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Qm Symmetries And Conservation

1 · Prerequisites

2 · Summary

This page develops the symmetry and conservation structure of the declared massive model. The vocabulary is fixed in Symmetries of the declared dynamics, unitary symmetry groups and conserved sharp observables: a symmetry of the declared dynamics is a unitary that preserves the energy domain and commutes with the energy operator; a unitary symmetry group is a group of such symmetries; and a conserved sharp observable is one whose outcome law is time-independent in every preparation. The model's candidate actions are the spatial translations, the spatial rotations with their coordinate-rotation generators, the spatial parity action and the gauge multipliers of the prescribed-coupling model; none of them is a symmetry by definition, and the theorems decide for which potentials or couplings each is one.

The supporting mathematics is local to this page. Unitary conjugation, generator transport and strong commutation of self-adjoint observables proves the equivalence between commutation of the flows, strong spectral commutation and invariance of the domain of a self-adjoint observable, and records the transport of spectral objects under unitary conjugation. The spatial rotation action on the component-valued wave-function space defines the rotation action $U (R) f = f \circ R^{- 1}$ and computes its conjugations of the position, momentum and bounded-potential operators; The spatial parity action on the component-valued wave-function space does the same for the parity involution $Π$ , including $Π Q_{j} Π = - Q_{j}$ and $Π P_{k} Π = - P_{k}$ ; Coordinate-rotation generators on the Schwartz classes and their commutators identifies the coordinate-rotation generators $L_{i} = \sum_{j, k} ε_{i j k} Q_{j} P_{k}$ on the Schwartz classes, proves that each Schwartz class is a core for its generator in the graph norm, and establishes the commutation relations $[L_{i}, L_{k}] = i ℏ \sum_{j} ε_{i k j} L_{j}$ there; and The free flow as a unimodular multiplier in the momentum representation exhibits the free flow as the unimodular multiplier $M_{f_{t}}$ , $f_{t} (p) = e^{- i t ∣ p ∣^{2} / (2 m ℏ)}$ , in the momentum representation.

On this basis Conservation of a symmetry generator: commuting flows, energy compatibility and conserved outcome laws proves the equivalence of flow commutation, spectral commutation and domain invariance, and derives the conservation of the outcome law and of the moments of a self-adjoint observable from these hypotheses, including the case of a self-adjoint unitary symmetry. Translation invariance of bounded potentials and conservation of momentum conjugates the bounded-potential operator by the translations, proves that a translation is a symmetry exactly for potentials invariant under it almost everywhere, shows that invariance under every translation forces an almost-everywhere constant potential, and derives momentum conservation for that case. Rotational and parity invariance of bounded-potential models: symmetries and conservation of angular momentum and parity laws proves the analogous statements for rotations — with conservation of the generators $L_{i}$ — and for parity, whose conservation is read through the self-adjoint unitary clause of the conservation theorem. Prescribed external electromagnetic coupling P5a: the minimal-coupling model and its gauge transformations states the prescribed-coupling assumption P5a with the admissible bounded-coefficient potentials, its units, the field definitions and the gauge transformation of potentials and state phase; and Gauge covariance of the prescribed-coupling model: unitary equivalence, the symmetry criterion and the transported propagator proves the conjugation identity transporting the minimal-coupling operator to the gauge-transformed potentials, the exact criterion under which the gauge multiplier is a symmetry of the fixed declared dynamics, the global-phase case, and the unitary equivalence of the transformed time-dependent model carrying the transported propagator. Scope and empirical status of the symmetry and conservation items records the conditional status of all of these statements, the empirical status of the model parameters and of any symmetry identification for a laboratory system, the excluded realizations and the supplier note on the operator development consumed by the gauge items.

The page is the prerequisite of the angular-momentum and spin development, which adds the orbital spherical theory and the intrinsic spin representation, and of the examples companion, which collects the free-model computation, the parity selection rule, the vanishing-translation counterexample and the central-potential instance.

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passOpen item page →

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).

LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passOpen item page →

The spatial rotation action on the component-valued wave-function space

Statement

Assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The Axiom of Choice), the choice hypotheses of the cited spectral suppliers. Fix an integer $r \geq 1$ and $ℏ > 0$ , let $H^{(r)} = L^{2} (R^{3}; C^{r})$ with inner product linear in the first argument (Complex Lp classes and Euclidean test-function conventions, The space $L^{p} (μ)$ as the quotient by null functions, Hilbert space), let $Q_{j}, P_{k}$ be the componentwise position and momentum operators (Componentwise position and momentum operators on the component-valued wave-function space clause (a)), and let $S^{(r)} = S (R^{3}; C^{r})$ and $W^{m, 2} (R^{3}; C^{r})$ be the componentwise Schwartz and Sobolev classes (Schwartz space and its seminorms, Integer-order Sobolev spaces and their norms). Let $S O (3)$ be the special orthogonal group with its Euclidean topology (The orthogonal group $O (3)$ and the rotation group $S O (3)$ , $S O (3)$ is a subgroup of the general linear group, with $R^{- 1} = R^{T}$ ) and for $R \in S O (3)$ define, componentwise, $(U (R) f)_{l} (x) : = f_{l} (R^{- 1} x) (x \in R^{3}, l = 1, \dots, r) .$ Let $U^{(r)}$ be the unitary momentum representation of The momentum representation of the componentwise kinematics and $M_{R}$ the bounded operator on $L^{2} (R^{3}, d p; C^{r})$ given by $(M_{R} g) (p) : = g (R^{- 1} p)$ . Then:

(a) $U (R)$ is a well-defined unitary operator for every $R$ , with $U (I) = i d$ , $U (R S) = U (R) U (S)$ and $U (R)^{- 1} = U (R^{- 1})$ for all $R, S \in S O (3)$ ; and for every $f \in H^{(r)}$ the orbit map $R \mapsto U (R) f$ is continuous on $S O (3)$ (Continuity of a map between metric spaces, at a point and globally, in the $ε$ - $δ$ form). Moreover $U (R) S^{(r)} = S^{(r)}$ and $U (R) W^{m, 2} (R^{3}; C^{r}) = W^{m, 2} (R^{3}; C^{r})$ for $m = 1, 2$ ;

(b) $U^{(r)} U (R) U^{(r) - 1} = M_{R}$ . For $a \in R^{3}$ define the maximal directional operators $Q (a) : = M_{a \cdot x}, P (a) : = U^{(r) - 1} M_{a \cdot p} U^{(r)},$ where each multiplier has its maximal domain, namely the classes for which its product is in $L^{2}$ . Thus $D (Q (a)) = {f : (a \cdot x) f \in L^{2}}, D (P (a)) = {f : (a \cdot p) U^{(r)} f \in L^{2}} .$ They are self-adjoint (for $a = 0$ they are the zero operator on all of $H^{(r)}$ ). On the common coordinate domains they agree with $\sum_{l} a_{l} Q_{l}$ and $\sum_{l} a_{l} P_{l}$ , respectively. The full conjugation identities, including domains, are $U (R) Q_{j} U (R)^{- 1} = Q (R e_{j}), U (R) P_{k} U (R)^{- 1} = P (R e_{k}),$ $U (R) D (Q_{j}) = D (Q (R e_{j})), U (R) D (P_{k}) = D (P (R e_{k})) .$ In particular, an individual coordinate domain is preserved when $R e_{j} = \pm e_{j}$ (respectively $R e_{k} = \pm e_{k}$ ); no such preservation is asserted for an arbitrary rotation. The expressions $\sum_{l} R_{l j} Q_{l}$ and $\sum_{l} R_{l k} P_{l}$ in full operator identities mean these maximal directional realizations.

(c) with the free operator $H_{0} : = \frac{1}{2 m_{0}} \sum_{k = 1}^{3} P_{k}^{2}$ for a parameter $m_{0} > 0$ , whose domain is $W^{2, 2} (R^{3}; C^{r})$ , one has $U (R) H_{0} U (R)^{- 1} = H_{0}$ ; and for a real bounded measurable $V : R^{3} \to R$ with multiplier $M_{V}$ (multiplication by $V$ ) and $V_{R} (x) : = V (R^{- 1} x)$ one has $U (R) M_{V} U (R)^{- 1} = M_{V_{R}}$ , so that $U (R) H_{V} U (R)^{- 1} = H_{V_{R}}$ for the free-plus- $V$ operator $H_{V}$ with domain $W^{2, 2} (R^{3}; C^{r})$ .

Facts & Assumptions

Given: The Axiom of Countable Choice and the Axiom of Choice; $r \geq 1$ ; $ℏ > 0$ ; the space $H^{(r)}$ ; the componentwise operators $Q_{j}, P_{k}$ ; $R, S \in S O (3)$ ; $f \in H^{(r)}$ ; $ψ \in S^{(r)}$ ; the momentum unitary $U^{(r)}$ and the multipliers $M_{R}$ ; a real bounded measurable $V$ .

[F1]

Componentwise kinematics. $Q_{j}, P_{k}$ are self-adjoint with domains $D (Q_{j}) = {f : x_{j} f_{l} \in L^{2} \forall l}$ , $D (P_{k}) = {f : D_{k} f_{l} \in L^{2} \forall l}$ , and on $S^{(r)}$ one has $Q_{j} f = x_{j} f$ , $P_{k} f = - i ℏ D_{k} f$ componentwise; $S^{(r)} \subseteq D (Q_{j}) \cap D (P_{k})$ and both operators map $S^{(r)}$ into itself (Componentwise position and momentum operators on the component-valued wave-function space clauses (a),(c), Schwartz space and its seminorms).

[F2]

Momentum representation. $U^{(r)}$ is unitary and satisfies $U^{(r)} D (P_{k}) = D (M_{p_{k}})$ and $U^{(r)} P_{k} = M_{p_{k}} U^{(r)}$ ; on Schwartz classes it acts by the componentwise Fourier transform with normalization $(2 π ℏ)^{- 3 / 2}$ , and it maps $S^{(r)}$ onto itself (The momentum representation of the componentwise kinematics clauses (a),(b), Componentwise position and momentum operators on the component-valued wave-function space clause (b), Schwartz space and its seminorms).

[F3]

Substitution and components. For an orthogonal $R$ and $f \in L^{1} (R^{3}; C)$ one has $\int f (R^{- 1} x) d x = \int f (y) d y$ , because the linear substitution scales Lebesgue measure by $∣ \det R ∣ = 1$ (A linear map $T$ of $R^{n}$ sends Lebesgue measurable sets to Lebesgue measurable sets, with $λ_{n} (T [E]) = ∣ \det T ∣ λ_{n} (E)$ when $T$ is invertible and $T [E]$ Lebesgue null when it is not, $S O (3)$ is a subgroup of the general linear group, with $R^{- 1} = R^{T}$ ). A class $f = (f_{l})$ of $H^{(r)}$ is determined by its components, and componentwise operations preserve the $L^{2}$ norm: $∥ f ∥^{2} = \sum_{l} ∥ f_{l} ∥^{2}$ (Unitary identifications of the component-valued wave-function space clause (a), The induced length is a norm).

[F4]

Schwartz calculus. $ψ \in S^{(r)}$ means each component $ψ_{l}$ is smooth with $\sup_{x} ∣ x^{α} \partial^{β} ψ_{l} (x) ∣ < \infty$ for all multi-indices $α, β$ ; hence each component and each of its derivatives is bounded, and $S^{(r)} \subseteq W^{2, 2} (R^{3}; C^{r})$ (Schwartz space and its seminorms, Integer-order Sobolev spaces and their norms). For an invertible linear substitution $A$ with $∣ A x ∣ \leq ∥ A ∥ ∣ x ∣$ , the chain rule gives $\partial^{β} (ψ_{l} \circ A) (x) = \sum_{∣ γ ∣ = ∣ β ∣} c_{γ} \partial^{γ} ψ_{l} (A x)$ with constants depending only on $A, β$ , so the seminorms of $ψ_{l} \circ A$ are bounded by finitely many seminorms of $ψ_{l}$ ; in particular $S^{(r)}$ is invariant under composition with an invertible linear map.

[F5]

Sobolev and Fourier characterization. For a scalar position-space class $f$ , let $g = U^{(1)} f$ be its momentum representative in the normalization of [F2], with $p = 2 π ℏ ξ$ . Then $f \in W^{1, 2}$ exactly when $p_{k} g \in L^{2}$ for all $k$ , and $f \in W^{2, 2}$ exactly when $∣ p ∣^{2} g \in L^{2}$ . Moreover $∥ f ∥_{W^{2, 2}}$ is equivalent to $∥ (1 + ∣ p ∣^{2}) g ∥_{2}$ for fixed $ℏ > 0$ . These statements apply componentwise (Integer-order W^{k,2} and H^k agree with equivalent norms, Plancherel theorem, Integer-order Sobolev spaces and their norms).

[F6]

Density and extension. $S^{(r)}$ is dense in $H^{(r)}$ (Schwartz space is dense in L2); a bounded linear operator on a dense subspace extends uniquely to the whole space with the same norm (A bounded linear map from a dense normed subspace into a Banach space extends uniquely with the same norm); if a sequence $g_{n} \to g$ pointwise and $∣ g_{n} ∣ \leq h$ with $h$ integrable, then $\int g_{n} \to \int g$ (Dominated convergence); and for $u, v \in R^{3}$ one has $∣ u \cdot v ∣ \leq ∣ u ∣ ∣ v ∣$ (Cauchy–Schwarz: $∣ ⟨ u, v ⟩ ∣ \leq ∥ u ∥ ∥ v ∥$ , with equality exactly for linearly dependent vectors).

Proof

technique · direct; define the rotation action componentwise, prove unitarity and the representation property by orthogonal substitution, transport the action to the momentum representation for the operator conjugations, and obtain strong continuity from dominated convergence and density

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

1.1F3givenalgebra

[Well-definedness and unitarity] Let $R \in S O (3)$ . For a representative $f$ of a class in $H^{(r)}$ the function $x \mapsto f (R^{- 1} x)$ is measurable and its components satisfy $\int ∣ f_{l} (R^{- 1} x) ∣^{2} d x = \int ∣ f_{l} (y) ∣^{2} d y$ by [F3], so the assignment is well defined on classes, linear, and isometric; it is surjective with inverse given by $R^{- 1}$ because composition with $R^{- 1}$ undoes it on representatives, so $U (R)$ is a unitary operator and $U (R)^{- 1} = U (R^{- 1})$ .

2.1step 1.1F3F4algebra

[Representation and Schwartz invariance] For $R, S \in S O (3)$ and $f$ one has $(U (R) U (S) f)_{l} (x) = (U (S) f)_{l} (R^{- 1} x) = f_{l} (S^{- 1} R^{- 1} x) = f_{l} ((R S)^{- 1} x) = (U (R S) f)_{l} (x)$ , since $(R S)^{- 1} = S^{- 1} R^{- 1}$ by [F3]; and $U (I) = i d$ . For $ψ \in S^{(r)}$ each component $ψ_{l} \circ R^{- 1}$ is smooth with all seminorms finite by [F4] applied to $A = R^{- 1}$ , and the same argument with $R$ in place of $R^{- 1}$ gives $U (R) S^{(r)} = S^{(r)}$ .

2.2step 1.1F1givenalgebra

[Conjugation of position] For $f \in H^{(r)}$ , the condition $U (R)^{- 1} f \in D (Q_{j})$ is, by orthogonal substitution, exactly $(\sum_{l} R_{l j} x_{l}) f \in L^{2}$ . On this domain the componentwise computation gives $(U (R) Q_{j} U (R)^{- 1} f) (x) = (R^{- 1} x)_{j} f (x) = (R e_{j} \cdot x) f (x)$ . Thus the conjugate is exactly the maximal multiplier $Q (R e_{j})$ , and $U (R) D (Q_{j}) = D (Q (R e_{j}))$ . More generally $Q (a)$ is self-adjoint: for $a \neq 0$ choose a rotation taking $e_{j}$ to $a / ∣ a ∣$ , and the same computation makes $Q (a)$ a real scalar multiple of a unitary conjugate of the self-adjoint $Q_{j}$ ; the case $a = 0$ is immediate. On the common coordinate domain multiplication distributes to give $Q (a) f = \sum_{l} a_{l} Q_{l} f$ .

3.1step 1.1step 2.1F4F6givenalgebra

[Strong continuity] Let $ψ \in S^{(r)}$ and $R_{0} \in S O (3)$ . For $R$ in a bounded neighbourhood of $R_{0}$ one has $∣ R^{- 1} x ∣ \geq \frac{1}{2} ∣ x ∣$ and $∣ R_{0}^{- 1} x ∣ \geq \frac{1}{2} ∣ x ∣$ , so with the radial majorant $h (x) : = \sup_{∣ y ∣ \geq ∣ x ∣ / 2} \max_{l} ∣ ψ_{l} (y) ∣$ one has $∣ (U (R) ψ - U (R_{0}) ψ)_{l} (x) ∣^{2} \leq 4 h (x)^{2}$ for every $x$ , where $h \in L^{2}$ because $ψ$ is Schwartz; moreover $(U (R) ψ)_{l} (x) \to (U (R_{0}) ψ)_{l} (x)$ for every $x$ as $R \to R_{0}$ , by continuity of $ψ_{l}$ and of $R \mapsto R^{- 1} x$ . Dominated convergence gives $∥ U (R) ψ - U (R_{0}) ψ ∥ \to 0$ , and for $ε / 3$ -close $ψ$ to $f$ in the sense of [F6] the triangle inequality and unitarity of step 1.1 give $∥ U (R) f - U (R_{0}) f ∥ \leq 2 ∥ f - ψ ∥ + ∥ U (R) ψ - U (R_{0}) ψ ∥ < ε$ for $R$ close to $R_{0}$ ; hence the orbit map is continuous.

3.2step 2.1F2F3F6givenalgebra

[Momentum-space form of the action] Let $ψ \in S^{(r)}$ and compute componentwise for $p \in R^{3}$ , using the substitution $y = R^{- 1} x$ , $x = R y$ of [F3] and $R^{T} = R^{- 1}$ : $(U^{(r)} U (R) ψ)_{l} (p) = (2 π ℏ)^{- 3 / 2} \int e^{- i p \cdot x / ℏ} ψ_{l} (R^{- 1} x) d x = (2 π ℏ)^{- 3 / 2} \int e^{- i p \cdot R y / ℏ} ψ_{l} (y) d y = (2 π ℏ)^{- 3 / 2} \int e^{- i R^{T} p \cdot y / ℏ} ψ_{l} (y) d y = (U^{(r)} ψ)_{l} (R^{- 1} p) = (M_{R} U^{(r)} ψ)_{l} (p)$ . Both $U^{(r)} U (R) U^{(r) - 1}$ and $M_{R}$ are bounded operators (products of unitaries), and they agree on the dense subspace $U^{(r)} S^{(r)} = S^{(r)}$ of [F2]; [F6] extends the identity to all of $L^{2} (R^{3}, d p; C^{r})$ .

4.1step 3.2F2F5givenalgebra

[Sobolev invariance] By [F2] and [F5], $W^{1, 2} (R^{3}; C^{r})$ and $W^{2, 2} (R^{3}; C^{r})$ correspond in momentum space to the conditions $p_{k} g \in L^{2}$ for every $k$ , respectively $∣ p ∣^{2} g \in L^{2}$ . By step 3.2 the rotated representative is $g_{R} (p) = g (R^{- 1} p)$ . Set $q = R^{- 1} p$ , so $d p = d q$ and $p_{k} = \sum_{l} R_{k l} q_{l}$ ; therefore each $p_{k} g_{R} (p)$ is the rotation-substituted finite linear combination $\sum_{l} R_{k l} q_{l} g (q)$ , which is in $L^{2}$ when every $q_{l} g (q)$ is, and the converse follows by applying the same argument to $R^{- 1}$ . For the second-order condition, $∣ p ∣^{2} g_{R} (p) = ∣ q ∣^{2} g (q)$ under the same measure-preserving substitution, and the converse again follows with $R^{- 1}$ . Thus $U (R) W^{m, 2} = W^{m, 2}$ for $m = 1, 2$ .

4.2step 3.2step 2.2F2givenalgebra

[Conjugation of momentum] By [F2] and step 3.2, the momentum representation of $U (R) P_{k} U (R)^{- 1}$ is $M_{R} M_{p_{k}} M_{R}^{- 1}$ . Orthogonal substitution shows that its domain is exactly ${g : (R e_{k} \cdot p) g \in L^{2}}$ , and its action there is multiplication by $(R^{- 1} p)_{k} = \sum_{l} R_{l k} p_{l}$ . Transporting by $U^{(r)}$ gives exactly $P (R e_{k})$ and $U (R) D (P_{k}) = D (P (R e_{k}))$ . For any nonzero $a$ , the same rotation-and-scaling argument as step 2.2 makes $M_{a \cdot p}$ and hence $P (a)$ self-adjoint; for $a = 0$ this is immediate. On the common momentum domain $P (a) f = \sum_{l} a_{l} P_{l} f$ by [F2] and distribution of the multiplier. This proves the full directional identities of (b).

5.1step 4.1step 4.2F1givenalgebra

[Free operator and multipliers] Since $U (R) P_{k} U (R)^{- 1} = \sum_{l} R_{l k} P_{l}$ on the maximal directional domains by step 4.2 and $U (R) W^{2, 2} = W^{2, 2}$ by step 4.1, also $U (R) P_{k}^{2} U (R)^{- 1} = \sum_{l, l^{'}} R_{l k} R_{l^{'} k} P_{l} P_{l^{'}}$ on $W^{2, 2}$ ; summing over $k$ and using $\sum_{k} R_{l k} R_{l^{'} k} = δ_{l l^{'}}$ for orthogonal $R$ gives $U (R) (\sum_{k} P_{k}^{2}) U (R)^{- 1} = \sum_{l} P_{l}^{2}$ , so $U (R) H_{0} U (R)^{- 1} = H_{0}$ . For $f \in H^{(r)}$ one has $(U (R) M_{V} U (R)^{- 1} f)_{l} (x) = V (R^{- 1} x) f_{l} (x)$ by step 1.1, so $U (R) M_{V} U (R)^{- 1} = M_{V_{R}}$ on all of $H^{(r)}$ ; adding the two identities on the common domain $W^{2, 2}$ gives $U (R) H_{V} U (R)^{- 1} = H_{V_{R}}$ , which is (c).

6.1step 1.1step 2.1step 3.1step 3.2step 4.1step 4.2step 2.2step 5.1∎

[Assembly] Steps 1.1, 2.1, 3.1 prove unitarity, the representation property, the Schwartz invariance, strong continuity and, with step 4.1, the Sobolev invariance of (a); steps 3.2, 4.2, 2.2 prove (b); step 5.1 proves (c).

Remarks

Convention. With $U (R) f = f \circ R^{- 1}$ the position operator transforms with the inverse matrix, $U (R) Q_{j} U (R)^{- 1} = \sum_{l} R_{l j} Q_{l}$ ; this is exactly the conjugation identity computed in Binney's (4.26) with $U^{†} (α) x U (α) = R (α) x$ . The invariant statement used by the physics items is the multiplier identity $U (R) M_{V} U (R)^{- 1} = M_{V \circ R^{- 1}}$ and the criterion $V \circ R^{- 1} = V$ .
Scope. The lemma concerns the spatial factor only. No rotation action on the internal factor $C^{r}$ and no SU(2) lift are introduced; those belong to the orbital-and-spin development.

LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passOpen item page →

The spatial parity action on the component-valued wave-function space

Statement

Assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The Axiom of Choice). Fix an integer $r \geq 1$ and $ℏ > 0$ , let $H^{(r)} = L^{2} (R^{3}; C^{r})$ with inner product linear in the first argument (Complex Lp classes and Euclidean test-function conventions, The space $L^{p} (μ)$ as the quotient by null functions, Hilbert space), let $Q_{j}, P_{k}$ be the componentwise position and momentum operators (Componentwise position and momentum operators on the component-valued wave-function space clause (a)), and let $S^{(r)} = S (R^{3}; C^{r})$ and $W^{m, 2} (R^{3}; C^{r})$ be the componentwise Schwartz and Sobolev classes (Schwartz space and its seminorms, Integer-order Sobolev spaces and their norms). Define the parity action $(Π f)_{l} (x) : = f_{l} (- x) (x \in R^{3}, l = 1, \dots, r),$ let $U^{(r)}$ be the unitary momentum representation of The momentum representation of the componentwise kinematics, and let $M_{- 1}$ be the reflection operator on $L^{2} (R^{3}, d p; C^{r})$ defined by $(M_{- 1} g) (p) : = g (- p)$ . This composition operator is not pointwise multiplication by a function; reflection substitution makes it unitary. Then:

(a) $Π$ is a well-defined unitary operator with $Π^{2} = i d$ and $Π^{*} = Π$ ; moreover $Π S^{(r)} = S^{(r)}$ and $Π W^{m, 2} (R^{3}; C^{r}) = W^{m, 2} (R^{3}; C^{r})$ for $m = 1, 2$ ;

(b) $Π D (Q_{j}) = D (Q_{j})$ and $Π Q_{j} Π = - Q_{j}$ ; $Π D (P_{k}) = D (P_{k})$ and $Π P_{k} Π = - P_{k}$ ;

(d) with $H_{0} = \frac{1}{2 m_{0}} \sum_{k} P_{k}^{2}$ for a parameter $m_{0} > 0$ and a real bounded measurable $V : R^{3} \to R$ , writing $V_{P} (x) : = V (- x)$ and $H_{V} = H_{0} + M_{V}$ , one has $Π H_{0} Π = H_{0}$ and $Π M_{V} Π = M_{V_{P}}$ , so $Π H_{V} Π = H_{V_{P}}$ .

Facts & Assumptions

Given: The Axiom of Countable Choice and the Axiom of Choice; $r \geq 1$ ; $ℏ > 0$ ; the space $H^{(r)}$ ; the componentwise operators $Q_{j}, P_{k}$ ; $f \in H^{(r)}$ ; $ψ \in S^{(r)}$ ; the momentum unitary $U^{(r)}$ and the reflection operator $M_{- 1}$ ; a real bounded measurable $V$ ; a parameter $m_{0} > 0$ .

[F1]

Componentwise kinematics. $Q_{j}, P_{k}$ are self-adjoint with the maximal componentwise domains, $S^{(r)}$ lies in both domains and is invariant under both operators, and on $S^{(r)}$ one has $Q_{j} f = x_{j} f$ and $P_{k} f = - i ℏ D_{k} f$ (Componentwise position and momentum operators on the component-valued wave-function space clauses (a),(c), Schwartz space and its seminorms).

[F2]

Momentum representation. $U^{(r)}$ is unitary with $U^{(r)} D (P_{k}) = D (M_{p_{k}})$ , $U^{(r)} P_{k} = M_{p_{k}} U^{(r)}$ , acting on Schwartz classes by the componentwise Fourier transform with normalization $(2 π ℏ)^{- 3 / 2}$ , and $U^{(r)} S^{(r)} = S^{(r)}$ (The momentum representation of the componentwise kinematics clauses (a),(b), Componentwise position and momentum operators on the component-valued wave-function space clause (b)).

[F3]

Reflection substitution. For $f \in L^{1} (R^{3}; C)$ one has $\int f (- x) d x = \int f (y) d y$ , since $x \mapsto - x$ is a linear substitution of determinant $- 1$ and absolute determinant $1$ (A linear map $T$ of $R^{n}$ sends Lebesgue measurable sets to Lebesgue measurable sets, with $λ_{n} (T [E]) = ∣ \det T ∣ λ_{n} (E)$ when $T$ is invertible and $T [E]$ Lebesgue null when it is not). A class in $H^{(r)}$ is determined by its components, $∥ f ∥^{2} = \sum_{l} ∥ f_{l} ∥^{2}$ , and componentwise operations commute with the $L^{2}$ class formation (Unitary identifications of the component-valued wave-function space clause (a), The induced length is a norm).

[F4]

Schwartz invariance. If $ψ \in S^{(r)}$ , then $ψ_{l} (- \cdot)$ is smooth and $\sup_{x} ∣ x^{α} \partial^{β} ψ_{l} (- x) ∣ = \sup_{y} ∣ (- y)^{α} \partial^{β} ψ_{l} (y) ∣$ , which is finite; hence $Π ψ \in S^{(r)}$ and by symmetry $Π S^{(r)} = S^{(r)}$ (Schwartz space and its seminorms).

[F5]

Sobolev and Fourier characterization. Under the momentum variable $p = 2 π ℏ ξ$ of [F2], $W^{1, 2}$ and $W^{2, 2}$ are the sets of classes with $p_{k} U^{(r)} f \in L^{2}$ for all $k$ , respectively with $∣ p ∣^{2} U^{(r)} f \in L^{2}$ (Integer-order W^{k,2} and H^k agree with equivalent norms, Plancherel theorem, Integer-order Sobolev spaces and their norms).

[F6]

Density and extension. $S^{(r)}$ is dense in $H^{(r)}$ (Schwartz space is dense in L2); a bounded linear operator on a dense subspace extends uniquely to the whole space (A bounded linear map from a dense normed subspace into a Banach space extends uniquely with the same norm).

Proof

Proof technique: direct; verify unitarity and involution by the reflection substitution, transport the action to the momentum representation for the operator conjugations, and read the Hamiltonian identities off the operator conjugations.

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

1.1F3givenalgebra

[Unitarity and involution] For $f \in H^{(r)}$ the function $x \mapsto f (- x)$ is measurable with $\int ∣ f_{l} (- x) ∣^{2} d x = \int ∣ f_{l} (y) ∣^{2} d y$ by [F3], so the assignment is well defined on classes, linear and isometric; applying it twice returns the original representative, so $Π^{2} = i d$ , $Π$ is bijective and hence unitary with $Π^{- 1} = Π$ . Being unitary and an involution, $Π$ is also self-adjoint: $⟨ Π f, g ⟩ = ⟨ f, Π g ⟩$ follows by the same substitution, $Π^{*} = Π$ .

1.2F4given

[Schwartz invariance] [F4] gives $Π S^{(r)} = S^{(r)}$ directly, with the substitution $y = - x$ used only to identify the reflected exponents.

2.1step 1.1F1givenalgebra

[Conjugation of position] Let $f \in H^{(r)}$ and compute $(Π Q_{j} Π f)_{l} (x) = (Q_{j} Π f)_{l} (- x) = (- x)_{j} (Π f)_{l} (- x) = - x_{j} f_{l} (x)$ , so $Π Q_{j} Π = - Q_{j}$ as an identity of operators with domain $D (Q_{j})$ , since $Π$ maps $D (Q_{j})$ onto itself (multiplication by $- x_{j}$ is bounded exactly where $x_{j}$ is) and $Π^{- 1} = Π$ by step 1.1.

2.2step 1.2F2F3F6givenalgebra

[Momentum-space form] Let $ψ \in S^{(r)}$ and compute componentwise with the substitution $y = - x$ of [F3]: $(U^{(r)} Π ψ)_{l} (p) = (2 π ℏ)^{- 3 / 2} \int e^{- i p \cdot x / ℏ} ψ_{l} (- x) d x = (2 π ℏ)^{- 3 / 2} \int e^{i p \cdot y / ℏ} ψ_{l} (y) d y = (U^{(r)} ψ)_{l} (- p) = (M_{- 1} U^{(r)} ψ)_{l} (p)$ . Both $U^{(r)} Π U^{(r) - 1}$ and $M_{- 1}$ are bounded, and they agree on the dense subspace $U^{(r)} S^{(r)} = S^{(r)}$ by [F2]; [F6] extends the identity to all of $L^{2} (R^{3}, d p; C^{r})$ .

3.1step 2.2F2F5givenalgebra

[Conjugation of momentum] By [F2] and step 2.2 the conjugate of $P_{k}$ corresponds to $M_{- 1} M_{p_{k}} M_{- 1} = M_{p_{k} \circ (- i d)} = M_{- p_{k}}$ , because $(M_{- 1} M_{p_{k}} M_{- 1} g) (p) = p_{k} g (p) \cdot (- 1)$ ; hence $U^{(r)} Π P_{k} Π U^{(r) - 1} = U^{(r)} (- P_{k}) U^{(r) - 1}$ and, conjugating by the unitary $U^{(r)}$ , $Π P_{k} Π = - P_{k}$ including domains by [F5] and step 2.2.

3.2step 2.2F5givenalgebra

[Sobolev invariance] By [F5] the sets $W^{1, 2}, W^{2, 2}$ are described by the conditions $p_{k} g \in L^{2}$ and $∣ p ∣^{2} g \in L^{2}$ on the momentum representative; by step 2.2 this representative is transformed by $g \mapsto g (- \cdot)$ , and $∣ (- p) ∣ = ∣ p ∣$ with the coordinate conditions again unchanged; hence $Π W^{m, 2} = W^{m, 2}$ for $m = 1, 2$ .

4.1step 3.1step 3.2F1givenalgebra

[Hamiltonian conjugation] Since $Π P_{k} Π = - P_{k}$ with domain preservation by step 3.1 and $Π W^{2, 2} = W^{2, 2}$ by step 3.2, applying the identity twice gives $Π P_{k}^{2} Π = P_{k}^{2}$ on $W^{2, 2}$ , hence $Π H_{0} Π = H_{0}$ ; and for $f \in H^{(r)}$ one has $(Π M_{V} Π f)_{l} (x) = V (- x) f_{l} (x)$ by step 1.1, so $Π M_{V} Π = M_{V_{P}}$ on all of $H^{(r)}$ . Adding on the common domain gives $Π H_{V} Π = H_{V_{P}}$ .

5.1step 1.1step 1.2step 2.1step 2.2step 3.1step 3.2step 4.1∎

[Assembly] Steps 1.1 and 1.2 prove (a) with step 3.2 for the Sobolev part, steps 2.1, 2.2, 3.1 prove (b) and (c), and step 4.1 proves (d).

Remarks

Discrete transformation. The declared parity action is the two-element group generated by an involution; no continuous one-parameter group or Stone generator is assigned to it here; its commutation with the dynamics is decided directly by the conjugation identity $Π H_{V} Π = H_{V_{P}}$ .
Scope. Parity acts on the spatial argument only; no internal-factor sign and no SU(2) lift is introduced.

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Coordinate-rotation generators on the Schwartz classes and their commutators

Statement

Assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The Axiom of Choice). Fix an integer $r \geq 1$ and $ℏ > 0$ , let $H^{(r)} = L^{2} (R^{3}; C^{r})$ with the componentwise operators $Q_{j}, P_{k}$ and the Schwartz class $S^{(r)} = S (R^{3}; C^{r})$ (Componentwise position and momentum operators on the component-valued wave-function space, Schwartz space and its seminorms), and let $U (R)$ for $R \in S O (3)$ be the spatial rotation action of The spatial rotation action on the component-valued wave-function space (so that $(U (R) f) (x) = f (R^{- 1} x)$ componentwise and $U$ is a strongly continuous unitary representation). For $i = 1, 2, 3$ define the rotation matrices $R_{1} (t) : = (\begin{array}{ccc} 1 & 0 & 0 \\ 0 & \cos t & - \sin t \\ 0 & \sin t & \cos t \end{array}), R_{2} (t) : = (\begin{array}{ccc} \cos t & 0 & \sin t \\ 0 & 1 & 0 \\ - \sin t & 0 & \cos t \end{array}), R_{3} (t) : = (\begin{array}{ccc} \cos t & - \sin t & 0 \\ \sin t & \cos t & 0 \\ 0 & 0 & 1 \end{array}),$ put $U_{i} (t) : = U (R_{i} (t))$ , and let $L_{i}$ be the self-adjoint operator with $U_{i} (t) = e^{- i t L_{i} / ℏ}$ furnished by Stone's theorem (Stone's theorem: unitary groups and self-adjoint generators, Infinitesimal generator of a unitary group). Then:

(a) for each $i$ the family $t \mapsto U_{i} (t)$ is a strongly continuous one-parameter unitary group with $U_{i} (s + t) = U_{i} (s) U_{i} (t)$ and $U_{i} (0) = i d$ (Strongly continuous one-parameter unitary group);

(b) for every $ψ \in S^{(r)}$ the curve $t \mapsto U_{i} (t) ψ$ is differentiable at $t = 0$ into $H^{(r)}$ with $\frac{d}{d t} ∣_{t = 0} U_{i} (t) ψ = - \sum_{j = 1}^{3} (e_{i} \times x)_{j} D_{j} ψ,$ and consequently $S^{(r)} \subseteq D (L_{i})$ and $L_{i} ψ = - i ℏ \sum_{j = 1}^{3} (e_{i} \times x)_{j} D_{j} ψ = \sum_{j, k = 1}^{3} ε_{i j k} Q_{j} P_{k} ψ (ψ \in S^{(r)});$

(c) $L_{i} S^{(r)} \subseteq S^{(r)}$ for each $i$ , and on $S^{(r)}$ the commutation relations hold: $L_{i} L_{k} ψ - L_{k} L_{i} ψ = i ℏ \sum_{j = 1}^{3} ε_{i k j} L_{j} ψ (ψ \in S^{(r)}) .$ (d) For each $i$ the Schwartz class is a core for $L_{i}$ : on $D (L_{i})$ the graph norm $∥ ψ ∥_{L_{i}} : = (∥ ψ ∥_{H^{(r)}}^{2} + ℏ^{- 2} ∥ L_{i} ψ ∥_{H^{(r)}}^{2})^{1 / 2}$ makes $S^{(r)}$ dense in $D (L_{i})$ (equivalently, the graph closure of $L_{i} ∣_{S^{(r)}}$ is $L_{i}$ ). This norm is equivalent to the ordinary graph norm because $ℏ > 0$ is fixed.

Facts & Assumptions

Given: The Axiom of Countable Choice and the Axiom of Choice; $r \geq 1$ ; $ℏ > 0$ ; the space $H^{(r)}$ ; the componentwise operators $Q_{j}, P_{k}$ ; the rotation action $U$ and the groups $U_{i}$ ; the generators $L_{i}$ ; a class $ψ \in S^{(r)}$ ; real $s, t$ and $h \neq 0$ .

[F1]

Rotation action. $(U (R) f) (x) = f (R^{- 1} x)$ componentwise defines a strongly continuous unitary representation of $S O (3)$ on $H^{(r)}$ ; for $R$ near the identity and $ψ \in S^{(r)}$ the reflected class $ψ \circ R^{- 1}$ is Schwartz, and $R \mapsto U (R) ψ$ is continuous (The spatial rotation action on the component-valued wave-function space clauses (a),(b), Continuity of a map between metric spaces, at a point and globally, in the $ε$ - $δ$ form).

[F2]

Matrices. $t \mapsto R_{i} (t)$ is a continuous group homomorphism $(R, +) \to S O (3)$ with $R_{i} (0) = I$ , $R_{i} (s) R_{i} (t) = R_{i} (s + t)$ and $R_{i} (t)^{- 1} = R_{i} (- t)$ ; and $\frac{d}{d t} ∣_{t = 0} R_{i} (t) = G_{i}$ , where $G_{1} x = (0, - x_{3}, x_{2})$ , $G_{2} x = (x_{3}, 0, - x_{1})$ and $G_{3} x = (- x_{2}, x_{1}, 0)$ for $x \in R^{3}$ , so that $G_{i} x = e_{i} \times x$ . These follow by direct computation from the addition formulas and the derivatives $\sin^{'} = \cos$ , $\cos^{'} = - \sin$ and the multiplication rules for $3 \times 3$ matrices (The addition formulas for sine and cosine, The derivatives of sine and cosine are cosine and minus sine, Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , The cross product in $R^{3}$ , The orthogonal group $O (3)$ and the rotation group $S O (3)$ , $S O (3)$ is a subgroup of the general linear group, with $R^{- 1} = R^{T}$ ).

[F3]

Componentwise operators and derivatives. $Q_{j}, P_{k}$ are self-adjoint with maximal componentwise domains, they map $S^{(r)}$ into itself, and $D_{j}$ denotes the weak derivative, which on Schwartz classes agrees with the classical partial derivative (Componentwise position and momentum operators on the component-valued wave-function space clause (a), Weak derivative of a locally integrable function, Schwartz space and its seminorms).

[F4]

Stone's theorem. $T \mapsto e^{i t T}$ is a bijection from self-adjoint operators onto strongly continuous one-parameter unitary groups, and the domain $D (T)$ is exactly the set of vectors where $t \mapsto e^{i t T} x$ is differentiable at $0$ , with derivative $i T x$ (Stone's theorem: unitary groups and self-adjoint generators, Infinitesimal generator of a unitary group).

[F5]

Schwartz decay and dominated convergence. For $ψ \in S^{(r)}$ each component and each of its first partial derivatives is bounded and decays faster than any polynomial; hence for fixed $x$ the map $t \mapsto ψ_{l} (R_{i} (t)^{- 1} x)$ is differentiable with derivative $- \sum_{j} (G_{i} x)_{j} (D_{j} ψ_{l}) (x)$ at $t = 0$ , and the difference quotients are dominated by a fixed Schwartz-decaying function near $t = 0$ (Schwartz space and its seminorms, The mean value inequality: if $f : [a, b] \to R^{m}$ is continuous and differentiable on $(a, b)$ with $∥ f^{'} ∥_{2} \leq M$ , then $∥ f (b) - f (a) ∥_{2} \leq M (b - a)$ , Dominated convergence, Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ ).

[F6]

Canonical commutation relations. On $S^{(r)}$ one has $(Q_{j} P_{k} - P_{k} Q_{j}) ψ = i ℏ δ_{j k} ψ$ (Canonical commutation relations of position and momentum on the Schwartz core clause (a)).

[F7]

Schwartz density. The componentwise Schwartz classes are dense in $H^{(r)}$ (Schwartz space is dense in L2).

Proof

Proof technique: direct; read the group law and continuity from the rotation representation, differentiate the orbit in the position representation along the coordinate subgroups, identify the generator with the angular-momentum expression on Schwartz classes, and compute the commutators from the canonical commutation relations.

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

1.1F1F2givenalgebra

[Group law and strong continuity] By [F1], $U_{i} (s + t) = U (R_{i} (s + t)) = U (R_{i} (s) R_{i} (t)) = U (R_{i} (s)) U (R_{i} (t)) = U_{i} (s) U_{i} (t)$ using the group law of [F2] and the representation property of [F1]; $U_{i} (0) = U (I) = i d$ ; each $U_{i} (t)$ is unitary; and strong continuity of $t \mapsto U_{i} (t) ψ$ is the composition of the continuous map $t \mapsto R_{i} (t)$ of [F2] with the continuous orbit map of [F1]. This proves (a).

1.2F1F2F5givenalgebra

[Differentiation of the orbit on Schwartz classes] Fix $ψ \in S^{(r)}$ , $i$ and one component. By [F2], $R_{i} (h)^{- 1} = R_{i} (- h) = I - h G_{i} + O (h^{2})$ entrywise, so $R_{i} (h)^{- 1} x = x - h G_{i} x + O (h^{2})$ uniformly for $x$ in a compact set; hence $ψ_{l} (R_{i} (h)^{- 1} x) - ψ_{l} (x) = - h \sum_{j} (G_{i} x)_{j} (D_{j} ψ_{l}) (x) + o (h)$ pointwise as $h \to 0$ by the differentiability of $ψ_{l}$ and the chain rule [F5], with $(G_{i} x)_{j} = (e_{i} \times x)_{j}$ by [F2]. The difference quotients are dominated near $h = 0$ by a fixed Schwartz-decaying function because $ψ_{l}$ and its first derivatives decay faster than any polynomial and $∣ x - h G_{i} x + O (h^{2}) ∣ \leq 2 ∣ x ∣$ for small $h$ ; dominated convergence [F5] therefore gives convergence in $H^{(r)}$ , so the orbit is differentiable at $0$ with the displayed derivative.

2.1step 1.2F2F3F4givenalgebra

[Generator on Schwartz classes] By step 1.2 the difference quotient of $t \mapsto U_{i} (t) ψ$ at $0$ converges, so $ψ \in D (L_{i})$ and $- \frac{i}{ℏ} L_{i} ψ = - \sum_{j} (e_{i} \times x)_{j} D_{j} ψ$ by [F4]; multiplying by $i ℏ$ gives $L_{i} ψ = - i ℏ \sum_{j} (e_{i} \times x)_{j} D_{j} ψ$ . Since $(e_{i} \times x)_{j} = x_{j^{'}}$ for the two indices $j^{'}$ complementary to $i$ with the signs of [F2], this reads $L_{i} ψ = \sum_{j, k} ε_{i j k} Q_{j} P_{k} ψ$ : for example $L_{3} ψ = - i ℏ (- x_{2} D_{1} + x_{1} D_{2}) ψ = x_{1} (- i ℏ D_{2}) ψ - x_{2} (- i ℏ D_{1}) ψ = (Q_{1} P_{2} - Q_{2} P_{1}) ψ$ , and the other two cases are the cyclic permutations. This proves (b).

2.2step 1.1F1F4F7givenalgebra

[Schwartz classes are a graph core] Fix $i$ and choose a nonnegative $η \in C_{c}^{\infty} (R)$ supported in $[- 1, 1]$ with $\int_{R} η (t) d t = 1$ ; for example, normalize $e^{- 1 / (1 - t^{2})}$ on $∣ t ∣ < 1$ and extend it by zero. Its derivatives at the endpoints vanish because each is a finite sum of powers of $(1 - t^{2})^{- 1}$ times the exponential, and $u^{- N} e^{- 1 / u} \to 0$ as $u ↓ 0$ for every $N$ . Put $η_{ϵ} (t) = ϵ^{- 1} η (t / ϵ)$ and, for $f \in H^{(r)}$ , define $S_{ϵ} f : = \int_{R} η_{ϵ} (t) U_{i} (t) f d t$ , where the integral is the norm limit of Riemann sums for the continuous $H^{(r)}$ -valued integrand on its compact support; completeness of $H^{(r)}$ gives this limit. If $f \in S^{(r)}$ , then $S_{ϵ} f \in S^{(r)}$ : its representative is $\int η_{ϵ} (t) f (R_{i} (- t) x) d t$ , and on the compact $t$ -support every spatial derivative and polynomially weighted derivative is bounded uniformly by finitely many Schwartz seminorms, so differentiation under the integral gives finite seminorms of every order. By the group law and the substitution $s = t + h$ , $\frac{U_{i} (h) S_{ϵ} f - S_{ϵ} f}{h} = \int_{R} \frac{η_{ϵ} (s - h) - η_{ϵ} (s)}{h} U_{i} (s) f d s ⟶ - \int_{R} η_{ϵ}^{'} (s) U_{i} (s) f d s$ in $H^{(r)}$ : the scalar difference quotients converge uniformly to $- η_{ϵ}^{'}$ and have support in a common compact interval for small $h$ . The Stone-generator domain characterization in [F4] therefore gives $S_{ϵ} f \in D (L_{i})$ and $L_{i} S_{ϵ} f = - i ℏ \int η_{ϵ}^{'} (s) U_{i} (s) f d s$ , so $ℏ^{- 1} ∥ L_{i} S_{ϵ} f ∥ \leq ϵ^{- 1} ∥ η^{'} ∥_{1} ∥ f ∥$ . If $f \in D (L_{i})$ , instead write the quotient as $\int η_{ϵ} (t) U_{i} (t) \frac{U_{i} (h) f - f}{h} d t$ ; its integrand converges uniformly in $t$ in $H^{(r)}$ to $- \frac{i}{ℏ} η_{ϵ} (t) U_{i} (t) L_{i} f$ , so $L_{i} S_{ϵ} f = S_{ϵ} L_{i} f$ . Strong continuity and nonnegativity of $η$ now give $∥ S_{ϵ} f - f ∥ \leq \sup_{∣ t ∣ \leq ϵ} ∥ U_{i} (t) f - f ∥ \to 0$ , and the same estimate for $L_{i} f$ shows $S_{ϵ} f \to f$ in $∥ \cdot ∥_{L_{i}}$ . Finally take $ϵ_{n} = 1 / n$ and choose $φ_{n} \in S^{(r)}$ with $∥ φ_{n} - f ∥ < [n (1 + ϵ_{n}^{- 1} ∥ η^{'} ∥_{1})]^{- 1}$ , possible by [F7]. The bounds above give $∥ S_{ϵ_{n}} (φ_{n} - f) ∥_{L_{i}} \to 0$ , while $S_{ϵ_{n}} φ_{n} \in S^{(r)}$ ; hence $S^{(r)}$ is dense in $D (L_{i})$ in the stated graph norm.

3.1step 2.1F3F5givenalgebra

[Invariance of the Schwartz class] The expression $L_{i} ψ = - i ℏ \sum_{j} (e_{i} \times x)_{j} D_{j} ψ$ is a finite sum of products of a linear polynomial with a first weak derivative of a Schwartz class; hence $L_{i} ψ \in S^{(r)}$ by the closure of $S^{(r)}$ under polynomial multiplication and differentiation [F5], so $L_{i} S^{(r)} \subseteq S^{(r)}$ .

4.1step 2.1step 3.1F3F6givenalgebra

[Commutators on the Schwartz class] Let $ψ \in S^{(r)}$ and use the product rule for commutators $[A B, C D] = A [B, C] D + A C [B, D] + [A, C] D B + C [A, D] B$ , verified by expanding both sides. By step 2.1, $L_{1} = Q_{2} P_{3} - Q_{3} P_{2}$ and $L_{2} = Q_{3} P_{1} - Q_{1} P_{3}$ , and by [F3] all displayed products are defined on $S^{(r)}$ . Using [F6] and bilinearity, only the terms containing a nonvanishing commutator contribute: $[Q_{2} P_{3}, Q_{3} P_{1}] = Q_{2} [P_{3}, Q_{3}] P_{1} = - i ℏ Q_{2} P_{1}$ , and $[Q_{3} P_{2}, Q_{1} P_{3}] = Q_{1} [Q_{3}, P_{3}] P_{2} = i ℏ Q_{1} P_{2}$ , while $[Q_{2} P_{3}, Q_{1} P_{3}] = 0$ and $[Q_{3} P_{2}, Q_{3} P_{1}] = 0$ because in the remaining products every commutator is of coordinate with coordinate or of momentum with momentum, namely $[P_{3}, Q_{1}] = [Q_{2}, Q_{1}] = [Q_{2}, P_{3}] = 0$ for the first and $[P_{2}, Q_{3}] = [Q_{3}, Q_{3}] = [Q_{3}, P_{1}] = 0$ for the second. Hence $[L_{1}, L_{2}] ψ = - i ℏ Q_{2} P_{1} ψ + i ℏ Q_{1} P_{2} ψ = i ℏ (Q_{1} P_{2} - Q_{2} P_{1}) ψ = i ℏ L_{3} ψ$ . The index triple $(1, 2, 3)$ and the three coordinate directions enter the definitions of $L_{1}, L_{2}, L_{3}$ cyclically symmetrically, and the canonical commutation relations are cyclically invariant, so the two cyclic permutations of this computation give $[L_{2}, L_{3}] = i ℏ L_{1}$ and $[L_{3}, L_{1}] = i ℏ L_{2}$ on $S^{(r)}$ ; that is $[L_{i}, L_{k}] ψ = i ℏ \sum_{j} ε_{i k j} L_{j} ψ$ for all $i, k$ . This proves (c).

5.1step 1.1step 1.2step 2.1step 3.1step 2.2step 4.1∎

[Assembly] Step 1.1 proves (a), steps 1.2 and 2.1 prove (b), step 3.1 proves the Schwartz invariance in (c), steps 2.2 and 4.1 prove the graph-core and commutator assertions, and the fixed definition of the graph norm in the Statement completes (d).

Remarks

Sign and normalization. All three generators are fixed by the convention $U_{i} (t) = e^{- i t L_{i} / ℏ}$ already used for the energy flow; with it, $L_{i} = \sum_{j, k} ε_{i j k} Q_{j} P_{k}$ on $S^{(r)}$ , the classical expression $L = x \times p$ read with $P = (- i ℏ D)$ .
Core and commutators. The graph-core statement concerns each coordinate generator separately and does not assert a joint graph core for products or for a sum such as $\sum_{i} L_{i}^{2}$ .

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The free flow as a unimodular multiplier in the momentum representation

Statement

Assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The Axiom of Choice) and the choice hypotheses of the cited spectral suppliers. Fix an integer $r \geq 1$ , a mass parameter $m > 0$ and $ℏ > 0$ , let $H^{(r)} = L^{2} (R^{3}; C^{r})$ with the componentwise operators $Q_{j}, P_{k}$ of Componentwise position and momentum operators on the component-valued wave-function space, let $U^{(r)}$ be the momentum unitary of The momentum representation of the componentwise kinematics, and let $H_{0} : = \frac{1}{2 m} \sum_{k = 1}^{3} P_{k}^{2}, D (H_{0}) = W^{2, 2} (R^{3}; C^{r}),$ which is self-adjoint with the displayed domain (Real multiplication operators and free plus bounded-potential Schrödinger operators clause (c) applied componentwise, Componentwise position and momentum operators on the component-valued wave-function space clause (a)). Write $M_{a}$ for multiplication by a measurable function $a$ on $L^{2} (R^{3}, d p; C^{r})$ and put $f_{t} (p) : = e^{- i t ∣ p ∣^{2} / (2 m ℏ)}$ (The complex exponential by its power series). Then:

(a) $U^{(r)} H_{0} U^{(r) - 1} = M_{∣ p ∣^{2} / (2 m)}$ as self-adjoint operators, with domain $D (M_{∣ p ∣^{2} / (2 m)}) = {g \in L^{2} : ∣ p ∣^{2} g \in L^{2}}$ ;

(b) $U^{(r)} e^{- i t H_{0} / ℏ} U^{(r) - 1} = M_{f_{t}}$ for every $t \in R$ ;

(c) for every $ψ \in H^{(r)}$ , every $t$ and almost every $p$ , $(U^{(r)} e^{- i t H_{0} / ℏ} ψ) (p) = f_{t} (p) (U^{(r)} ψ) (p), ∣ (U^{(r)} e^{- i t H_{0} / ℏ} ψ) (p) ∣ = ∣ (U^{(r)} ψ) (p) ∣;$ consequently the spectral outcome law of each $P_{k}$ is independent of $t$ ; and $e^{- i t H_{0} / ℏ} S^{(r)} = S^{(r)}$ for every $t$ , where $S^{(r)} = S (R^{3}; C^{r})$ .

Facts & Assumptions

Given: The Axiom of Countable Choice and the Axiom of Choice; integers $r \geq 1$ ; $m, ℏ > 0$ ; the space $H^{(r)}$ ; the componentwise operators $Q_{j}, P_{k}$ ; the momentum unitary $U^{(r)}$ ; the free operator $H_{0}$ ; a vector $ψ \in H^{(r)}$ ; a Schwartz class $u \in S^{(r)}$ ; real $t$ .

[F1]

Free operator. $H_{0} = \frac{1}{2 m} \sum_{k} P_{k}^{2}$ with domain $W^{2, 2} (R^{3}; C^{r})$ is self-adjoint; it is the componentwise assembly of the scalar free operators of Real multiplication operators and free plus bounded-potential Schrödinger operators clause (b), which are self-adjoint with domain $W^{2, 2} (R^{3}; C)$ per component and equal $- \frac{ℏ^{2}}{2 m} Δ$ on Schwartz classes (Real multiplication operators and free plus bounded-potential Schrödinger operators clauses (b),(c), Componentwise position and momentum operators on the component-valued wave-function space clause (a), Integer-order Sobolev spaces and their norms).

[F2]

Momentum representation. $U^{(r)}$ is unitary, $U^{(r)} D (P_{k}) = D (M_{p_{k}})$ with $U^{(r)} P_{k} = M_{p_{k}} U^{(r)}$ , and $U^{(r)}$ maps $S^{(r)}$ onto itself and acts there by the componentwise Fourier transform with normalization $(2 π ℏ)^{- 3 / 2}$ (The momentum representation of the componentwise kinematics clauses (a),(b), Componentwise position and momentum operators on the component-valued wave-function space clause (b), Schwartz space and its seminorms).

[F3]

Domain in momentum form. With $p$ the momentum variable of [F2], $W^{2, 2} (R^{3}; C^{r}) = {ψ \in H^{(r)} : ∣ p ∣^{2} U^{(r)} ψ \in L^{2}}$ and $W^{1, 2} (R^{3}; C^{r}) = {ψ : p_{k} U^{(r)} ψ \in L^{2} \forall k}$ as sets, by the Fourier characterization of integer-order Sobolev spaces read in the normalization of [F2] (Integer-order W^{k,2} and H^k agree with equivalent norms, Plancherel theorem, Integer-order Sobolev spaces and their norms).

[F4]

Conjugation calculus. For a unitary $W$ and a self-adjoint $K$ with spectral PVM $E_{K}$ , the operator $W K W^{- 1}$ with domain $W D (K)$ is self-adjoint with PVM $W E_{K} (\cdot) W^{- 1}$ and flow $e^{- i t W K W^{- 1} / ℏ} = W e^{- i t K / ℏ} W^{- 1}$ (Unitary conjugation, generator transport and strong commutation of self-adjoint observables clause (a)), and if two self-adjoint operators have a common domain $D$ and agree on a dense linear subspace of $D$ , then they are equal: their difference is symmetric on $D$ and vanishes on a dense subspace, hence is zero (Symmetric, self-adjoint and essentially self-adjoint operators).

[F5]

Multiplier spectral data. For a real measurable finite-valued function $a$ the multiplication operator $M_{a}$ on $L^{2} (R^{3}, d p; C^{r})$ is self-adjoint with domain ${g : a g \in L^{2}}$ and spectral PVM $M_{1_{a^{- 1} (B)}}$ ; for bounded Borel $h$ the bounded calculus satisfies $Φ_{M_{a}} (h) = M_{h \circ a}$ and $∥ M_{g} ∥ = {ess sup}_{p} ∣ g (p) ∣$ for bounded measurable $g$ ; these are the componentwise versions of the scalar multiplier facts (Spectral measure of a real L2 multiplication operator, Spectral theorem for unbounded self-adjoint operators (PVM form), Bounded borel pvm integral, Unbounded Borel functional calculus: domains, products, spectral mapping, Integral of a measurable function against a projection-valued measure).

[F6]

Density. $S^{(r)}$ is dense in $H^{(r)}$ (Schwartz space is dense in L2).

Proof

technique · direct; identify the momentum-space form of the free operator as a multiplier, obtain the flow by the conjugation calculus and read the modulus invariance and Schwartz preservation off the multiplier formula

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

1.1F1F2F3F4F5F6givenalgebra

[Momentum form of the free operator] Put $M : = M_{∣ p ∣^{2} / (2 m)}$ , self-adjoint with domain $D (M) = U^{(r)} W^{2, 2}$ by [F3] and [F5], and put $A : = U^{(r) - 1} M U^{(r)} = U^{(r) *} M U^{(r)}$ ; by [F4] $A$ is self-adjoint with domain $D (A) = U^{(r) - 1} D (M) = W^{2, 2}$ . On a Schwartz class $u \in S^{(r)}$ one has $P_{k}^{2} u \in S^{(r)}$ by [F2] and hence $H_{0} u = \frac{1}{2 m} \sum_{k} P_{k}^{2} u$ by [F1], while $U^{(r)} H_{0} u = M_{∣ p ∣^{2} / (2 m)} U^{(r)} u$ by [F2]; thus $A u = H_{0} u$ for every $u \in S^{(r)}$ . Both operators are self-adjoint with the common domain $W^{2, 2}$ and agree on the dense subspace $S^{(r)}$ [F6], so $A = H_{0}$ by [F4], which is (a).

2.1step 1.1F4F5givenalgebra

[Flow] Applying the flow identity of [F4] to $A = U^{(r) - 1} M U^{(r)} = H_{0}$ gives $U^{(r) - 1} e^{- i t M / ℏ} U^{(r)} = e^{- i t H_{0} / ℏ}$ , hence by the functional calculus of [F5] $M_{f_{t}} = e^{- i t M / ℏ}$ , $f_{t} (p) = e^{- i t ∣ p ∣^{2} / (2 m ℏ)}$ , and therefore $U^{(r)} e^{- i t H_{0} / ℏ} U^{(r) - 1} = M_{f_{t}}$ , which is (b).

3.1step 2.1F2givenalgebra

[Pointwise phase and modulus] Writing $g : = U^{(r)} ψ$ , step 2.1 gives $U^{(r)} e^{- i t H_{0} / ℏ} ψ = M_{f_{t}} g$ with $∣ f_{t} (p) ∣ = 1$ for every $p$ , so the two displays of (c) hold for almost every $p$ .

3.2step 2.1F2F5givenalgebra

[Schwartz preservation] Let $u \in S^{(r)}$ ; by [F2], $g : = U^{(r)} u \in S^{(r)}$ . The function $p \mapsto f_{t} (p) g (p)$ is smooth, and every seminorm $\sup_{p} ∣ p^{α} \partial^{β} (f_{t} g) (p) ∣$ is finite because $f_{t}$ is a unimodular exponential of a polynomial and hence has derivatives of polynomial growth while $g$ is Schwartz; thus $f_{t} g \in S^{(r)}$ , and $U^{(r) - 1} (f_{t} g) \in S^{(r)}$ by [F2] again. Applying the same argument to $- t$ gives the reverse inclusion, so $e^{- i t H_{0} / ℏ} S^{(r)} = S^{(r)}$ .

4.1step 3.1F2F5givenalgebra

[Momentum laws] By [F2], $E_{P_{k}} (B) = U^{(r) - 1} M_{1_{B} (p_{k})} U^{(r)}$ , so for a unit $ψ$ the law is $μ_{ψ}^{P_{k}} (B) = ∥ M_{1_{B} (p_{k})} U^{(r)} ψ ∥^{2} = \int_{{p_{k} \in B}} ∣ (U^{(r)} ψ) (p) ∣^{2} d p$ ; by step 3.1 the modulus $∣ (U^{(r)} e^{- i t H_{0} / ℏ} ψ) (p) ∣$ equals $∣ (U^{(r)} ψ) (p) ∣$ almost everywhere, so the integral is independent of $t$ .

5.1step 1.1step 2.1step 3.1step 3.2step 4.1∎

[Assembly] Step 1.1 proves (a), step 2.1 proves (b), and steps 3.1, 3.2 and 4.1 prove (c): the pointwise phase and modulus, the Schwartz invariance and the momentum laws.

Remarks

What is not claimed. No free propagator integral kernel, no pointwise trajectory of the particle, and no assertion that the momentum representative's modulus is measurable-free (it is an $L^{2}$ class; the invariance is that of the class of the modulus) appear in this lemma.
Scope. The statement is purely mathematical, about the free operator on $L^{2} (R^{3}; C^{r})$ ; the physical reading of the momentum laws is that of the observable and probability postulate.

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Symmetries of the declared dynamics, unitary symmetry groups and conserved sharp observables

Physical scope

The symmetry and conservation vocabulary of the declared framework model: symmetries of the dynamics of a declared isolated system as unitaries preserving the energy domain and commuting with the energy operator, unitary symmetry groups and their strong continuity, one-parameter symmetry groups and their Stone generators, and the reading of a conserved sharp observable as one whose outcome law is time-independent in every preparation. The item fixes terminology and candidate actions of the model (translations, spatial rotations, parity, and explicitly defined gauge phase multipliers on the spinless scalar space); it asserts no symmetry of any concrete model potential or external coupling, no antiunitary or superselection structure, no apparatus realization and no empirical result.

Definition

(a) Adopted readings. Adopt the state and composition postulate P1, the observable and probability postulate P2 in the reading of Assigned sharp observables and the outcome law of the framework, and the autonomous-dynamics postulate P4 in the reading of Assigned energy observable, unitary flow and the two pictures of autonomous evolution for a declared isolated system with state space $H$ (The state space of the framework: preparations, pure rays and density operators) and assigned energy observable $(H_{o p}, u_{H})$ with spectral projection-valued measure $E_{H}$ and flow $U (t) = \exp (- i t H_{o p} / ℏ)$ (State and composition postulate P1: Hilbert spaces, preparations and composite systems, Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes, Autonomous dynamics postulate P4: self-adjoint generators and unitary evolution). Units are those of Dimensions, SI unit conventions and quantity types of the framework.

(b) Symmetry of the dynamics. A symmetry of the declared dynamics, or simply a symmetry, is a unitary operator $S \in B (H)$ with $S D (H_{o p}) = D (H_{o p}), S H_{o p} = H_{o p} S on D (H_{o p}) .$ The reading is the model's: a symmetry is a unitary that preserves the assigned energy domain and commutes with the assigned energy operator on that domain; the equivalent formulation by commutation with the flow $U (t)$ , and the concomitant conservation statements, are proved results of this page and are not part of this definition. Neither the energy operator nor its potential or coupling terms are assumed to possess any symmetry here.

(c) Unitary symmetry group. A unitary symmetry group of the declared system is a group $G$ together with a map $g \mapsto S_{g}$ from $G$ into the unitary operators of $H$ such that $S_{e} = I$ , $S_{g h} = S_{g} S_{h}$ and every $S_{g}$ is a symmetry in the sense of (b). When $G$ carries a topology, the group is strongly continuous when $g \mapsto S_{g} f$ is continuous for every $f \in H$ ; for the one-parameter case the concrete condition is that of Strongly continuous one-parameter unitary group. The model's standard candidate spatial symmetry groups and additional unitary symmetry actions, which must satisfy clause (b) before any one of them is called a symmetry, include:

the spatial translations $T_{a}$ of Componentwise position and momentum operators on the component-valued wave-function space clause (c), forming the group $R^{3}$ , with generator data supplied by Componentwise position and momentum operators on the component-valued wave-function space clause (d);
the spatial rotations $U (R)$ of The spatial rotation action on the component-valued wave-function space, forming the group $S O (3)$ with its Euclidean topology, and their coordinate one-parameter subgroups, whose generators are the operators $L_{i}$ of Coordinate-rotation generators on the Schwartz classes and their commutators;
the spatial parity action $Π$ of The spatial parity action on the component-valued wave-function space, forming the two-element group ${I, Π}$ ;
in the spinless scalar case $H = L^{2} (R^{3}; C)$ , the gauge phase multipliers $G_{χ} (t)$ . No action on an internal factor is defined here. Given an open interval $I$ , a real Borel function $χ : R^{3} \times I \to R$ with unit $J s C^{- 1}$ , a charge parameter $q \in R$ with unit C, and $ℏ > 0$ with unit $J s$ , define $(G_{χ} (t) ψ) (x) : = \exp ⁣ (i q χ (x, t) / ℏ) ψ (x) for almost every x, ψ \in L^{2} (R^{3}; C) .$ For each fixed $t$ , this is a unitary operator. The real coefficients in the defining power series give $\overline{\exp (i θ)} = \exp (- i θ)$ for real $θ$ ; by absolute convergence, the Cauchy product for $\exp (i θ) \exp (- i θ)$ groups the coefficient of degree $k$ as $\frac{1}{k!} \sum_{j = 0}^{k} (\binom{k}{j}) (- 1)^{k - j}$ ; the finite sum is $(1 - 1)^{k}$ , hence is $1$ for $k = 0$ and $0$ for $k \geq 1$ . Therefore $∣ \exp (i θ) ∣ = 1$ and the inverse multiplier is $\exp ⁣ (- i q χ (\cdot, t) / ℏ)$ . This definition asserts only its action on $L^{2}$ ; it assumes no regularity of $χ$ and does not imply preservation of an unbounded energy domain. The gauge-covariance result treats only the additional regularity class stated by the prescribed-coupling postulate.

(d) One-parameter symmetry group and generator. A one-parameter symmetry group is a strongly continuous one-parameter unitary group $t \mapsto S_{t} = e^{- i t A / ℏ}$ with every $S_{t}$ a symmetry in the sense of (b), where $A$ is the self-adjoint generator furnished by Stone's theorem (Stone's theorem: unitary groups and self-adjoint generators); its declared unit is that of the spectral coordinate of $A$ . When $S_{t} = U (R_{i} (t))$ for a coordinate rotation subgroup, $A$ is the operator $L_{i}$ above; when $S_{t} = T_{t e_{j}}$ , $A$ is the momentum component $P_{j}$ in the reading of Componentwise position and momentum operators on the component-valued wave-function space clause (d).

(e) Conserved sharp observables. Let $(A, u_{A})$ be an assigned sharp observable of the framework (Assigned sharp observables and the outcome law of the framework). It is conserved along the flow when its outcome law is independent of time in every preparation: for every preparation $ρ$ (The state space: density operators, pure and mixed states, and purity) and every Borel $B \subseteq R$ , $μ_{ρ (t)}^{A} (B) = μ_{ρ (0)}^{A} (B), ρ (t) = U (t) ρ (0) U (t)^{*} .$ The conserved object is thus the outcome law of the assigned observable — a probability measure on the Borel sets — and, where the moments exist, its moments; the definition does not say that a single numerical value is revealed at each time, and it identifies no laboratory quantity with the pair $(A, u_{A})$ .

(f) Status and scope. Clauses (a)-(e) fix the vocabulary of this page within P1, P2 and P4 and the cited readings. They assert no symmetry of any concrete model: whether $T_{a}$ , $U (R)$ or $Π$ is a symmetry of a given bounded-potential dynamics is decided by the conditional results of this page. The gauge definition asserts only an $L^{2}$ unitary; a symmetry claim must separately verify the energy-domain and commutation conditions of clause (b). Antiunitary symmetries, superselection sectors, symmetries of preparations as opposed to dynamics, and any empirical claim about which symmetries a laboratory system possesses are outside this definition. The parameter $a$ of a translation has the unit of length (metre) of Dimensions, SI unit conventions and quantity types of the framework; the rotation group and parity act on the spatial factor only, in the readings of The spatial rotation action on the component-valued wave-function space and The spatial parity action on the component-valued wave-function space, and the prescribed-coupling reading is that of Bounded-coefficient magnetic Schrodinger operator.

Remarks

Why the energy domain appears. For an unbounded energy operator, the statement " $S$ commutes with $H_{o p}$ " needs the domain equality $S D (H_{o p}) = D (H_{o p})$ to make the operator identity meaningful; this is the same convention under which the covariance theorems of the kinematics page are stated.
Candidate actions are not symmetries by definition. The translations, rotations, parity and gauge multipliers are defined as unitaries or unitary families; the definition records them as the model's candidate symmetry actions, and the theorems decide for which potentials or couplings each candidate is actually a symmetry.

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Prescribed external electromagnetic coupling P5a: the minimal-coupling model and its gauge transformations

Physical scope

The prescribed-external-coupling assumption P5a of the framework: a spinless particle of mass m>0 and charge q in a named Newtonian inertial frame with admissible prescribed potentials (A,Phi) of the bounded-coefficient class, whose energy operator is the minimal-coupling magnetic Schrodinger operator on the domain W^{2,2}; the adopted field definitions, the units, and the adopted gauge transformation of potentials and state phase. Adoption is a model assumption for the stated potential class; it asserts no Maxwell equation or source model, no time-dependent well-posedness beyond the stated class, no derivation from P1-P5, and no identification of any laboratory arrangement or measured field with the declared data.

Postulate

Adopt as a physical assumption of the framework, additional to P1-P5 and valid only for the model of this page, the following prescribed-coupling statement for a spinless particle. Assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The Axiom of Choice), the choice hypotheses of the cited magnetic-operator suppliers.

(i) Frame and particle data. Fix a named Newtonian inertial frame in the class adopted by Kinematics and massive-particle model postulate P5: Newtonian frames and one-particle operators (P5) and a declared spinless particle of mass parameter $m > 0$ and charge parameter $q \in R$ ; the model state space is the scalar component of the massive model, $L^{2} (R^{3}; C)$ , with the position and momentum operators of the massive model. The frame, the mass and the charge are part of the model specification; the frame is not derived from the potentials, and $m$ and $q$ are empirical parameters of a species when one is identified.

(ii) Admissible prescribed potentials. The external data are an admissible pair $(A_{e x t}, Φ_{e x t})$ in the sense of Bounded-coefficient magnetic Schrodinger operator: real classes $A_{e x t} = (A_{1}, A_{2}, A_{3})$ with $A_{j} \in L^{\infty} (R^{3}; R)$ whose weak first derivatives $D_{k} A_{j}$ are bounded, and $Φ_{e x t} \in L^{\infty} (R^{3}; R)$ (Weak derivative of a locally integrable function, Integer-order Sobolev spaces and their norms). The potentials are prescribed: no field equation, source model or back-reaction is assumed.

(iii) Energy operator. The model's energy operator is the minimal-coupling magnetic Schrodinger operator $H_{e x t} : = K_{A_{e x t}, Φ_{e x t}}, D (H_{e x t}) = W^{2, 2} (R^{3}; C),$ of Bounded-coefficient magnetic Schrodinger operator, that is $H_{e x t} = \frac{1}{2 m} \sum_{j = 1}^{3} (P_{j} - q A_{e x t, j} (Q))^{2} + q Φ_{e x t} (Q)$ on the displayed domain. This operator is self-adjoint with domain $W^{2, 2} (R^{3}; C)$ by Self-adjointness of the bounded-coefficient magnetic Schrodinger operator, so it satisfies the self-adjointness requirement of the dynamics postulate P4 in the reading of Assigned energy observable, unitary flow and the two pictures of autonomous evolution; the adoption here is the identification of this operator as the energy of the declared particle, which is a model assumption and not a consequence of P1-P5.

(iv) Units. Use the units of Dimensions, SI unit conventions and quantity types of the framework; the additional assignments are: $A_{e x t}$ in $T m = V s m^{- 1}$ , $Φ_{e x t}$ in V, $q$ in C, so $q A_{e x t}$ has momentum units and $q Φ_{e x t}$ energy units. The exponential of clause (vi) uses The complex exponential by its power series and $χ$ is measured in $J s C^{- 1}$ , so the phase $q χ / ℏ$ is dimensionless.

(v) Field definitions. The coefficient classes in (ii) are static spatial data on $Ω : = R^{3}$ . For any open time interval $I \subseteq R$ lift them to ${\tilde{A}}_{e x t} (x, t) : = A_{e x t} (x)$ and ${\tilde{Φ}}_{e x t} (x, t) : = Φ_{e x t} (x)$ on $Ω \times I$ . Define $B : = (D_{2} A_{e x t, 3} - D_{3} A_{e x t, 2}, D_{3} A_{e x t, 1} - D_{1} A_{e x t, 3}, D_{1} A_{e x t, 2} - D_{2} A_{e x t, 1}) \in L^{\infty} (Ω \times I; R^{3}) in T .$ $E : = - \nabla_{x} {\tilde{Φ}}_{e x t} - \partial_{t} {\tilde{A}}_{e x t} \in D^{'} (Ω \times I; R^{3}) in V m^{- 1} .$ The curl uses the weak spatial derivatives already required in (ii); $\partial_{t} {\tilde{A}}_{e x t} = 0$ . Here $D^{'}$ denotes vector-valued distributions: each component is a continuous linear functional on $C_{c}^{\infty} (Ω \times I)$ , with continuity meaning that for every compact $K \subset Ω \times I$ it is bounded by a constant times the suprema on $K$ of finitely many derivatives of each test function supported in $K$ . The displayed derivatives are distributional derivatives. Explicitly, for every real test function $φ \in C_{c}^{\infty} (Ω \times I)$ and component $j$ , $⟨ E_{j}, φ ⟩ = \int_{Ω \times I} Φ_{e x t} (x) \partial_{x_{j}} φ (x, t) d x d t + \int_{Ω \times I} A_{e x t, j} (x) \partial_{t} φ (x, t) d x d t .$ The second integral vanishes for the static lift; the first is well defined because $Φ_{e x t} \in L^{\infty}$ and $φ$ has compact support. The same distributional convention applies to the locally integrable time-dependent coefficients in clause (vi). These are definitions of the model's field symbols; they assert no Maxwell equation, no relation to measured fields, and no coupling of the particle back on the sources.

(vi) Gauge transformations. Let $I \subseteq R$ be an open interval. A real function $χ : Ω \times I \to R$ is admissible for this clause when (1) for each $t \in I$ , $χ (\cdot, t)$ is $C^{2}$ and its first and second spatial derivatives are bounded uniformly in $t$ , and (2) for each $x \in Ω$ , $t \mapsto χ (x, t)$ is differentiable on $I$ with $\partial_{t} χ$ bounded on $Ω \times I$ . These bounds make $χ$ jointly Lipschitz and continuous; its time derivative is measurable as a pointwise limit of time difference quotients, and its spatial derivatives are jointly measurable as limits of spatial difference quotients and agree with their weak derivatives. For every admissible $χ$ , use the unitary $G_{χ} (t)$ defined in Symmetries of the declared dynamics, unitary symmetry groups and conserved sharp observables clause (c) on $L^{2} (Ω; C)$ . The model relates the pair $(A_{e x t}, Φ_{e x t})$ to $A_{e x t}^{'} = A_{e x t} + \nabla χ, Φ_{e x t}^{'} = Φ_{e x t} - \partial_{t} χ,$ and the state by $ψ^{'} = G_{χ} (t) ψ = e^{i q χ / ℏ} ψ$ . The operator consequences, the criterion under which the multiplier is a symmetry of a fixed model, and the propagator transported to the transformed pair are proved on this page for the stated regularity; the reading that the two coefficient pairs describe the same physical situation is part of this assumption, not a theorem, and the theorem statements of this page are independent of that reading.

(vii) Status and scope. Clauses (i)-(vi) are adopted model assumptions for the declared potential class only. They assert nothing about a laboratory realization, about field sources, about singular or unbounded potentials, about time-dependent data beyond the hypotheses of the gauge items, about statistical or measurement models, or about the accuracy of the spinless minimal-coupling description of any species; the identification of a physical arrangement with the model is a tested identification in the sense of P7.

Status and scope

What is assumed and what is proved. The minimal-coupling form of the energy operator, the field definitions and the gauge transformation formulas are adopted; self-adjointness on $W^{2, 2}$ is a mathematical theorem cited from the magnetic-operator development, and the covariance, symmetry and propagator consequences are proved on this page.
Bounded coefficients only. The assumption is restricted to admissible pairs with bounded potentials and bounded weak first derivatives; Coulomb, singular, unbounded and general time-dependent potentials are separate model realizations requiring their own self-adjointness and evolution statements.
No Maxwell theory. The field symbols of clause (v) are definitions; no Maxwell equation, constitutive relation or measured field value appears.

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Conservation of a symmetry generator: commuting flows, energy compatibility and conserved outcome laws

Physical scope

A conditional consequence of the state, observable and autonomous-dynamics postulates P1, P2 and P4 for a declared isolated system: for a self-adjoint assigned sharp observable A whose one-parameter unitary group commutes with the flow, the three formulations (commuting flows, strong commutation of the spectral measures, and invariance of the domain of A under the flow with U(t)*AU(t)=A) are equivalent, the outcome law of A is time-independent in every preparation, its mean and second moment are conserved whenever they exist, and a unitary symmetry of the dynamics commutes with the flow and, when self-adjoint and assigned as a sharp observable with dimensionless unit, has a time-independent outcome law. The statement asserts no symmetry of any concrete model, no definite value of a conserved observable, no apparatus realization, no equilibration statement and no empirical result, and uses the strong-commutation hypothesis rather than a formal commutator on a small domain.

Statement

Adopt the state and composition postulate P1, the observable and probability postulate P2 in the reading of Assigned sharp observables and the outcome law of the framework, and the autonomous-dynamics postulate P4 in the reading of Assigned energy observable, unitary flow and the two pictures of autonomous evolution for a declared isolated system with state space $H$ , assigned energy observable $(H_{o p}, u_{H})$ and flow $U (t) = \exp (- i t H_{o p} / ℏ)$ (State and composition postulate P1: Hilbert spaces, preparations and composite systems, Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes, Autonomous dynamics postulate P4: self-adjoint generators and unitary evolution); use the symmetry vocabulary of Symmetries of the declared dynamics, unitary symmetry groups and conserved sharp observables. 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). Let $(A, u_{A})$ be an assigned sharp observable of the system with spectral projection-valued measure $E_{A}$ and put $W (s) : = e^{- i s A / ℏ}$ . Then the following are equivalent:

(i) $W (s) U (t) = U (t) W (s)$ for all $s, t \in R$ ;

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

(iii) $U (t) D (A) = D (A)$ and $U (t) A U (t)^{*} = A$ for all $t \in R$ .

When they hold:

(a) Conserved law. The outcome law of $(A, u_{A})$ is time-independent: for every preparation $ρ$ , every Borel $B \subseteq R$ and every $t$ , $μ_{ρ (t)}^{A} (B) = μ_{ρ (0)}^{A} (B), μ_{ψ (t)}^{A} (B) = μ_{ψ (0)}^{A} (B) for a unit ψ with ψ (t) = U (t) ψ (0) .$

(b) Conserved moments. If $ψ \in D (A)$ is a unit vector, then the mean and the second moment of the law of $A$ in the pure preparation $[ψ (t)]$ are finite and independent of $t$ , with time-zero values $⟨ A ψ, ψ ⟩$ and $∥ A ψ ∥^{2}$ ; if a moment is infinite at $t = 0$ , no equality of finite quantities is claimed.

(c) Unitary symmetries of the dynamics. If a unitary operator $S$ is a symmetry in the sense of clause (b) of Symmetries of the declared dynamics, unitary symmetry groups and conserved sharp observables, that is $S D (H_{o p}) = D (H_{o p})$ and $S H_{o p} = H_{o p} S$ , then $S U (t) = U (t) S$ for every $t$ ; if in addition $S$ is self-adjoint and $(S, u_{S})$ is assigned as a sharp observable of the system with declared dimensionless unit $u_{S} = 1$ , then its outcome law is time-independent in every preparation.

(d) Status and exclusions. The statement is conditional on P1, P2 and P4 and the cited mathematics. Its symmetry hypothesis is the strong commutation of the flows or, equivalently, of the spectral measures, not a formal commutator identity on a small domain; no definite value of a conserved observable is asserted; no converse from the constancy of laws to the commutation conditions is claimed for unbounded $A$ ; and no concrete model potential, apparatus, measurement record or empirical result is introduced.

Facts & Assumptions

Given: The postulates P1, P2, P4 in the stated readings; the Axioms of Choice, Countable Choice and Dependent Choice; the assigned energy observable $(H_{o p}, u_{H})$ , its PVM $E_{H}$ and flow $U (t)$ ; an assigned sharp observable $(A, u_{A})$ with PVM $E_{A}$ and group $W (s)$ ; a unitary symmetry $S$ in clause (c), and for its outcome-law assertion a self-adjoint $S$ assigned as $(S, 1)$ ; a preparation $ρ$ ; a unit vector $ψ$ ; a Borel set $B$ ; real $s, t$ .

[F1]

Model readings. $H_{o p}$ and $A$ are densely defined self-adjoint operators with regular spectral PVMs $E_{H}, E_{A}$ and flows $U (t) = e^{- i t H_{o p} / ℏ}$ , $W (s) = e^{- i s A / ℏ}$ ; $ρ (t) = U (t) ρ U (t)^{*}$ and $ψ (t) = U (t) ψ$ are the preparation evolutions of P4 (Assigned energy observable, unitary flow and the two pictures of autonomous evolution clauses (a)-(c), The Schroedinger-picture flow: unitarity, domain invariance and the domain-qualified equation clauses (a),(d), Assigned sharp observables and the outcome law of the framework clause (b), Spectral theorem for unbounded self-adjoint operators (PVM form), Strongly continuous one-parameter unitary group, A self-adjoint operator generates a strongly continuous unitary group).

[F2]

Conjugation and commutation calculus. For self-adjoint $K, A$ with flows $U_{K}, U_{A}$ the conditions (i), (ii), (iii) of Unitary conjugation, generator transport and strong commutation of self-adjoint observables clause (b) are equivalent; under them $E_{A} (B) U_{K} (t) = U_{K} (t) E_{A} (B)$ for all $B, t$ (clause (d)); and for a unitary $W$ one has $E_{W K W^{*}} (B) = W E_{K} (B) W^{*}$ and $e^{- i t W K W^{*} / ℏ} = W e^{- i t K / ℏ} W^{*}$ (clause (a)).

[F3]

Conserved spectral laws. If a PVM $F$ commutes with $E_{H}$ in the strong spectral sense, then $tr (U (t) ρ U (t)^{*} F (S)) = tr (ρ F (S))$ for every density operator $ρ$ and every $S$ , and $∥ F (S) U (t) ψ ∥^{2} = ∥ F (S) ψ ∥^{2}$ for every unit $ψ$ (Conserved spectral probability laws for commuting observables and stationary eigenrays clause (b)).

[F4]

Outcome laws and moments. $μ_{ρ}^{A} (B) = tr (ρ E_{A} (B))$ and $μ_{ψ}^{A} (B) = ∥ E_{A} (B) ψ ∥^{2}$ ; for a unit $ψ \in D (A)$ the mean and second moment of the law exist and equal $⟨ A ψ, ψ ⟩$ and $∥ A ψ ∥^{2}$ (Assigned sharp observables and the outcome law of the framework clauses (b),(e), Mean, second moment and variance of the scalar spectral law, Density operators and scalar spectral probability clause (a), Trace class operator).

[F5]

Energies and unitaries. A self-adjoint unitary $S$ has spectral PVM supported on ${- 1, 1}$ and its own flow $e^{- i s S / ℏ}$ (Spectral theorem for unbounded self-adjoint operators (PVM form), Integral of a measurable function against a projection-valued measure, Self-adjoint, positive, unitary and normal operators).

Proof

Proof technique: direct; translate the model data into the mathematical conjugation calculus, transfer the conservation theorem for commuting spectral families, and handle the discrete unitary symmetry by the generator comparison.

Given: The data of the statement and [F1]-[F5].

1.1F1F2given

[Translation of the model data] By [F1] the operators $H_{o p}$ and $A$ are self-adjoint on the Hilbert space $H$ of the declared system, with flows $U$ and $W$ exactly as in [F2]; hence the equivalence of (i), (ii) and (iii) below is the equivalence stated in [F2] for the pair $K : = H_{o p}$ , $A$ : condition (i) is the commutation of the two flows, condition (ii) the strong commutation of $E_{H}$ and $E_{A}$ , and condition (iii) the domain invariance and conjugation identity for $A$ under $U$ . This proves the equivalence.

2.1step 1.1F3F4givenalgebra

[Conserved law] Assume (ii), so that $F : = E_{A}$ commutes with $E_{H}$ in the strong spectral sense. Applying [F3] with $S : = B$ to the preparation $ρ$ gives $tr (U (t) ρ U (t)^{*} E_{A} (B)) = tr (ρ E_{A} (B))$ , and by [F4] this is $μ_{ρ (t)}^{A} (B) = μ_{ρ (0)}^{A} (B)$ . The pure form follows from the second display of [F3] with the unit vector $ψ (0)$ and [F4]. This proves (a).

2.2step 1.1F2givenalgebra

[Unitary symmetries commute with the flow] Let $S$ be unitary with $S D (H_{o p}) = D (H_{o p})$ and $S H_{o p} = H_{o p} S$ , that is $S H_{o p} S^{*} = H_{o p}$ with domain $S D (H_{o p}) = D (H_{o p})$ . By [F2] applied with $W : = S$ and $K : = H_{o p}$ one has $e^{- i t S H_{o p} S^{*} / ℏ} = S e^{- i t H_{o p} / ℏ} S^{*}$ for every $t$ ; the left-hand operator is $e^{- i t H_{o p} / ℏ} = U (t)$ and the right-hand one is $S U (t) S^{*}$ , so $S U (t) S^{*} = U (t)$ and, multiplying on the right by $S$ , $S U (t) = U (t) S$ .

3.1step 2.1F4givenalgebra

[Conserved moments] Let $ψ \in D (A)$ be a unit vector. By [F4] the mean and second moment of the law of $A$ in $[ψ (0)]$ exist and equal $⟨ A ψ, ψ ⟩$ and $∥ A ψ ∥^{2}$ ; since by step 2.1 the law itself is the same measure at every time, its moments are the same extended real numbers at every time, hence finite and time-independent with the displayed values. If the second moment is infinite at $t = 0$ , [F4] does not apply and no equality is claimed. This proves (b).

3.2step 2.2step 1.1step 2.1F2F5givenalgebra

[Self-adjoint symmetries have conserved laws] Let $S$ be unitary and self-adjoint with $S H_{o p} S^{*} = H_{o p}$ and assigned sharp observable $(S, 1)$ . By step 2.2, $S U (t) = U (t) S$ ; multiplying on the left by $U (t)^{*}$ gives $U (t)^{*} S U (t) = S$ , and conjugating this identity by the unitary $U (t)$ gives $U (t) S U (t)^{*} = S$ as well. Apply the flow identity of [F2] with $W : = U (t)^{*}$ and $K : = S$ , at time parameter $s$ : $e^{- i s U (t)^{*} S U (t) / ℏ} = U (t)^{*} e^{- i s S / ℏ} U (t)$ ; since $U (t)^{*} S U (t) = S$ , the left side is $e^{- i s S / ℏ}$ , so $U (t)$ commutes with $e^{- i s S / ℏ}$ for every $s$ . Writing $W_{S} (s) : = e^{- i s S / ℏ}$ , this says $W_{S} (s) U (t) = U (t) W_{S} (s)$ for all $s, t$ , which is condition (i) for the pair $(S, H_{o p})$ ; by the equivalence of step 1.1 condition (ii) holds for that pair, and applying step 2.1 to the self-adjoint operator $S$ gives the time-independence of the law of $(S, u_{S})$ in every preparation. This proves (c).

4.1step 1.1step 2.1step 3.1step 2.2step 3.2∎

[Assembly and scope] Step 1.1 proves the equivalence, steps 2.1 and 3.1 prove (a) and (b), step 2.2 proves the first part of (c) and step 3.2 its second part; clause (d) records the hypotheses actually used (P1, P2, P4, strong commutation of flows or spectral measures) and the exclusions (no definite value, no unbounded converse, no concrete model or empirical claim).

Remarks

Energy conservation is the special case $A = H_{o p}$ . With $A = H_{o p}$ the conditions (i)-(iii) hold trivially, and clause (a) reproduces the conservation of the energy law proved on the autonomous-dynamics page; the content here is the transfer to any other generator.
Commutation on a small domain is excluded. The equivalence proved is between the flow commutation (i) and the strong spectral commutation (ii); a formal identity $[A, H_{o p}] ψ = 0$ on a small domain is neither a hypothesis nor a conclusion.

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Translation invariance of bounded potentials and conservation of momentum

Physical scope

A conditional consequence of the framework postulates for the componentwise massive model with a real bounded measurable potential V: the translation T_a is a symmetry of the bounded-potential operator H_V exactly when V is invariant under translation by a, the full translation group is a symmetry group exactly when V is almost everywhere constant, and for such translation-invariant models the momentum components are conserved observables by the generator-conservation theorem. The statement concerns real bounded measurable potentials only; it asserts no momentum conservation for non-invariant potentials, no statement about singular or unbounded potentials, no lattice or discrete-group conservation claim beyond the fixed-translation criterion, no apparatus realization and no empirical result.

Statement

Adopt the postulates P1, P2, P4, P5 and P6 in the readings of The massive spin-s particle model of the framework: state space, operators, units and frame readings, Assigned sharp observables and the outcome law of the framework, Assigned energy observable, unitary flow and the two pictures of autonomous evolution and Symmetries of the declared dynamics, unitary symmetry groups and conserved sharp observables for a declared particle of mass parameter $m > 0$ and spin label $s$ (State and composition postulate P1: Hilbert spaces, preparations and composite systems, Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes, Autonomous dynamics postulate P4: self-adjoint generators and unitary evolution, Kinematics and massive-particle model postulate P5: Newtonian frames and one-particle operators), and 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). Let $H_{s}$ be the model's state space with componentwise operators $Q_{j}, P_{k}$ (Componentwise position and momentum operators on the component-valued wave-function space), let $V : R^{3} \to R$ be real, bounded and measurable, and let $H_{V} : = \frac{1}{2 m} \sum_{k = 1}^{3} P_{k}^{2} + M_{V}, D (H_{V}) = W^{2, 2} (R^{3}; C^{2 s + 1}), V_{a} (x) : = V (x - a),$ be the model's bounded-potential energy operator with $M_{V}$ multiplication by $V$ and $T_{a} : = T_{a}^{(2 s + 1)}$ the componentwise translations (Real multiplication operators and free plus bounded-potential Schrödinger operators clause (c), Translation covariance of the massive spin-s kinematics and of its bounded-potential dynamics). Then:

(a) for every $a \in R^{3}$ : $T_{a} H_{V} T_{a}^{- 1} = H_{V_{a}},$ and $T_{a}$ is a symmetry of $H_{V}$ in the sense of Symmetries of the declared dynamics, unitary symmetry groups and conserved sharp observables clause (b) if and only if $V_{a} = V$ almost everywhere;

(b) if $T_{a}$ is a symmetry for every $a \in R^{3}$ , then $V$ is almost everywhere constant; in that case $H_{V} = H_{0} + c I$ on $W^{2, 2}$ for $H_{0} = \frac{1}{2 m} \sum_{k} P_{k}^{2}$ and $c$ the constant, and the translations form a strongly continuous unitary group of symmetries;

(c) Momentum conservation for translation-invariant models. Let $V$ be almost everywhere constant and let $c$ be that constant. Then for every $k \in {1, 2, 3}$ the momentum component $P_{k}$ is a conserved observable of the model with energy $H_{V}$ : for every preparation $ρ$ , every Borel $B \subseteq R$ and every $t$ , $μ_{ρ (t)}^{P_{k}} (B) = μ_{ρ (0)}^{P_{k}} (B);$ if $ψ (0) \in D (P_{k})$ is a unit vector, the mean and second moment of the law of $P_{k}$ are conserved with time-zero values $⟨ P_{k} ψ, ψ ⟩$ and $∥ P_{k} ψ ∥^{2}$ ; and the coordinate translations satisfy $T_{t e_{k}} = e^{- i t P_{k} / ℏ}$ for all $t$ , so the generators of the one-parameter symmetry groups are the momentum components;

(d) Scope. The statement concerns the componentwise massive model with real bounded measurable potentials. It asserts no momentum conservation for a potential that is not translation invariant — the translation-covariance theorem transports the position law and conjugates the dynamics, but does not by itself conserve momentum — and it makes no claim about singular, unbounded, boundary or relativistic models. Clause (a) includes bounded periodic potentials and their fixed period translations; no additional discrete-group conservation theorem, apparatus or empirical result is asserted.

Facts & Assumptions

Given: The postulates P1, P2, P4, P5, P6 in the stated readings; the Axioms of Choice, Countable Choice and Dependent Choice; the state space $H_{s}$ ; the operators $Q_{j}, P_{k}$ and translations $T_{a}$ ; a real bounded measurable $V$ ; the operator $H_{V}$ ; a preparation $ρ$ ; a unit vector $ψ$ ; a Borel set $B$ ; $a \in R^{3}$ ; $k \in {1, 2, 3}$ ; $t \in R$ .

[F1]

Translations and the bounded-potential operator. $T_{a}$ is unitary with $T_{a} D (Q_{j}) = D (Q_{j})$ , $T_{a} Q_{j} T_{a}^{- 1} = Q_{j} - a_{j} I$ , $T_{a} D (P_{k}) = D (P_{k})$ , $T_{a} P_{k} T_{a}^{- 1} = P_{k}$ , and $T_{a} W^{2, 2} = W^{2, 2}$ ; $T_{a} H_{V} T_{a}^{- 1} = H_{V_{a}}$ ; and $T_{t e_{k}} = e^{- i t P_{k} / ℏ}$ with the generator identification of the translation group (Translation covariance of the massive spin-s kinematics and of its bounded-potential dynamics clause (c), Componentwise position and momentum operators on the component-valued wave-function space clauses (c),(d)).

[F2]

Multipliers. $M_{V}$ is the bounded self-adjoint multiplication operator by the real bounded class $V$ ; for two real bounded measurable functions $W_{1}, W_{2}$ one has $M_{W_{1}} = M_{W_{2}}$ if and only if $W_{1} = W_{2}$ almost everywhere; and $H_{V} = H_{0} + M_{V}$ is self-adjoint with domain $W^{2, 2}$ and generates a strongly continuous unitary group (Real multiplication operators and free plus bounded-potential Schrödinger operators clauses (a),(c), The space $L^{p} (μ)$ as the quotient by null functions).

[F3]

Mollifiers and Lebesgue points. Let $ρ (x) = q (∣ x ∣)$ be the unit-mass radial mollifier supported in $B_{1} (0)$ and $ρ_{ε} (x) = ε^{- 3} ρ (x / ε)$ its family (The mollifier family generated by a unit-mass smooth bump, A radial mollifier family in Rn). Since $q$ is continuous with compact support, $Φ (r) : = \sup_{u \geq r} ∣ q (u) ∣$ is bounded, nonincreasing, vanishes for $r \geq 1$ , and satisfies $∣ ρ (y) ∣ \leq Φ (∣ y ∣)$ and $\int_{R^{3}} Φ (∣ y ∣) d y < \infty$ . The radial-majorant recovery lemma therefore applies: for $f \in L^{1} (R^{3})$ and a Lebesgue point $x$ of $f$ , $(f * ρ_{ε}) (x) \to f (x)$ as $ε ↓ 0$ (Lebesgue-point convergence for radial-majorized kernels, Lebesgue points and the Lebesgue set of an $L_{l o c}^{1}$ class); almost every point is a Lebesgue point of $f$ (Almost every point is a Lebesgue point of a locally integrable function). For bounded measurable $V$ , each integral against $ρ_{ε} (x - \cdot)$ is absolutely convergent because the kernel is bounded with compact support.

[F4]

Conservation theorem. For a declared isolated system with energy $H_{o p}$ and flow $U (t)$ , an assigned sharp observable $(A, u_{A})$ with $e^{- i s A / ℏ} U (t) = U (t) e^{- i s A / ℏ}$ for all $s, t$ has a time-independent outcome law, conserved moments for vectors in $D (A)$ , and a unitary symmetry $S$ of the dynamics satisfies $S U (t) = U (t) S$ (Conservation of a symmetry generator: commuting flows, energy compatibility and conserved outcome laws clauses (a)-(c)).

[F5]

Outcome laws. The outcome law of an assigned sharp observable is $μ_{ρ}^{A} (B) = tr (ρ E_{A} (B))$ , respectively $μ_{ψ}^{A} (B) = ∥ E_{A} (B) ψ ∥^{2}$ , with the spectral PVMs of the model's self-adjoint operators (Assigned sharp observables and the outcome law of the framework clauses (b),(e), Spectral theorem for unbounded self-adjoint operators (PVM form), Density operators and scalar spectral probability clause (a)).

Proof

Proof technique: direct; read the symmetry criterion off the conjugation identity for the bounded-potential operator, prove that a potential invariant under every translation is almost everywhere constant by mollification and Lebesgue-point convergence, and instantiate the conservation theorem for the momentum components of an invariant model.

Given: The data of the statement and [F1]-[F5].

1.1F1F2givenalgebra

[Symmetry criterion] The identity $T_{a} H_{V} T_{a}^{- 1} = H_{V_{a}}$ is [F1], and $T_{a}$ is a symmetry exactly when $T_{a} H_{V} T_{a}^{- 1} = H_{V}$ with $T_{a} D (H_{V}) = D (H_{V}) = W^{2, 2}$ , which by the identity holds exactly when $H_{V_{a}} = H_{V}$ . Subtracting the common free part $H_{0}$ (self-adjoint with domain $W^{2, 2}$ by [F2]) gives $M_{V_{a}} = M_{V}$ as bounded operators, and by [F2] this is equivalent to $V_{a} = V$ almost everywhere. This proves (a).

1.2F3givenalgebra

[Invariance makes the mollifications constant] Assume $V_{a} = V$ almost everywhere for every $a \in R^{3}$ , fix $ε > 0$ and $x, a \in R^{3}$ . By [F3] the defining integral is absolutely convergent, and the substitution $z = y + a$ gives $(V * ρ_{ε}) (x - a) = \int V (y) ρ_{ε} (x - a - y) d y = \int V (z - a) ρ_{ε} (x - z) d z$ . Since $V (z - a) = V (z)$ for almost every $z$ and the kernel is integrable with compact support, this equals $\int V (z) ρ_{ε} (x - z) d z = (V * ρ_{ε}) (x)$ . Thus $V * ρ_{ε}$ is translation invariant pointwise and therefore is constant, equal to some $c_{ε}$ .

2.1step 1.2F1F2F3givenalgebra

[Lebesgue points and constancy] For each positive integer $n$ put $B_{n} : = B (0, n)$ and $f_{n} : = V 1_{B (0, n + 1)} \in L^{1} (R^{3})$ . By [F3] almost every $x \in B_{n}$ is a Lebesgue point of $f_{n}$ , so choose one such point $x_{n} \in B_{n}$ ; for $0 < ε < 1$ the support of $ρ_{ε} (x_{n} - \cdot)$ lies in $B (0, n + 1)$ , and the recovery lemma gives $c_{ε} = (V * ρ_{ε}) (x_{n}) = (f_{n} * ρ_{ε}) (x_{n}) \to f_{n} (x_{n}) = V (x_{n}) = : c_{n}$ . For every Lebesgue point $x \in B_{n}$ of $f_{n}$ , the same support argument and recovery lemma give $V (x) = \lim_{ε ↓ 0} c_{ε} = c_{n}$ ; hence $V = c_{n}$ almost everywhere on $B_{n}$ . The balls $B_{n}$ are nested and each has positive measure, so their constants agree and $V$ is almost everywhere constant on $R^{3}$ . Calling that constant $c$ , [F2] gives $H_{V} = H_{0} + c I$ , and the translations are symmetries by (a) and form a strongly continuous unitary group by [F1]. This proves (b).

3.1step 2.1F1F4F5givenalgebra

[Momentum conservation] Let $V = c$ almost everywhere, so $H_{V} = H_{0} + c I$ on $W^{2, 2}$ . For fixed $k$ , the group $W (s) : = e^{- i s P_{k} / ℏ} = T_{s e_{k}}$ of [F1] consists of unitaries $S$ with $S D (H_{V}) = D (H_{V})$ and $S H_{V} S^{*} = H_{V}$ : the identity for $P_{k}$ in [F1] gives $T_{s e_{k}} P_{k} T_{s e_{k}}^{- 1} = P_{k}$ and hence the same for $P_{k}^{2}$ and for $H_{V}$ with its domain. By [F4] applied to the unitary symmetry $S = T_{s e_{k}}$ , $W (s) U (t) = U (t) W (s)$ for all $s, t$ , where $U$ is the flow of $H_{V}$ ; hence the conservation conclusion of [F4] applies with $A : = P_{k}$ , giving the displayed time-independence of the momentum law for every preparation and, for $ψ (0) \in D (P_{k})$ , the conservation of the mean and second moment with the time-zero values of [F4] and [F5]. The generator statement is [F1]. This proves (c).

4.1step 1.1step 1.2step 2.1step 3.1∎

[Assembly and scope] Steps 1.1, 1.2, 2.1 and 3.1 prove (a)-(c); clause (d) records the bounded measurable class of potentials, the explicit exclusion of momentum conservation for non-invariant potentials, and the absence of discrete-symmetry, singular-potential, apparatus and empirical claims.

Remarks

Where translation invariance fails. If a bounded measurable potential is not almost everywhere constant, the contrapositive of clause (b) gives some nonzero translation $a$ for which $V_{a}$ and $V$ are not equal almost everywhere, and then $T_{a}$ is not a symmetry; the companion page exhibits such a potential. Covariance is not conservation.
The role of boundedness. The proof of the constancy criterion uses only boundedness and measurability of $V$ ; no regularity, decay or periodicity is assumed, and no conclusion is drawn for potentials outside this class.

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Rotational and parity invariance of bounded-potential models: symmetries and conservation of angular momentum and parity laws

Physical scope

Conditional symmetry and conservation statements for the massive spin-s model with a real bounded measurable potential: spatial rotations conjugate the bounded-potential energy operator to the rotated potential, rotational invariance makes every rotation a symmetry and the coordinate-rotation generators conserved sharp observables, and parity invariance makes the parity action a symmetry with a time-independent parity outcome law. The item asserts no symmetry of any concrete potential outside the stated invariance hypotheses, no spinorial or internal rotation action, no SU(2) lift, no apparatus and no empirical result.

Statement

Adopt the postulates P1, P2, P4, P5 and P6 in the readings of The massive spin-s particle model of the framework: state space, operators, units and frame readings, Assigned sharp observables and the outcome law of the framework, Assigned energy observable, unitary flow and the two pictures of autonomous evolution and Symmetries of the declared dynamics, unitary symmetry groups and conserved sharp observables for a declared particle of mass parameter $m > 0$ and spin label $s$ (State and composition postulate P1: Hilbert spaces, preparations and composite systems, Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes, Autonomous dynamics postulate P4: self-adjoint generators and unitary evolution, Kinematics and massive-particle model postulate P5: Newtonian frames and one-particle operators, Identical-particle and spin assignments P6: permutation sectors and finite-spin representations), and 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). Let $H_{s} = L^{2} (R^{3}; C^{r})$ , $r = 2 s + 1$ , be the model's state space with componentwise operators $Q_{j}, P_{k}$ , let $U (R)$ be the spatial rotation action and $Π$ the spatial parity action (The spatial rotation action on the component-valued wave-function space, The spatial parity action on the component-valued wave-function space), let $U_{i} (t) = U (R_{i} (t))$ be the coordinate rotation subgroups with Stone generators $L_{i}$ on the Schwartz classes (Coordinate-rotation generators on the Schwartz classes and their commutators), let $V : R^{3} \to R$ be real, bounded and measurable, and let $H_{V} : = \frac{1}{2 m} \sum_{k = 1}^{3} P_{k}^{2} + M_{V}, D (H_{V}) = W^{2, 2} (R^{3}; C^{r}), V_{R} (x) : = V (R^{- 1} x), V_{P} (x) : = V (- x),$ be the model's bounded-potential energy operator with flow $U (t) = \exp (- i t H_{V} / ℏ)$ in the reading of Assigned energy observable, unitary flow and the two pictures of autonomous evolution clause (b) (Real multiplication operators and free plus bounded-potential Schrödinger operators clause (c) applied componentwise, Componentwise position and momentum operators on the component-valued wave-function space clause (a)). Then:

(a) for every $R \in S O (3)$ one has $U (R) H_{V} U (R)^{- 1} = H_{V_{R}}$ , and $U (R)$ is a symmetry of $H_{V}$ in the sense of Symmetries of the declared dynamics, unitary symmetry groups and conserved sharp observables clause (b) if and only if $V_{R} = V$ almost everywhere;

(b) if $V_{R} = V$ almost everywhere for every $R \in S O (3)$ , then every $U (R)$ is a symmetry, the map $R \mapsto U (R)$ is a strongly continuous unitary symmetry group of the declared dynamics, and each generator $L_{i}$ is a conserved observable of the model with energy $H_{V}$ in the unit $J s$ of the model's action (Dimensions, SI unit conventions and quantity types of the framework): for every preparation $ρ$ , every Borel $B \subseteq R$ and every $t$ , $μ_{ρ (t)}^{L_{i}} (B) = μ_{ρ (0)}^{L_{i}} (B),$ and for a unit vector $ψ (0) \in D (L_{i})$ the mean and second moment of the law of $L_{i}$ are finite and independent of $t$ , with time-zero values $⟨ L_{i} ψ, ψ ⟩$ and $∥ L_{i} ψ ∥^{2}$ ;

(c) if $V (- x) = V (x)$ for almost every $x$ , then $Π H_{V} Π = H_{V}$ on $W^{2, 2}$ , so $Π$ is a symmetry of $H_{V}$ , and $Π$ is a symmetry if and only if $V_{P} = V$ almost everywhere; moreover, reading the self-adjoint unitary $Π$ as an assigned sharp observable with its spectral coordinate in the dimensionless unit of its eigenvalues $\pm 1$ , the outcome law of $Π$ is time-independent in every preparation: $μ_{ρ (t)}^{Π} (B) = μ_{ρ (0)}^{Π} (B) (B \subseteq R Borel, t \in R);$

(d) Scope. The statement is conditional on P1, P2, P4, P5 and P6 and the cited mathematics, and it concerns the componentwise massive model with real bounded measurable potentials. Rotations and parity act on the spatial factor only: no SU(2) lift, no rotation of the internal factor and no spin angular momentum are introduced. The rotational and parity invariance hypotheses are exactly the displayed almost-everywhere identities; no concrete potential, apparatus, measured spectrum or empirical result is asserted, and no characterization of the invariant potentials beyond these identities is claimed.

Facts & Assumptions

Given: The postulates P1, P2, P4, P5, P6 in the stated readings; the Axioms of Choice, Countable Choice and Dependent Choice; the state space $H_{s}$ ; the operators $Q_{j}, P_{k}$ ; the rotation action $U$ and its coordinate subgroups $U_{i}$ with generators $L_{i}$ ; the parity action $Π$ ; a real bounded measurable $V$ ; the operator $H_{V}$ with flow $U (t)$ ; a preparation $ρ$ ; a unit vector $ψ$ ; a Borel set $B$ ; $R \in S O (3)$ ; $i \in {1, 2, 3}$ and $t, s \in R$ .

[F1]

Model. $H_{V}$ is self-adjoint with domain $D (H_{V}) = W^{2, 2} (R^{3}; C^{r})$ and generates the strongly continuous unitary group $U (t) = e^{- i t H_{V} / ℏ}$ (Real multiplication operators and free plus bounded-potential Schrödinger operators clause (c), Componentwise position and momentum operators on the component-valued wave-function space clause (a), The massive spin-s particle model of the framework: state space, operators, units and frame readings, A self-adjoint operator generates a strongly continuous unitary group). A symmetry of the declared dynamics with energy $H_{V}$ is a unitary $S$ with $S D (H_{V}) = D (H_{V})$ and $S H_{V} = H_{V} S$ on $D (H_{V})$ ; a unitary symmetry group is a group of such symmetries; a one-parameter symmetry group has all its members symmetries and its generator is furnished by Stone's theorem (Symmetries of the declared dynamics, unitary symmetry groups and conserved sharp observables clauses (b)-(d), Stone's theorem: unitary groups and self-adjoint generators, Strongly continuous one-parameter unitary group).

[F2]

Rotation action. $U (R)$ is unitary, $U (I) = i d$ , $U (R S) = U (R) U (S)$ , $R \mapsto U (R) f$ is continuous for every $f$ , and $U (R) S^{(r)} = S^{(r)}$ , $U (R) W^{2, 2} (R^{3}; C^{r}) = W^{2, 2} (R^{3}; C^{r})$ ; moreover $U (R) H_{0} U (R)^{- 1} = H_{0}$ for the free operator $H_{0} = \frac{1}{2 m} \sum_{k} P_{k}^{2}$ and $U (R) M_{V} U (R)^{- 1} = M_{V_{R}}$ , hence $U (R) H_{V} U (R)^{- 1} = H_{V_{R}}$ (The spatial rotation action on the component-valued wave-function space clauses (a)-(c), The orthogonal group $O (3)$ and the rotation group $S O (3)$ , $S O (3)$ is a subgroup of the general linear group, with $R^{- 1} = R^{T}$ ).

[F3]

Multiplier equality. For real bounded measurable functions $V_{1}, V_{2}$ one has $M_{V_{1}} = M_{V_{2}}$ if and only if $V_{1} = V_{2}$ almost everywhere (Real multiplication operators and free plus bounded-potential Schrödinger operators clause (a), The space $L^{p} (μ)$ as the quotient by null functions).

[F4]

Parity action. $Π$ is unitary, $Π^{2} = i d$ and $Π^{*} = Π$ ; $Π W^{2, 2} (R^{3}; C^{r}) = W^{2, 2} (R^{3}; C^{r})$ ; $Π H_{0} Π = H_{0}$ and $Π M_{V} Π = M_{V_{P}}$ , hence $Π H_{V} Π = H_{V_{P}}$ (The spatial parity action on the component-valued wave-function space clauses (a),(d)).

[F5]

Rotation generators. For each $i$ , $t \mapsto U_{i} (t)$ is a strongly continuous one-parameter unitary group with $U_{i} (t) = e^{- i t L_{i} / ℏ}$ for a self-adjoint generator $L_{i}$ , and $L_{i} ψ = - i ℏ \sum_{j} (e_{i} \times x)_{j} D_{j} ψ$ on $ψ \in S^{(r)}$ (Coordinate-rotation generators on the Schwartz classes and their commutators clauses (a),(b), Infinitesimal generator of a unitary group).

[F6]

Conservation theorem. For a declared isolated system with energy $H_{o p}$ and flow $U (t)$ , the following are equivalent: commutation of the flows $W (s) U (t) = U (t) W (s)$ of an assigned self-adjoint observable $(A, u_{A})$ with $W (s) = e^{- i s A / ℏ}$ , strong spectral commutation of the PVMs, and domain preservation with $U (t) A U (t)^{*} = A$ ; when they hold the outcome law of $A$ is time-independent in every preparation and the mean and second moment are conserved for unit vectors in $D (A)$ ; a unitary symmetry $S$ satisfies $S U (t) = U (t) S$ for every $t$ ; and a self-adjoint unitary symmetry assigned as $(S, 1)$ has a time-independent outcome law (Conservation of a symmetry generator: commuting flows, energy compatibility and conserved outcome laws clauses (a)-(c)).

[F7]

Outcome laws and moments. For an assigned sharp observable $A$ one has $μ_{ρ}^{A} (B) = tr (ρ E_{A} (B))$ and $μ_{ψ}^{A} (B) = ∥ E_{A} (B) ψ ∥^{2}$ , and for a unit vector $ψ \in D (A)$ the mean and second moment of the law exist and equal $⟨ A ψ, ψ ⟩$ and $∥ A ψ ∥^{2}$ (Assigned sharp observables and the outcome law of the framework clauses (b),(e), Mean, second moment and variance of the scalar spectral law, Density operators and scalar spectral probability clause (a), Trace class operator, The state space of the framework: preparations, pure rays and density operators, The state space: density operators, pure and mixed states, and purity, Spectral theorem for unbounded self-adjoint operators (PVM form)).

Proof

Proof technique: direct; read the rotation covariance off the rotation action, convert the symmetry definition into the operator identity and apply the multiplier criterion, transfer the conservation theorem to the coordinate rotation generators, and handle parity as the discrete self-adjoint unitary symmetry.

Given: The data of the statement and [F1]-[F7].

1.1F1F2F3givenalgebra

[Rotation covariance and symmetry criterion] By [F2] one has $U (R) H_{V} U (R)^{- 1} = H_{V_{R}}$ and $U (R) D (H_{V}) = D (H_{V})$ ; by [F1] the unitary $U (R)$ is a symmetry exactly when also $U (R) H_{V} U (R)^{- 1} = H_{V}$ as operators on $W^{2, 2}$ , that is exactly when $H_{V_{R}} = H_{V}$ . Both sides are self-adjoint with the same domain $W^{2, 2}$ , so subtracting the common free part $H_{0}$ of [F2] gives the equivalent condition $M_{V_{R}} = M_{V}$ , which by [F3] is equivalent to $V_{R} = V$ almost everywhere. This proves (a).

2.1step 1.1F2given

[Invariance gives a unitary symmetry group] Assume $V_{R} = V$ almost everywhere for every $R \in S O (3)$ . By step 1.1 each $U (R)$ is a symmetry; by [F2] the map $R \mapsto U (R)$ satisfies $U (I) = i d$ , $U (R S) = U (R) U (S)$ and strong continuity, so it is a strongly continuous unitary symmetry group in the sense of [F1].

2.2step 1.1F1F3F4F6F7givenalgebra

[Parity symmetry and conserved parity law] Assume $V_{P} = V$ almost everywhere. By [F4] $Π H_{V} Π = H_{V_{P}} = H_{V}$ on $W^{2, 2}$ and $Π W^{2, 2} = W^{2, 2}$ , $Π$ being unitary and self-adjoint; hence by [F1] $Π$ is a symmetry. Conversely, if $Π$ is a symmetry, then $H_{V_{P}} = Π H_{V} Π = H_{V}$ , and subtracting the common free part gives $M_{V_{P}} = M_{V}$ , so $V_{P} = V$ almost everywhere by [F3]. Reading $Π$ as a self-adjoint unitary assigned sharp observable, applying [F6] clause (c) to the unitary symmetry $S : = Π$ gives $Π U (t) = U (t) Π$ for every $t$ and the time-independence of the outcome law of the assigned sharp observable $(Π, u)$ in every preparation, which by [F7] is the displayed identity $μ_{ρ (t)}^{Π} (B) = μ_{ρ (0)}^{Π} (B)$ ; the spectral coordinate of $Π$ takes the eigenvalues $\pm 1$ in the dimensionless unit. This proves (c).

3.1step 2.1F5F6F7givenalgebra

[Conservation of the rotation generators] Fix $i$ and $s \in R$ . Since $R_{i} (s) \in S O (3)$ , step 2.1 makes $U_{i} (s) = U (R_{i} (s))$ a symmetry, so by [F6] the first part of clause (c) gives $U_{i} (s) U (t) = U (t) U_{i} (s)$ for every $t \in R$ . By [F5], $W (s) : = e^{- i s L_{i} / ℏ} = U_{i} (s)$ ; hence the flow-commutation condition (i) of [F6] holds for the self-adjoint generator $L_{i}$ and the flow $U$ of $H_{V}$ . Applying [F6] clauses (a),(b) to that pair and reading the laws through [F7] gives the time-independence of $μ_{ρ (t)}^{L_{i}} (B)$ in every preparation and the conservation of the mean and second moment for $ψ (0) \in D (L_{i})$ with the time-zero values $⟨ L_{i} ψ, ψ ⟩$ and $∥ L_{i} ψ ∥^{2}$ ; the unit $J s$ is the unit of the spectral coordinate of $L_{i}$ in the model's units of [F5] and Dimensions, SI unit conventions and quantity types of the framework. This proves (b).

4.1step 1.1step 2.1step 3.1step 2.2∎

[Assembly and scope] Step 1.1 proves (a); step 2.1 proves the symmetry-group assertion and step 3.1 the conservation assertions of (b); step 2.2 proves (c); clause (d) records the spatial-factor-only reading of the actions, the exact almost-everywhere form of the invariance hypotheses and the exclusion of any concrete-potential, spinorial, apparatus or empirical claim.

Remarks

Invariance is the hypothesis, not a universal property. No meaning of “generic” is used here. The companion counterexample $V = 1_{B (0, 1)}$ has no nonzero translation symmetry, but is invariant under every rotation and under parity because the Euclidean unit ball is rotation invariant and centrally symmetric. It therefore illustrates failure of translation symmetry only; no corresponding claim about a generic potential is made.
Discrete parity has no generator. $Π$ is an involution, so its conservation is read directly from the commutation with the flow through the self-adjoint unitary clause of the conservation theorem; no one-parameter group and no Stone generator is attached to it.

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Gauge covariance of the prescribed-coupling model: unitary equivalence, the symmetry criterion and the transported propagator

Physical scope

The gauge structure of the prescribed-coupling model P5a for a spinless particle with admissible bounded-coefficient potentials: the conjugation identity transporting the minimal-coupling operator to the gauge-transformed potentials, the exact criterion under which the gauge multiplier is a symmetry of the fixed declared dynamics, the global-phase case, and the unitary equivalence of the transformed time-dependent model carrying the transported propagator. The item is conditional on the postulates and the cited operator analysis; it asserts no Maxwell equation, no physical identification of gauges, no claim about nonconstant gauges beyond the displayed operator criterion, and no apparatus or empirical content.

Statement

Adopt the postulates P1, P2 and P4 in the reading of Symmetries of the declared dynamics, unitary symmetry groups and conserved sharp observables clause (a), for a declared isolated system whose energy operator is the prescribed-coupling minimal-coupling operator of Prescribed external electromagnetic coupling P5a: the minimal-coupling model and its gauge transformations: a declared spinless particle of mass parameter $m > 0$ and charge parameter $q \in R$ , with admissible prescribed potentials $(A_{e x t}, Φ_{e x t})$ and energy operator $K_{A_{e x t}, Φ_{e x t}}$ on the domain $W^{2, 2} (R^{3}; C)$ (Bounded-coefficient magnetic Schrodinger operator). Assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The Axiom of Choice). Let $I \subseteq R$ be an open interval and let $χ : R^{3} \times I \to R$ be a real gauge function satisfying the two regularity hypotheses of Gauge multiplication preserves H2 and transports the magnetic momentum (for every $t$ , $χ (\cdot, t)$ is $C^{2}$ with bounded first and second spatial derivatives, uniformly in $t$ ; and $t \mapsto χ (x, t)$ is differentiable with bounded $\partial_{t} χ$ ). Put $θ (x, t) : = \frac{q}{ℏ} χ (x, t), A_{j}^{'} (\cdot, t) : = A_{j} + D_{j} χ (\cdot, t), Φ^{'} (\cdot, t) : = Φ_{e x t} - \partial_{t} χ (\cdot, t),$ let $G (t)$ be multiplication by $e^{i θ (\cdot, t)}$ , and put $K : = K_{A_{e x t}, Φ_{e x t}}$ , $H^{'} (t) : = K_{A^{'} (t), Φ^{'} (t)}$ and $U (τ) : = e^{- i τ K / ℏ}$ . Then:

(a) Gauge conjugation and unitary equivalence. If $χ$ does not depend on $t$ , then $G : = G (t)$ is independent of $t$ , is unitary with $G W^{2, 2} (R^{3}; C) = W^{2, 2} (R^{3}; C)$ , and $G K G^{- 1} = K_{A_{e x t} + \nabla χ, Φ_{e x t}}, e^{- i t K_{A_{e x t} + \nabla χ, Φ_{e x t}} / ℏ} = G e^{- i t K / ℏ} G^{- 1} (t \in R) :$ the two prescribed-coupling models are related by the fixed unitary $G$ .

(b) Symmetry criterion and the global-phase case. For every $t \in I$ the operator $G (t)$ is unitary, preserves $W^{2, 2}$ , and is a symmetry of the declared dynamics with energy $K$ in the sense of Symmetries of the declared dynamics, unitary symmetry groups and conserved sharp observables clause (b) if and only if $K_{A^{'} (t), Φ_{e x t}} = K_{A_{e x t}, Φ_{e x t}},$ that is, if and only if transporting the vector potential while keeping the original scalar potential reproduces the original operator. In particular, if $χ$ is constant, then $A^{'} (t) = A_{e x t}$ , $Φ^{'} (t) = Φ_{e x t}$ and $G (t) = e^{i θ} I$ for the constant $θ = q χ / ℏ$ , so that every $G (t)$ is a symmetry of the declared dynamics (a global phase). No claim is made in the nonconstant case beyond the displayed criterion.

(c) Transported propagator and the time-dependent model. For the general gauge function of the statement and every $u_{0} \in W^{2, 2}$ and $s \in I$ , the curve $u (t) : = G (t) U (t - s) G (s)^{- 1} u_{0} (t \in I)$ lies in $W^{2, 2}$ , is differentiable into $L^{2}$ on $I$ , satisfies $u (s) = u_{0}$ and $i ℏ u^{'} (t) = H^{'} (t) u (t)$ for every $t$ , and is the unique such curve; the two-parameter family $U^{'} (t, s) : = G (t) U (t - s) G (s)^{- 1}$ consists of unitaries mapping $W^{2, 2}$ onto $W^{2, 2}$ , with $U^{'} (t, t) = I$ and $U^{'} (t, r) U^{'} (r, s) = U^{'} (t, s)$ . Moreover $H^{'} (t) = G (t) K G (t)^{- 1} - q \partial_{t} χ (\cdot, t),$ so the family $G (t)$ relates the autonomous model with energy $K$ to the transformed model whose generator is $G (t) K G (t)^{- 1} - q \partial_{t} χ$ ; a genuinely time-dependent family $G (t)$ is a family of unitaries indexed by time and is not claimed to be a symmetry of the autonomous dynamics except at those $t$ where the operator identity of (b) holds.

(d) Scope. The statement concerns the prescribed spinless minimal-coupling model with admissible bounded-coefficient potentials and the stated gauge regularity, and is conditional on P1, P2, P4 and P5a and the cited operator analysis. It asserts no Maxwell equation, no source or back-reaction model, no identification of the two coefficient pairs as descriptions of one physical situation (that reading is part of P5a), no statement about singular or unbounded potentials, and no apparatus or empirical result; the gauge multipliers are unitary transformations between coefficient classes and are symmetries of the fixed dynamics exactly under the criterion displayed in (b).

Facts & Assumptions

Given: The postulates P1, P2, P4 and P5a in the stated readings; the Axioms of Countable Choice and of Choice; the potentials $(A_{e x t}, Φ_{e x t})$ , $m > 0$ , $q \in R$ , the interval $I$ and the gauge function $χ$ ; the operators $K$ , $H^{'} (t)$ , the multipliers $G (t)$ , the flow $U (τ)$ ; times $s, t \in I$ , $τ \in R$ ; a vector $u_{0} \in W^{2, 2}$ .

[F1]

Declared model. The declared system's energy operator is $K$ , a self-adjoint operator with domain $D (K) = W^{2, 2} (R^{3}; C)$ that generates the strongly continuous unitary group $U (τ) = e^{- i τ K / ℏ}$ (Prescribed external electromagnetic coupling P5a: the minimal-coupling model and its gauge transformations clauses (i)-(iii), Self-adjointness of the bounded-coefficient magnetic Schrodinger operator clause (c), Bounded-coefficient magnetic Schrodinger operator); a symmetry of the declared dynamics is a unitary $S$ with $S D (K) = D (K)$ and $S K = K S$ on $D (K)$ , and a unitary symmetry group is a group of such symmetries (Symmetries of the declared dynamics, unitary symmetry groups and conserved sharp observables clauses (b),(c)).

[F2]

Gauge multipliers and covariance. For every $t \in I$ the operator $G (t)$ is unitary with $G (t) W^{k, 2} (R^{3}; C) = W^{k, 2} (R^{3}; C)$ for $k = 1, 2$ ; for every $u \in W^{1, 2}$ one has $(P_{j} - q A_{j}^{'} (\cdot, t)) G (t) u = G (t) (P_{j} - q A_{j}) u$ ; and $t \mapsto G (t) h$ is continuous with $\frac{d}{d t} G (t) h = \frac{i q}{ℏ} \partial_{t} χ (\cdot, t) G (t) h$ for every $h \in L^{2}$ (Gauge multiplication preserves H2 and transports the magnetic momentum clauses (a),(c),(d)). In particular $G (t) W^{2, 2} (R^{3}; C) = W^{2, 2} (R^{3}; C)$ , so $G (t) D (K) = D (K)$ by [F1].

[F3]

Transported operator and propagator. With $A_{j}^{'} (\cdot, t) = A_{j} + D_{j} χ (\cdot, t)$ and $Φ^{'} (\cdot, t) = Φ_{e x t} - \partial_{t} χ (\cdot, t)$ the pair $(A^{'} (t), Φ^{'} (t))$ is admissible and $H^{'} (t) = K_{A^{'} (t), Φ^{'} (t)}$ is self-adjoint with domain $W^{2, 2}$ (Self-adjointness of the bounded-coefficient magnetic Schrodinger operator, Bounded-coefficient magnetic Schrodinger operator); on $W^{2, 2}$ one has the operator identities $G (t) K G (t)^{- 1} = K_{A^{'} (t), Φ_{e x t}}$ and $H^{'} (t) = G (t) K G (t)^{- 1} - q \partial_{t} χ (\cdot, t)$ (Gauge transported propagator for the bounded-coefficient magnetic operator clause (a)); and for every $u_{0} \in W^{2, 2}$ and $s \in I$ the curve $u (t) = G (t) U (t - s) G (s)^{- 1} u_{0}$ is a pointwise $W^{2, 2}$ -valued, $L^{2}$ -differentiable solution of $i ℏ u^{'} (t) = H^{'} (t) u (t)$ with $u (s) = u_{0}$ , while $U^{'} (t, s) = G (t) U (t - s) G (s)^{- 1}$ is unitary, maps $W^{2, 2}$ onto $W^{2, 2}$ and satisfies the composition law (Gauge transported propagator for the bounded-coefficient magnetic operator clauses (b),(c)); uniqueness in this pointwise-domain class is established locally in proof step 1.2, without invoking the supplier's additional $W^{2, 2}$ -continuity condition.

[F4]

Conjugation calculus. For a unitary $W$ and a self-adjoint $K$ the operator $W K W^{- 1}$ with domain $W D (K)$ is self-adjoint, and its flow satisfies $e^{- i t W K W^{- 1} / ℏ} = W e^{- i t K / ℏ} W^{- 1}$ for every $t$ (Unitary conjugation, generator transport and strong commutation of self-adjoint observables clause (a)).

Proof

Proof technique: direct; read the transported operator identity and the domain preservation off the gauge suppliers, turn the symmetry definition into the operator equality, obtain the flow identity from the conjugation calculus, and quote the transported propagator for the general time-dependent case.

Given: The data of the statement and [F1]-[F4].

1.1F1F2F3givenalgebra

[Transported operator identity and domain preservation] Fix $t \in I$ . By [F2] $G (t)$ is unitary and $G (t) W^{2, 2} = W^{2, 2}$ , so $G (t) D (K) = D (K)$ by [F1]; and by [F3] $G (t) K G (t)^{- 1} = K_{A^{'} (t), Φ_{e x t}}$ as an identity of self-adjoint operators with domain $W^{2, 2}$ . If $χ$ does not depend on $t$ , then $A_{j}^{'} (\cdot, t) = A_{j} + D_{j} χ$ is independent of $t$ and $Φ^{'} (t) = Φ_{e x t}$ , and $G (t)$ is independent of $t$ ; hence in that case $G K G^{- 1} = K_{A_{e x t} + \nabla χ, Φ_{e x t}}$ with domain $W^{2, 2}$ .

1.2F3givenalgebra

[Uniqueness in the stated solution class] Let $u_{1}, u_{2} : I \to L^{2} (R^{3}; C)$ be any two curves differentiable into $L^{2}$ , taking values in $W^{2, 2}$ at every $t$ , solving $i ℏ u_{j}^{'} (t) = H^{'} (t) u_{j} (t)$ for every $t$ , and agreeing at $s$ . Put $w : = u_{1} - u_{2}$ . Then $w (t) \in W^{2, 2}$ , $w (s) = 0$ , and $w^{'} (t) = - \frac{i}{ℏ} H^{'} (t) w (t)$ . By [F3], for each $t$ the operator $H^{'} (t)$ is self-adjoint on $W^{2, 2}$ , so $⟨ H^{'} (t) w (t), w (t) ⟩$ is real. Differentiability into $L^{2}$ and the first-linear inner-product convention give $\frac{d}{d t} ∥ w (t) ∥_{2}^{2} = 2 Re ⟨ w^{'} (t), w (t) ⟩ = 2 Re (- \frac{i}{ℏ} ⟨ H^{'} (t) w (t), w (t) ⟩) = 0.$ Hence $∥ w (t) ∥_{2} = ∥ w (s) ∥_{2} = 0$ for every $t \in I$ , so $u_{1} = u_{2}$ . This proves uniqueness without requiring continuity into $W^{2, 2}$ .

2.1step 1.1F1F2givenalgebra

[Symmetry criterion] By [F1] the unitary $G (t)$ is a symmetry of the declared dynamics with energy $K$ exactly when $G (t) D (K) = D (K)$ and $G (t) K G (t)^{- 1} = K$ on $D (K)$ ; the domain condition holds by step 1.1, and the operator identity holds exactly when $K_{A^{'} (t), Φ_{e x t}} = K$ , again by step 1.1. This is the criterion of (b). If $χ$ is constant, then $D_{j} χ = 0$ as a weak derivative and $\partial_{t} χ = 0$ , so $A^{'} (t) = A_{e x t}$ , $Φ^{'} (t) = Φ_{e x t}$ and $K_{A^{'} (t), Φ_{e x t}} = K$ ; moreover $G (t) = e^{i θ} I$ with the constant $θ = q χ / ℏ$ , so $G (t)$ is unitary and commutes with $K$ , hence is a symmetry.

2.2step 1.1F4givenalgebra

[Flow identity for static gauges] Let $χ$ be independent of $t$ and put $G : = G (t)$ . By step 1.1, $G K G^{- 1} = K_{A_{e x t} + \nabla χ, Φ_{e x t}}$ ; applying the flow identity of [F4] to the unitary $G$ and the self-adjoint $K$ gives $e^{- i t G K G^{- 1} / ℏ} = G e^{- i t K / ℏ} G^{- 1}$ , which is the second display of (a). Together with step 1.1 this proves (a).

3.1step 2.1F3givenstep 1.2

[Transported propagator] Supplier [F3] gives the curve $u (t) = G (t) U (t - s) G (s)^{- 1} u_{0}$ , its pointwise membership in $W^{2, 2}$ , its $L^{2}$ differentiability and equation, the composition law for $U^{'} (t, s)$ , and preservation of $W^{2, 2}$ . Its uniqueness proof uses the stronger class of curves continuous into $W^{2, 2}$ ; the uniqueness in the class stated here is proved in step 1.2. The operator identity $H^{'} (t) = G (t) K G (t)^{- 1} - q \partial_{t} χ (\cdot, t)$ is also [F3]. By step 2.1 the unitary $G (t)$ is a symmetry of the autonomous dynamics at the times $t$ at which $K_{A^{'} (t), Φ_{e x t}} = K$ , and no symmetry statement is made outside those times; this is (c).

4.1step 1.1step 2.1step 2.2step 3.1step 1.2∎

[Assembly and scope] Steps 1.1 and 2.1 prove the transported operator identity, steps 2.1 and 2.2 prove (a) and (b), and steps 3.1 and 1.2 prove (c); clause (d) records that the statement is conditional on P1, P2, P4 and P5a, that the gauge multipliers are unitary equivalences or symmetries exactly under the displayed operator criterion, and that no field equation, physical identification, apparatus or empirical claim is made.

Remarks

Gauge transformation is not automatically a symmetry. A gauge transformation replaces the coefficient pair; the multiplier is a symmetry of the fixed dynamics only when the transported magnetic operator with the original scalar potential coincides with the original operator. Constant gauges give the global phase, which is a symmetry; the nonconstant case is left to the criterion, and no general classification of the pairs for which the criterion holds is attempted here.
Two readings distinguished. The theorem proves the operator identities; the further reading that the gauge-related coefficient pairs describe the same physical situation is part of the adopted postulate P5a and is not a mathematical consequence of the identities proved here.

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Scope and empirical status of the symmetry and conservation items

Physical scope

The reading of the symmetry-and-conservation page as a whole: the conditional status of its symmetry statements, the empirical status of the model parameters and of any symmetry identification for a laboratory system, and the limits of the conservation and gauge claims. The remark asserts no new mathematics or dynamics, records no experiment and no measured value.

What the page establishes

The mathematical lemmas of this page follow from their stated mathematical hypotheses; its physics theorems follow conditionally from their stated model postulates and mathematical hypotheses. Symmetries of the declared dynamics, unitary symmetry groups and conserved sharp observables fixes the vocabulary: a symmetry of the declared dynamics is a unitary preserving the energy domain and commuting with the energy operator, a unitary symmetry group is a group of such symmetries, a conserved assigned sharp observable $(A, u_{A})$ is one whose outcome law is time-independent in every preparation, and the translations, spatial rotations, parity and gauge multipliers are recorded as the model's candidate actions, not as symmetries. Unitary conjugation, generator transport and strong commutation of self-adjoint observables proves the equivalence between flow commutation, strong spectral commutation and domain invariance together with invariance of the observable under conjugation: for self-adjoint $K, A$ on a nonzero complex Hilbert space and $ℏ > 0$ , clause (b)(iii) requires both $U_{K} (t) D (A) = D (A)$ and $U_{K} (t) A U_{K} (t)^{*} = A$ for every $t \in R$ , where $U_{K} (t) = e^{- i t K / ℏ}$ . Domain invariance alone is insufficient; The spatial rotation action on the component-valued wave-function space, The spatial parity action on the component-valued wave-function space, Coordinate-rotation generators on the Schwartz classes and their commutators and The free flow as a unimodular multiplier in the momentum representation provide the concrete actions and their operator identities on the component-valued model. Conservation of a symmetry generator: commuting flows, energy compatibility and conserved outcome laws proves the equivalence of its three commutation conditions, deduces conservation of the outcome law and of domain-qualified moments, and covers self-adjoint unitary symmetries; Translation invariance of bounded potentials and conservation of momentum decides exactly which bounded potentials have the translations as symmetries and derives momentum conservation for the translation-invariant case; Rotational and parity invariance of bounded-potential models: symmetries and conservation of angular momentum and parity laws does the same for rotations (with conservation of the coordinate-rotation generators) and for parity; and Gauge covariance of the prescribed-coupling model: unitary equivalence, the symmetry criterion and the transported propagator transports the prescribed-coupling operator under gauge transformations of Prescribed external electromagnetic coupling P5a: the minimal-coupling model and its gauge transformations, exhibits the exact criterion under which a gauge multiplier is a symmetry of the fixed dynamics and identifies the global-phase case. The mathematical lemmas are conditional on their displayed mathematical hypotheses, including the stated choice assumptions; they do not depend on physical postulates. The physics theorems and examples are conditional on the postulates and model hypotheses stated in each item: the general conservation result uses P1, P2 and P4; the bounded- potential translation, rotation and parity results use P1, P2, P4, P5 and P6; and gauge covariance uses P1, P2, P4 and the prescribed-coupling assumption P5a. P5a is adopted and is not derived by the gauge theorem. None of these conditional results establishes that its assumptions describe a laboratory system.

Empirical status of the symmetry and conservation statements

The conservation laws proved here are model consequences: they hold for the declared massive model with the displayed potential class and follow from the invariance hypotheses as identities about that model. They are not reports of measurements and do not establish that any laboratory preparation is exactly described by the model. In particular, the mass parameter $m > 0$ , the charge parameter $q$ , the spin label $s$ and the potentials $(A_{e x t}, Φ_{e x t})$ are declared model data in the units of Dimensions, SI unit conventions and quantity types of the framework; the statement that a concrete species, field configuration or apparatus realizes them is a tested identification with uncertainty, external to these items. A selection rule or a conservation law deduced here becomes an empirical prediction only after a preparation map, a detection model, a sampling model and an uncertainty budget are supplied, none of which this page provides. The agreement of such an identified model with data is evidence about the combined model and apparatus assumptions; it is not a proof of the postulates.

What is not claimed

The page asserts no symmetry of any concrete potential outside the displayed almost-everywhere invariance hypotheses, no classification of centrally symmetric or parity-even systems beyond those hypotheses, no definite value of a conserved observable, and no converse from the constancy of outcome laws to the commutation conditions for unbounded observables. The rotation and parity actions act on the spatial factor only: no SU(2) lift, no rotation of the internal factor and no spin angular momentum appear here; antiunitary and time-reversal symmetries, superselection sectors, gauge freedom beyond the prescribed bounded-coefficient class, singular or unbounded potentials, confined and boundary realizations, arbitrary driven external fields, relativistic matter and full field theory are outside the scope of these items. Bounded periodic potentials remain within the bounded-potential class; no periodic-band classification is claimed. The stated time-dependent gauge transports are covered by Gauge covariance of the prescribed-coupling model: unitary equivalence, the symmetry criterion and the transported propagator clause (c), with its prescribed coefficient class and gauge regularity; they supply no general driven-field model.

Recorded supplier note

The gauge items of this page consume the magnetic-operator development of an earlier batch of this run. The batch-53 notes preserve historical reports of role-metadata and source-locator defects in earlier versions of those suppliers; those notes are not a description of their current text. The current files inspected for this review declare the mathematics dependencies with role mathematical-premise and have these source locators:

def-nrqm-magnetic-and-detection-magnetic-schrodinger-operator cites Teschl, Sections 6.1 and 7.2, and Binney and Skinner, Sections 3.3 and 3.3.1.
thm-nrqm-magnetic-and-detection-gauge-h2-domain-preservation cites Teschl, Section 7.2, equations (7.22)-(7.24) and Theorem 7.8, and Binney and Skinner, Section 3.3.1, equations (3.40)-(3.45).
thm-nrqm-magnetic-and-detection-gauge-transported-propagator cites Teschl, Section 5.1, Theorem 5.1, and Binney and Skinner, Section 3.3.1, equations (3.40)-(3.45).

The current Teschl text has the cited free-operator material in Section 7.2 (printed pages 167-168), the Kato-Rellich theorem in Section 6.1 (printed page 135), and Stone's theorem in Section 5.1 (printed pages 123-124). The current Binney text has the static gauge phase and covariant-momentum intertwining in Section 3.3.1 (printed pages 46-47). I checked these current locators against the hosted full texts. They provide background for the magnetic results; the bounded-coefficient operator and time-dependent propagator arguments are supplied locally in the magnetic items. The earlier false locators recorded in the batch notes are absent from the current files.

5 · Examples, experiments and counterexamples

None yet.

Qm Symmetries And Conservation

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

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

Statement

Facts & Assumptions

Proof

Remarks

The spatial rotation action on the component-valued wave-function space

Statement

Facts & Assumptions

Proof

Remarks

The spatial parity action on the component-valued wave-function space

Statement

Facts & Assumptions

Proof

Remarks

Coordinate-rotation generators on the Schwartz classes and their commutators

Statement

Facts & Assumptions

Proof

Remarks

The free flow as a unimodular multiplier in the momentum representation

Statement

Facts & Assumptions

Proof

Remarks

Symmetries of the declared dynamics, unitary symmetry groups and conserved sharp observables

Definition

Remarks

Prescribed external electromagnetic coupling P5a: the minimal-coupling model and its gauge transformations

Postulate

Status and scope

Conservation of a symmetry generator: commuting flows, energy compatibility and conserved outcome laws

Statement

Facts & Assumptions

Proof

Remarks

Translation invariance of bounded potentials and conservation of momentum

Statement

Facts & Assumptions

Proof

Remarks

Rotational and parity invariance of bounded-potential models: symmetries and conservation of angular momentum and parity laws

Statement

Facts & Assumptions

Proof

Remarks

Gauge covariance of the prescribed-coupling model: unitary equivalence, the symmetry criterion and the transported propagator

Statement

Facts & Assumptions

Proof

Remarks

Scope and empirical status of the symmetry and conservation items

What the page establishes

Empirical status of the symmetry and conservation statements

What is not claimed

Recorded supplier note

5 · Examples, experiments and counterexamples

Sources