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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.

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

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources