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Position and momentum of the massive model have no joint sharp observable

Physical scope

The incompatibility of the j-th position and j-th momentum of the massive spin-s particle model: their spectral projection-valued measures do not commute, so the pair possesses no joint sharp observable with the two spectral measures as marginals. The statement is a conditional consequence of P2, P5 and P6 and the cited mathematics; it asserts no approximate joint measurement, no apparatus, no error-disturbance relation and no empirical result, and it concerns the model's assigned sharp observables only.

Statement

Adopt the observable and probability postulate P2, the kinematics and massive-particle model postulate P5 and the internal-factor assignment P6, read for the spin- $s$ model of The massive spin-s particle model of the framework: state space, operators, units and frame readings with mass parameter $m > 0$ and spin label $s$ (Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes, 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 Countable Choice and the Axiom of Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The Axiom of Choice). Fix a coordinate index $j$ . Then:

(a) the spectral projection-valued measures $E_{Q_{j}}$ and $E_{P_{j}}$ of the model's $j$ -th position and $j$ -th momentum do not strongly commute: there are Borel sets $Λ, Σ \subseteq R$ with $E_{Q_{j}} (Λ) E_{P_{j}} (Σ) \neq E_{P_{j}} (Σ) E_{Q_{j}} (Λ)$ ;

(b) consequently the pair $(Q_{j}, P_{j})$ is not compatible: there is no regular projection-valued measure $H$ on the Borel sets of $R^{2}$ whose marginals are $E_{Q_{j}}$ and $E_{P_{j}}$ , and in particular the framework assigns to the pair no joint sharp outcome law in any preparation;

(c) Scope. The statement is conditional on P2, P5, P6 and the cited mathematics and concerns the model's assigned sharp operators only. It does not assert that no approximate or unsharp joint measurement exists, gives no error-disturbance relation, and records no apparatus or empirical result.

Facts & Assumptions

Given: The postulates P2, P5, P6 read through The massive spin-s particle model of the framework: state space, operators, units and frame readings; the Axiom of Countable Choice and the Axiom of Choice; the model operators $Q_{j}, P_{j}$ with their maximal domains; real $t \neq 0$ ; a nonzero class $ψ \in D (Q_{j})$ .

[F1]

Model operators and translations. $Q_{j}$ and $P_{j}$ are self-adjoint with the maximal componentwise domains $D (Q_{j}) = {f : x_{j} f_{l} \in L^{2} \forall l}$ and $D (P_{j}) = {f : D_{j} f_{l} \in L^{2} \forall l}$ , and the componentwise translation $T_{t e_{j}} f = f (\cdot - t e_{j})$ satisfies $T_{t e_{j}} D (Q_{j}) = D (Q_{j})$ , $T_{t e_{j}} Q_{j} T_{t e_{j}}^{- 1} = Q_{j} - t I$ and $T_{t e_{j}} = \exp (- \frac{i}{ℏ} t P_{j})$ , the exponential being the spectral-calculus flow of $P_{j}$ (Componentwise position and momentum operators on the component-valued wave-function space clauses (a), (c) and (d); The massive spin-s particle model of the framework: state space, operators, units and frame readings clause (b)).

[F2]

Bounded Borel calculus. If two self-adjoint operators $A, B$ have strongly commuting spectral projection-valued measures, then for every bounded Borel $f$ one has $f (B) D (A) \subseteq D (A)$ and $A f (B) x = f (B) A x$ for every $x \in D (A)$ (Bounded Borel calculus of strongly commuting self-adjoint operators clause (b)).

[F3]

PVM multiplicativity and compatibility vocabulary. Every projection-valued measure $H$ satisfies $H (S \cap T) = H (S) H (T) = H (T) H (S)$ for measurable $S, T$ (Projection valued measure). A joint sharp observable and compatibility mean, respectively, a regular PVM on $B (R^{2})$ with the two spectral marginals and the existence of such a PVM, as defined in the Statement of Compatible sharp observables, their joint law and the ideal joint instrument; only this terminology is used here, not that theorem's deductions under P1-P3.

[F4]

The exponential function. $\exp (- \frac{i}{ℏ} t p)$ is a bounded Borel function of $p \in R$ with values of modulus $1$ , so it is admissible in the bounded Borel functional calculus of a self-adjoint operator (Spectral theorem for unbounded self-adjoint operators (PVM form), Projection valued measure).

Proof

technique · suppose the two projection-valued measures commute and use the bounded Borel calculus to make the unitary translation group of the momentum commute with the position operator, contradicting the conjugation identity $TQ_{j}T^{-1}=Q_{j}-tI$

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

1.1givenassume-contraF1F2F4

[Supposition and its consequence] Assume for contradiction that $E_{Q_{j}}$ and $E_{P_{j}}$ strongly commute, and fix $t \neq 0$ . By [F4] the function $f (p) = \exp (- \frac{i}{ℏ} t p)$ is bounded Borel, and by [F1] its operator is $f (P_{j}) = T_{t e_{j}}$ . Applying [F2] with $A = Q_{j}$ , $B = P_{j}$ gives $T_{t e_{j}} D (Q_{j}) \subseteq D (Q_{j})$ and $T_{t e_{j}} Q_{j} η = Q_{j} T_{t e_{j}} η for every η \in D (Q_{j}) .$

2.1givenstep 1.1F1algebra

[The contradiction] Let $ψ \in D (Q_{j})$ be nonzero, which exists because $D (Q_{j})$ is dense, and put $η : = T_{t e_{j}}^{- 1} ψ$ ; by the domain invariance of [F1], $η \in D (Q_{j})$ . The conjugation identity of [F1] gives $T_{t e_{j}} Q_{j} η = (Q_{j} - t I) T_{t e_{j}} η = (Q_{j} - t I) ψ,$ while step 1.1 gives $T_{t e_{j}} Q_{j} η = Q_{j} T_{t e_{j}} η = Q_{j} ψ$ . Hence $(Q_{j} - t I) ψ = Q_{j} ψ$ , that is $t ψ = 0$ , which contradicts $ψ \neq 0$ and $t \neq 0$ . Therefore $E_{Q_{j}}$ and $E_{P_{j}}$ do not strongly commute, and there are Borel sets on which the two projections fail to commute. This proves (a).

3.1givenstep 2.1F3discharge-contradiction∎

[No joint sharp observable] If a regular PVM $H$ on $B (R^{2})$ had the two spectral marginals, then [F3], with $S = Λ \times R$ and $T = R \times Σ$ , would give $E_{Q_{j}} (Λ) E_{P_{j}} (Σ) = H (Λ \times Σ) = E_{P_{j}} (Σ) E_{Q_{j}} (Λ)$ for all Borel $Λ, Σ$ . This contradicts step 2.1. Thus no such PVM exists and, by the terminology in [F3], no joint sharp outcome law of the pair is assigned by the framework to any preparation. This proves (b). The argument uses only the stated postulates and mathematical PVM properties; it asserts no approximate or unsharp joint measurement, error-disturbance relation, apparatus model or empirical result, as required by (c).

Position and momentum of the massive model have no joint sharp observable

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources