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Unitary conjugation, generator transport and strong commutation of self-adjoint observables

Statement

Assume the Axiom of Choice, the Axiom of Countable Choice and the Axiom of Dependent Choice (The Axiom of Choice, The Axiom of Countable Choice ( ${A C}_{ω}$ ), The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain), the choice hypotheses of the cited spectral and Bochner suppliers. Fix $ℏ > 0$ , let $H \neq {0}$ be a complex Hilbert space (Hilbert space), let $K$ and $A$ be self-adjoint operators on $H$ with spectral projection-valued measures $E_{K}$ and $E_{A}$ (Symmetric, self-adjoint and essentially self-adjoint operators, Spectral theorem for unbounded self-adjoint operators (PVM form), Projection valued measure), let $W \in B (H)$ be unitary (A bounded linear operator between normed spaces, Self-adjoint, positive, unitary and normal operators), and put $U_{K} (t) : = e^{- i t K / ℏ}, U_{A} (s) : = e^{- i s A / ℏ} (s, t \in R)$ in the spectral calculus of Integral of a measurable function against a projection-valued measure, which are strongly continuous one-parameter unitary groups (A self-adjoint operator generates a strongly continuous unitary group, Strongly continuous one-parameter unitary group). Write $Φ_{K} (g) = \int g d E_{K}$ for bounded Borel $g : R \to C$ (Bounded borel pvm integral), and let $K_{W} : = W K W^{*}$ with $D (K_{W}) : = W D (K)$ . Then:

(a) $K_{W}$ is self-adjoint with spectral projection-valued measure $E_{K_{W}} (B) = W E_{K} (B) W^{*} (B \subseteq R Borel),$ its flow satisfies $e^{- i t K_{W} / ℏ} = W U_{K} (t) W^{*}$ for every $t$ , and the conjugated family $s \mapsto W U_{A} (s) W^{*}$ is a strongly continuous one-parameter unitary group whose infinitesimal generator is $- \frac{i}{ℏ} W A W^{*}$ with domain $W D (A)$ (Infinitesimal generator of a unitary group);

(b) the following are equivalent:

(i) $U_{A} (s) U_{K} (t) = U_{K} (t) U_{A} (s)$ for all $s, t \in R$ ;

(ii) $E_{A} (B) E_{K} (C) = E_{K} (C) E_{A} (B)$ for all Borel sets $B, C \subseteq R$ ;

(iii) $U_{K} (t) D (A) = D (A)$ and $U_{K} (t) A U_{K} (t)^{*} = A$ for every $t \in R$ ;

(c) if a bounded $B \in B (H)$ satisfies $B U_{K} (t) = U_{K} (t) B$ for every $t \in R$ , then $B Φ_{K} (g) = Φ_{K} (g) B$ for every bounded Borel $g : R \to C$ ; in particular $B E_{K} (C) = E_{K} (C) B$ for every Borel $C \subseteq R$ ;

(d) if (i)-(iii) of (b) hold, then $E_{A} (B) U_{K} (t) = U_{K} (t) E_{A} (B)$ for every Borel $B \subseteq R$ and every $t \in R$ .

Facts & Assumptions

Given: The Axiom of Choice, the Axiom of Countable Choice and the Axiom of Dependent Choice; $ℏ > 0$ ; a nonzero complex Hilbert space $H$ ; self-adjoint $K, A$ with spectral PVMs $E_{K}, E_{A}$ ; a unitary $W$ ; the flows $U_{K}, U_{A}$ ; bounded Borel functions $g, h$ ; vectors $x, y \in H$ ; real $s, t$ ; Borel sets $B, C \subseteq R$ .

[F1]

Flow calculus. For a self-adjoint operator $L$ with spectral PVM $E_{L}$ and flow $U_{L} (τ) = e^{- i τ L / ℏ}$ : $U_{L}$ is a strongly continuous one-parameter unitary group with $U_{L} (τ)^{*} = U_{L} (- τ)$ , $U_{L} (τ) D (L) = D (L)$ and $L U_{L} (τ) = U_{L} (τ) L$ on $D (L)$ ; the difference quotient $\frac{1}{τ} (U_{L} (τ) x - x)$ converges as $τ \to 0$ exactly for $x \in D (L)$ , with limit $- \frac{i}{ℏ} L x$ ; and for bounded Borel $f$ one has $f (L) U_{L} (τ) = U_{L} (τ) f (L)$ with $f (L) = Φ_{E_{L}} (f)$ (A self-adjoint operator generates a strongly continuous unitary group, Unbounded Borel functional calculus: domains, products, spectral mapping, Pvm integral is a star homomorphism).

[F2]

Spectral calculus. $Φ_{E_{L}}$ is a unital $*$ -homomorphism from the bounded Borel functions to $B (H)$ , $∥ Φ_{E_{L}} (f) ∥ \leq \sup ∣ f ∣$ , $E_{L} (B) = Φ_{E_{L}} (1_{B})$ , and the map $B \mapsto E_{L} (B)$ is a regular projection-valued measure whose scalar measures $⟨ E_{L} (\cdot) x, x ⟩$ are finite regular Borel measures (Bounded borel pvm integral, Pvm integral is a star homomorphism, Projection valued measure, Scalar and complex measures from a pvm, Regular Borel measure on an LCH space).

[F3]

Uniqueness of the spectral measure. If a regular projection-valued measure $F$ on the Borel sets of $R$ satisfies $\int λ d F (λ) x = L x$ for all $x \in D (L)$ and $D (L) = {x : \int λ^{2} d F_{x} (λ) < \infty}$ , then $F = E_{L}$ (Spectral theorem for unbounded self-adjoint operators (PVM form)).

[F4]

Stone's theorem. $T \mapsto e^{i t T}$ is a bijection from the self-adjoint operators onto the strongly continuous one-parameter unitary groups; two such groups that agree at every real time have the same values, hence the same generator, and a group is determined by its generator (Stone's theorem: unitary groups and self-adjoint generators, Infinitesimal generator of a unitary group).

[F5]

Laplace resolvents. For a strongly continuous one-parameter unitary group $U$ with infinitesimal generator $G$ and $λ > 0$ , the vectorwise Bochner integrals $Q_{\pm} (λ) x : = \int_{0}^{\infty} e^{- λ t} U (\pm t) x d t (x \in H)$ define bounded operators with $∥ Q_{\pm} (λ) ∥ \leq 1 / λ$ and $(λ \mp G) Q_{\pm} (λ) = I$ , $Q_{\pm} (λ) (λ \mp G) = I$ on $D (G)$ . Bounded linear maps on $H$ commute with these $H$ -valued integrals (Laplace resolvents of a unitary group, Bounded linear maps commute with Bochner integration).

[F6]

Density of the resolvent algebra. For a closed nonempty $K \subseteq R$ the $*$ -subalgebra generated by the single function $λ \mapsto (λ - z)^{- 1}$ , $z \in C ∖ K$ , is dense in the algebra $C_{\infty} (K)$ of continuous functions vanishing at infinity; and for a finite Radon measure $μ$ on $R$ the space $C_{c} (R) \subseteq C_{\infty} (R)$ is dense in $L^{p} (μ)$ for $1 \leq p < \infty$ (Complex Stone–Weierstrass dichotomy for separating self-adjoint algebras; the unital case is dense, C_c(X) is dense in L^p(mu) for a Radon measure).

Proof

Proof technique: direct; transport the spectral data through the unitary conjugation, prove the equivalence (i) - (iii) by comparing generators of conjugated groups, and obtain the commutation with spectral projections through the Laplace resolvent and the density of the resolvent algebra.

Given: The data of the statement and [F1]-[F6].

1.1F1givenalgebra

[Well-definedness, density and symmetry of $K_{W}$ ] The domain $D (K_{W}) = W D (K)$ is a linear subspace, dense in $H$ because $D (K)$ is dense and $W$ is surjective, and $K_{W} x = W K W^{*} x$ is defined and linear on it. For $x = W u$ , $y = W v$ with $u, v \in D (K)$ one has $⟨ K_{W} x, y ⟩ = ⟨ W K u, W v ⟩ = ⟨ K u, v ⟩ = ⟨ u, K v ⟩ = ⟨ x, K_{W} y ⟩$ because $W$ is unitary and $K$ is symmetric; hence $K_{W}$ is symmetric.

1.2F2givenalgebra

[(ii) implies (i)] Assume (ii). For simple Borel functions $g = \sum_{i} c_{i} 1_{B_{i}}$ and $h = \sum_{j} d_{j} 1_{C_{j}}$ the identities $E_{A} (B_{i}) E_{K} (C_{j}) = E_{K} (C_{j}) E_{A} (B_{i})$ give $Φ_{A} (g) Φ_{K} (h) = \sum_{i, j} c_{i} d_{j} E_{A} (B_{i}) E_{K} (C_{j}) = \sum_{i, j} c_{i} d_{j} E_{K} (C_{j}) E_{A} (B_{i}) = Φ_{K} (h) Φ_{A} (g)$ . Given bounded Borel $g, h$ , choose simple $g_{n} \to g$ , $h_{n} \to h$ uniformly; then $Φ_{A} (g_{n}) \to Φ_{A} (g)$ and $Φ_{K} (h_{n}) \to Φ_{K} (h)$ in operator norm by [F2], so the identities pass to the limits, giving $Φ_{A} (g) Φ_{K} (h) = Φ_{K} (h) Φ_{A} (g)$ . Taking $g (λ) = e^{- i s λ / ℏ}$ and $h (μ) = e^{- i t μ / ℏ}$ and applying [F2] gives $U_{A} (s) U_{K} (t) = U_{K} (t) U_{A} (s)$ for all $s, t$ , which is (i).

1.3F5givenalgebra

[Commutation with the flow implies commutation with the calculus] Let $B \in B (H)$ satisfy $B U_{K} (t) = U_{K} (t) B$ for all $t$ . Apply [F5] to $U_{K}$ and $G_{K} = - \frac{i}{ℏ} K$ with domain $D (K)$ . For every $x \in H$ and either sign, $B Q_{\pm} (λ) x = B \int_{0}^{\infty} e^{- λ t} U_{K} (\pm t) x d t = \int_{0}^{\infty} e^{- λ t} B U_{K} (\pm t) x d t = \int_{0}^{\infty} e^{- λ t} U_{K} (\pm t) B x d t = Q_{\pm} (λ) B x .$ Thus $B$ commutes with both resolvents, using only the vectorwise integrals supplied by [F5]. Now $Q_{+} (λ) = (λ + \frac{i}{ℏ} K)^{- 1} = \frac{ℏ}{i} (K - i ℏ λ)^{- 1}$ and $Q_{-} (λ) = (λ - \frac{i}{ℏ} K)^{- 1} = i ℏ (K + i ℏ λ)^{- 1}$ are nonzero scalar multiples of the resolvents of $K$ at $i ℏ λ$ and $- i ℏ λ$ . Choosing $λ = 1$ shows that $B$ commutes with $g_{\pm} (K)$ for $g_{\pm} (μ) = (μ \pm i ℏ)^{- 1}$ and hence with every polynomial in them and their conjugates, that is with every element of the $*$ -subalgebra generated by $g_{+}$ .

2.1step 1.1F1givenalgebra

[Adjoint and self-adjointness] Let $y \in D (K_{W}^{*})$ and put $z = K_{W}^{*} y$ . For every $u \in D (K)$ the vector $W u$ lies in $D (K_{W})$ with $K_{W} (W u) = W K u$ , so $⟨ K u, W^{*} y ⟩ = ⟨ W K u, y ⟩ = ⟨ K_{W} (W u), y ⟩ = ⟨ W u, z ⟩ = ⟨ u, W^{*} z ⟩$ ; the functional $u \mapsto ⟨ K u, W^{*} y ⟩$ is therefore bounded on $D (K)$ , whence $W^{*} y \in D (K^{*}) = D (K)$ and $K^{*} W^{*} y = K W^{*} y = W^{*} z$ . Applying the unitary $W$ gives $y \in W D (K) = D (K_{W})$ and $K_{W} y = z$ . Hence $D (K_{W}^{*}) \subseteq D (K_{W})$ with agreement there, and with step 1.1 the operator $K_{W}$ is self-adjoint.

2.2step 1.3F2F6algebra

[Continuous vanishing functions] Let $f \in C_{\infty} (σ (K))$ , a closed subset of $R$ . By [F6] there are elements $p_{n}$ of the $*$ -subalgebra generated by $μ \mapsto (μ - i ℏ)^{- 1}$ with $\sup_{σ (K)} ∣ p_{n} - f ∣ \to 0$ . Each $p_{n} (K) = Φ_{K} (p_{n})$ commutes with $B$ by step 1.3, and $∥ Φ_{K} (p_{n}) - Φ_{K} (f) ∥ \leq \sup_{σ (K)} ∣ p_{n} - f ∣ \to 0$ by the norm bound of [F2], so $Φ_{K} (f)$ commutes with $B$ .

3.1step 2.1F2F3givenalgebra

[Spectral measure transport] Put $F (B) : = W E_{K} (B) W^{*}$ for Borel $B \subseteq R$ . Each $F (B)$ is an orthogonal projection as the conjugate of one; $F (\emptyset) = 0$ and $F (R) = I$ ; finite additivity and strong countable additivity follow from those of $E_{K}$ and boundedness of $W$ ; and $F$ is regular because its scalar measure in $x$ is $⟨ F (B) x, x ⟩ = ⟨ E_{K} (B) W^{*} x, W^{*} x ⟩$ , a finite regular Borel measure by [F2]. For any $x \in H$ , its scalar measure for $F$ is the scalar measure of $E_{K}$ at $W^{*} x$ , so $\int λ^{2} d F_{x} (λ) < \infty$ exactly when $W^{*} x \in D (K)$ by the spectral theorem; this is exactly $x \in W D (K) = D (K_{W})$ . If $x = W u$ with $u \in D (K)$ , bounded spectral calculus first gives $\int_{[- n, n]} λ d F (λ) x = W \int_{[- n, n]} λ d E_{K} (λ) u$ for every $n$ , and the unbounded-integral definition lets $n \to \infty$ in $H$ on both sides, giving $\int λ d F (λ) x = W K u = K_{W} x$ . Thus $F$ satisfies both hypotheses of [F3] for $K_{W}$ , so $F = E_{K_{W}}$ .

3.2step 2.2F2F6givenalgebra

[Bounded Borel functions] Let $g$ be bounded Borel and fix $x \in H$ . The finite regular measure $μ : = E_{K, x} + E_{K, B x}$ is a sum of the scalar measures at $x$ and $B x$ , and is regular by [F2]. By [F6] choose $f_{n} \in C_{c} (R) \subseteq C_{\infty} (R)$ with $\int ∣ g - f_{n} ∣^{2} d μ \to 0$ . The PVM norm identity in [F2] then gives both $(Φ_{K} (g) - Φ_{K} (f_{n})) x \to 0$ and $(Φ_{K} (g) - Φ_{K} (f_{n})) B x \to 0$ in $H$ . Since $Φ_{K} (f_{n})$ commutes with $B$ by step 2.2, $∥ [B, Φ_{K} (g)] x ∥ \leq ∥ B ∥ ∥ (Φ_{K} (g) - Φ_{K} (f_{n})) x ∥ + ∥ (Φ_{K} (g) - Φ_{K} (f_{n})) B x ∥ ⟶ 0,$ hence $B Φ_{K} (g) = Φ_{K} (g) B$ . Taking $g = 1_{C}$ and using $E_{K} (C) = Φ_{K} (1_{C})$ from [F2] gives $B E_{K} (C) = E_{K} (C) B$ . This proves (c).

4.1step 3.1F1F2givenalgebra

[Flow and generator transport] By [F2] applied to the bounded Borel functions $f_{t} (λ) = e^{- i t λ / ℏ}$ , $e^{- i t K_{W} / ℏ} = Φ_{E_{K_{W}}} (f_{t}) = W Φ_{E_{K}} (f_{t}) W^{*} = W U_{K} (t) W^{*}$ , which is the displayed flow identity. For arbitrary $x \in H$ , the difference quotients of $s \mapsto W U_{A} (s) W^{*} x$ at $0$ equal $W \frac{1}{s} (U_{A} (s) W^{*} x - W^{*} x)$ . Since $W$ is an isometry, this limit exists exactly when $W^{*} x \in D (A)$ by [F1], and then it equals $- \frac{i}{ℏ} W A W^{*} x$ . Therefore the conjugated group has infinitesimal generator $- \frac{i}{ℏ} W A W^{*}$ on the exact domain $W D (A)$ ; it is strongly continuous because $W$ is bounded and $U_{A}$ is. This proves (a).

4.2step 3.1step 1.3step 3.2F2F3givenalgebra

[(iii) implies (ii)] Assume (iii) and fix $t$ . By (iii), $U_{K} (t) A U_{K} (t)^{*} = A$ ; by uniqueness of the spectral measure [F3] and step 3.1 applied with $W = U_{K} (t)$ , $E_{A} (B) = E_{U_{K} (t) A U_{K} (t)^{*}} (B) = U_{K} (t) E_{A} (B) U_{K} (t)^{*}$ for every Borel $B$ , so $E_{A} (B)$ commutes with $U_{K} (t)$ and with $U_{K} (t)^{*} = U_{K} (- t)$ . Since $E_{A} (B)$ is bounded and commutes with the flow $U_{K}$ , step (c) with $B : = E_{A} (B)$ gives $E_{A} (B) Φ_{K} (g) = Φ_{K} (g) E_{A} (B)$ for every bounded Borel $g$ ; taking $g = 1_{C}$ gives $E_{A} (B) E_{K} (C) = E_{K} (C) E_{A} (B)$ , which is (ii).

5.1step 4.1F1F4givenalgebra

[(i) implies (iii)] Assume (i), fix $t \in R$ , and put $V (s) : = U_{K} (t)^{*} U_{A} (s) U_{K} (t)$ ; by (i) $V (s) = U_{A} (s)$ for every $s$ . The family $V$ is a strongly continuous unitary group, and by step 4.1 with $W = U_{K} (t)^{*}$ its infinitesimal generator is $- \frac{i}{ℏ} U_{K} (t)^{*} A U_{K} (t)$ with domain $U_{K} (t)^{*} D (A)$ ; the right-hand side $U_{A}$ has generator $- \frac{i}{ℏ} A$ with domain $D (A)$ by [F1]. Since the two groups coincide, their generators coincide by [F4], so $U_{K} (t)^{*} D (A) = D (A)$ and $U_{K} (t)^{*} A U_{K} (t) = A$ . Applying the unitary $U_{K} (t)$ on both sides gives $U_{K} (t) D (A) = D (A)$ and $U_{K} (t) A U_{K} (t)^{*} = A$ , which is (iii).

5.2step 4.1F1F4givenalgebra

[(iii) implies (i)] Assume (iii) and fix $t$ . Put $V (s) : = U_{K} (t)^{*} U_{A} (s) U_{K} (t)$ . By step 4.1 with $W = U_{K} (t)^{*}$ , the generator of $V$ is $- \frac{i}{ℏ} U_{K} (t)^{*} A U_{K} (t) = - \frac{i}{ℏ} A$ with domain $U_{K} (t)^{*} D (A) = D (A)$ , which is the generator of $U_{A}$ by [F1]; hence $V = U_{A}$ by [F4] and $U_{A} (s) U_{K} (t) = U_{K} (t) U_{A} (s)$ for all $s$ .

6.1step 1.2step 5.1step 5.2step 4.2algebra∎

[Assembly] Steps 1.2, 5.1, 5.2 and 4.2 give the implications (ii) $\Rightarrow$ (i) $\Rightarrow$ (iii) $\Rightarrow$ (ii), so the three conditions are equivalent, which is (b). Under these conditions, step 4.2 in its first display already gives $E_{A} (B) U_{K} (t) = U_{K} (t) E_{A} (B)$ for all Borel $B$ and all $t$ , which is (d): the displayed identity $E_{A} (B) = U_{K} (t) E_{A} (B) U_{K} (t)^{*}$ multiplied on the right by $U_{K} (t)$ yields $E_{A} (B) U_{K} (t) = U_{K} (t) E_{A} (B)$ .

Remarks

Why the hypotheses are formulated with groups. The equivalence of (i), (ii) and (iii) is the precise sense in which "a self-adjoint quantity commutes with the energy" may be used: strong commutation of the spectral measures, commutation of the flows, and invariance of the domain of the quantity under the flow. A formal commutator identity on a small domain is not among the equivalent conditions and is not used.
Choice. The Axiom of Choice is the hypothesis of the spectral theorem and of Stone's theorem; Countable and Dependent Choice are the hypotheses of the Bochner and density suppliers used in clause (c).

Unitary conjugation, generator transport and strong commutation of self-adjoint observables

Statement

Facts & Assumptions

Proof

Remarks

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