥^{2}$ respectively; moments that are infinite at $t = 0$ remain outside the assertion;

(d) if $K ψ = λ ψ$ for some $λ \in R$ and unit $ψ \in D (K)$ , then $U (t) ψ = e^{- i t λ / ℏ} ψ$ for every $t$ , and every $S \mapsto ∥ F (S) ψ ∥^{2}$ of (b) agrees with $S \mapsto ∥ F (S) U (t) ψ ∥^{2}$ .

No equilibration, relaxation or convergence of the law of an arbitrary state is asserted, and no moment beyond those finite at $t = 0$ is claimed.

Facts & Assumptions

Given: a nonzero complex Hilbert space $H$ ; a self-adjoint $K$ with spectral PVM $E$ ; $U (t) = e^{- i t K / ℏ}$ ; in (b) a PVM $F$ commuting with $E$ and a density operator $ρ$ .

[F1]

For bounded Borel $h : R \to C$ the bounded integral $Φ_{E} (h) = \int h d E$ satisfies $∥ Φ_{E} (h) x ∥^{2} = \int ∣ h ∣^{2} d E_{x}$ , $Φ_{E} (h_{1} h_{2}) = Φ_{E} (h_{1}) Φ_{E} (h_{2})$ , and $Φ_{E} (h)$ is a norm limit of integrals of simple functions: there are simple Borel functions $s_{n}$ with $∥ h - s_{n} ∥_{\infty} \to 0$ and $∥ Φ_{E} (h) - \int s_{n} d E ∥ \to 0$ (Bounded borel pvm integral, Pvm integral is a star homomorphism).

[F2]

$U (t) = Φ_{E} (f_{t})$ with $f_{t} (λ) = e^{- i t λ / ℏ}$ , bounded with $∣ f_{t} ∣ = 1$ ; each $U (t)$ is unitary and $U (t) D (K) = D (K)$ (A self-adjoint operator generates a strongly continuous unitary group, Integral of a measurable function against a projection-valued measure).

[F3]

For a self-adjoint operator $T$ with PVM $E_{T}$ and $ψ \in D (T)$ : $∥ (T - μ) ψ ∥^{2} = \int (λ - μ)^{2} d E_{T, ψ} (λ)$ , $\int λ d E_{T, ψ} = ⟨ T ψ, ψ ⟩$ and $\int λ^{2} d E_{T, ψ} = ∥ T ψ ∥^{2}$ (The unbounded PVM integral is densely defined, closed and normal, Density operators and scalar spectral probability clause (c)).

[F4]

Trace cyclicity: $tr (B T) = tr (T B)$ for trace-class $T$ and bounded $B$ ; the trace-class operators form a two-sided operator ideal, so $U ρ U^{*}$ is trace class for a density operator $ρ$ (Nuclear bounded cyclicity for trace-class operators clause (b), Rank-one operator calculus).

[F5]

Measures on $R$ are determined by their values on Borel sets; a positive measure with $μ (R ∖ {λ}) = 0$ is the point mass at $λ$ (Scalar and complex measures from a pvm).

Proof

1.1givenF1F2algebra

[Commutation passes to bounded Borel functions] Assume $B$ commutes with every $E (C)$ , hence with every simple integral $\int s d E = \sum_{k} c_{k} E (B_{k})$ . Choose simple $s_{n} \to f_{t}$ uniformly by [F1]; then $\int s_{n} d E \to U (t)$ in operator norm, and $∥ B \int s_{n} d E - \int s_{n} d E B ∥ \leq ∥ B ∥ ∥ \int s_{n} d E - U (t) ∥ + ∥ \int s_{n} d E - U (t) ∥ ∥ B ∥ \to 0$ , while each commutator is zero; hence $B U (t) = U (t) B$ . This is (a).

1.2givenF1F2algebra

[Resolvent commutation and the group] Assume $S K \subseteq K S$ for a bounded self-adjoint $S$ . For $z \in C ∖ R$ the resolvent $R (z) = (K - z)^{- 1}$ is a bounded operator with range $D (K)$ , and $S (K - z) = (K - z) S$ on $D (K)$ implies $S R (z) = R (z) S$ . For $t \in R$ and $n \geq 1$ put $W_{n} (t) : = (1 + \frac{i t}{ℏ n} K)^{- 1}$ , which is defined and bounded because it equals $- \frac{i ℏ n}{t} (K - \frac{i ℏ n}{t})^{- 1}$ for $t \neq 0$ (and $W_{n} (0) = I$ ), so $S W_{n} (t) = W_{n} (t) S$ . By the spectral theorem, $(1 + \frac{i t λ}{ℏ n})^{- n} \to e^{- i t λ / ℏ}$ boundedly for every $λ$ , so $W_{n} (t)^{n} \to U (t)$ strongly by dominated convergence in the spectral measures [F1]; since every $W_{n} (t)^{n}$ commutes with $S$ , the strong limit $U (t)$ does, by $∥ (S U (t) - U (t) S) x ∥ \leq ∥ S ∥ ∥ (W_{n}^{n} - U (t)) x ∥ + ∥ (W_{n}^{n} - U (t)) S x ∥$ . This is (a').

2.1givenstep 1.1

[Commuting sharp observable] Under the hypotheses of (b), each $F (S)$ is bounded and commutes with every $E (C)$ , so step 1.1 with $B = F (S)$ gives $F (S) U (t) = U (t) F (S)$ for all $t, S$ . This is (b)(i).

3.1givenstep 2.1F4algebra

[Invariant law of a density operator] Let $ρ$ be a density operator. Then $U (t) ρ U (t)^{*}$ is trace class by [F4], and by trace cyclicity twice, $tr (U (t) ρ U (t)^{*} F (S)) = tr (ρ U (t)^{*} F (S) U (t))$ . By step 2.1, $U (t)^{*} F (S) U (t) = U (t)^{*} U (t) F (S) = F (S)$ ; substituting gives $tr (U (t) ρ U (t)^{*} F (S)) = tr (ρ F (S))$ , independent of $t$ . This is (b)(ii).

3.2givenstep 2.1F2algebra

[Invariant law of a pure state] For unit $ψ$ , $∥ F (S) U (t) ψ ∥^{2} = ∥ U (t) F (S) ψ ∥^{2}$ by step 2.1, and this equals $∥ F (S) ψ ∥^{2}$ because $U (t)$ is unitary. This is (b)(iii).

4.1givenstep 3.1step 3.2F5

[Energy law] Applying steps 3.1 and 3.2 with $F = E$ gives $tr (U (t) ρ U (t)^{*} E (C)) = tr (ρ E (C))$ and $∥ E (C) U (t) ψ ∥^{2} = ∥ E (C) ψ ∥^{2}$ for every Borel $C$ . Moreover $E_{K, U (t) ψ} (C) = ∥ E (C) U (t) ψ ∥^{2} = ∥ E (C) ψ ∥^{2} = E_{K, ψ} (C)$ for every $C$ , so the two scalar measures agree by [F5].

5.1givenstep 4.1F2F3

[Moment conservation] Let $ψ \in D (K)$ and $u_{0} : = ψ$ ; by [F2], $U (t) ψ \in D (K)$ for every $t$ . Since $E_{K, U (t) ψ} = E_{K, ψ}$ by step 4.1, the two integrals $\int λ^{k} d E_{K, U (t) ψ}$ and $\int λ^{k} d E_{K, ψ}$ coincide for $k = 1, 2$ (extended real values, both finite at $t = 0$ by [F3]); by [F3] the time-zero values are $⟨ K ψ, ψ ⟩$ and $∥ K ψ ∥^{2}$ . If a moment is infinite at $t = 0$ , no equality of finite quantities is claimed. This proves (c).

6.1givenstep 3.2F1F3F5algebra∎

[Stationary eigenrays] Let $K ψ = λ ψ$ with $∥ ψ ∥ = 1$ . By [F3], $∥ (K - λ) ψ ∥^{2} = \int (μ - λ)^{2} d E_{K, ψ} (μ) = 0$ , so the finite positive measure $E_{K, ψ}$ assigns mass zero to $R ∖ {λ}$ and is the point mass at $λ$ by [F5]. Then $∥ U (t) ψ - e^{- i t λ / ℏ} ψ ∥^{2} = \int ∣ e^{- i t μ / ℏ} - e^{- i t λ / ℏ} ∣^{2} d E_{K, ψ} (μ) = 0$ by [F1], so $U (t) ψ = e^{- i t λ / ℏ} ψ$ . The last assertion of (d) is step 3.2 with this $ψ$ .

Conserved spectral probability laws for commuting observables and stationary eigenrays

Statement

Facts & Assumptions

Proof

Depends on

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