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Position and momentum outcome laws of the massive spin-s model

Physical scope

The position and momentum outcome laws of the spin-s massive-particle model of the framework: the spectral projection-valued measures of the componentwise position and momentum operators, the joint position law with Born density the squared component norm, the momentum laws with density the squared momentum-representative norm, the atomlessness of exact values, and the invariance of the spatial laws under the internal factor. The statement is a conditional consequence of the observable and probability postulate P2 and the model postulates P5 and P6 and the cited mathematics; it asserts no joint position-momentum statistics, no dynamics, no apparatus or detector realization, no species identification and no empirical result.

Statement

Adopt the observable and probability postulate in the reading of Assigned sharp observables and the outcome law of the framework, the kinematics and massive-particle model postulate P5, and the internal-factor assignment of the postulate P6, all 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 $m > 0$ , spin label $s \in {0, \frac{1}{2}, 1, \dots}$ and $r = 2 s + 1$ (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). Let $H_{s}$ be the model's state space, $Q_{j}, P_{k}$ its componentwise position and momentum operators, $U^{(s)}$ its momentum transform, and let $μ_{ψ}^{A} (B) = ∥ E_{A} (B) ψ ∥^{2}$ be the outcome law of an assigned sharp observable $A$ in a unit vector $ψ$ (Assigned sharp observables and the outcome law of the framework clause (b)). Then:

(a) for each $j$ the spectral projection-valued measure of $Q_{j}$ is $E_{Q_{j}} (B) = M_{1_{x_{j}^{- 1} (B)}} (B \subseteq R Borel),$ the componentwise multiplication by the indicator, and $μ_{ψ}^{Q_{j}} (B) = \int_{x_{j}^{- 1} (B)} ∥ ψ (x) ∥_{C^{r}}^{2} d x;$ the law is atomless, $μ_{ψ}^{Q_{j}} ({λ}) = 0$ for every $λ \in R$ , and $σ (Q_{j}) = R$ ;

(b) there is a unique regular projection-valued measure $G$ on the Borel sets of $R^{3}$ with $G (B_{1} \times B_{2} \times B_{3}) = E_{Q_{1}} (B_{1}) E_{Q_{2}} (B_{2}) E_{Q_{3}} (B_{3})$ for all Borel $B_{1}, B_{2}, B_{3}$ , namely $G (B) = M_{1_{B}}$ , and its scalar law is the model's Born position law, $μ_{ψ}^{G} (B) = ∥ G (B) ψ ∥^{2} = \int_{B} ∥ ψ (x) ∥_{C^{r}}^{2} d x;$

(c) for each $k$ the spectral projection-valued measure of $P_{k}$ is $E_{P_{k}} (B) = U^{(s) - 1} M_{1_{B} (p_{k})} U^{(s)},$ and with $ψ_{p} : = U^{(s)} ψ$ $μ_{ψ}^{P_{k}} (B) = \int_{{p \in R^{3} : p_{k} \in B}} ∥ ψ_{p} (p) ∥_{C^{r}}^{2} d p;$ the law is atomless and $σ (P_{k}) = R$ ;

(d) for every unitary $V \in U (r)$ , with $V_{\times}$ the matrix action of Canonical commutation relations on the component-valued core and commutation with the internal factor clause (b), $μ_{V_{\times} ψ}^{Q_{j}} = μ_{ψ}^{Q_{j}}, μ_{V_{\times} ψ}^{P_{k}} = μ_{ψ}^{P_{k}};$

(e) Scope. The statement is conditional on P2, P5 and P6 and the cited mathematics, and extends the position and momentum laws of the massive model to the spin- $s$ state space. No joint law of a position and a momentum observable is asserted, no dynamics is used or claimed, and the identification of the operators with concrete laboratory apparatus is a tested identification under the preparation and apparatus postulate, not a consequence of this theorem; no empirical result is recorded. Exact values have probability zero although both spectra equal $R$ . The real outcome spaces remain $R$ ; zero singleton mass means atomlessness and supplies no normalizable exact-value eigenstate. Finite-resolution records use the displayed Borel bin probabilities.

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 state space $H_{s}$ with $r = 2 s + 1$ ; the operators $Q_{j}, P_{k}$ and the unitaries $Ψ_{0}, U^{(s)}$ ; a unit vector $ψ \in H_{s}$ ; Borel sets $B, B_{1}, B_{2}, B_{3} \subseteq R$ ; a Borel set $B^{'} \subseteq R^{3}$ ; a unitary matrix $V \in U (r)$ ; a real $λ$ .

[F1]

Outcome law. For an assigned sharp observable $A$ with spectral PVM $E_{A}$ the law in a unit vector is $μ_{ψ}^{A} (B) = ∥ E_{A} (B) ψ ∥^{2}$ , a probability measure on the Borel sets; the spectral support of $E_{A}$ is the spectrum $σ (A)$ , and $μ_{ψ}^{A} ({λ}) = 0$ unless $λ$ is an eigenvalue of $A$ (Assigned sharp observables and the outcome law of the framework clauses (b)-(d)). If $A$ has no eigenvalues its laws are atomless, with real outcome space $R$ and no normalizable exact-value eigenstate; zero singleton mass does not exclude real point realizations (clause (c) of the same definition, Projection valued measure).

[F2]

Componentwise operators and their spectral measures. $Q_{j}$ and $P_{k}$ are the componentwise operators of the model, self-adjoint with their maximal domains; $E_{Q_{j}} (B) = M_{1_{x_{j}^{- 1} (B)}}$ and $E_{P_{k}} (B) = F_{2}^{(r) *} M_{1_{{2 π ℏ ξ_{k} \in B}}} F_{2}^{(r)}$ are their spectral PVMs (Canonical commutation relations on the component-valued core and commutation with the internal factor clauses (b),(c), Componentwise position and momentum operators on the component-valued wave-function space clause (a)); the matrix action $V_{\times}$ commutes with these PVMs and is unitary for unitary $V$ (Canonical commutation relations on the component-valued core and commutation with the internal factor clauses (b)-(d)).

[F3]

Momentum representation. $U^{(s)} = U^{(\cdot)}$ for $r = 2 s + 1$ is unitary, $U^{(s)} D (P_{k}) = D (M_{p_{k}})$ and $U^{(s)} P_{k} = M_{p_{k}} U^{(s)}$ ; on Schwartz classes the displayed integral formula holds and the Plancherel identity $\int ∥ U^{(s)} f ∥^{2} d p = \int ∥ f ∥^{2} d x$ (The momentum representation of the componentwise kinematics clauses (a),(b)).

[F4]

Multiplication spectral measure. For a $σ$ -finite measure space and a real measurable finite-valued multiplier $m$ , the multiplication operator $M_{m}$ is self-adjoint with regular spectral PVM $M_{1_{m^{- 1} (B)}}$ and $σ (M_{m})$ equal to the essential range of $m$ (Spectral measure of a real L2 multiplication operator clauses (a),(c),(d)); a regular PVM representing a self-adjoint operator is its spectral PVM (Spectral theorem for unbounded self-adjoint operators (PVM form)).

[F5]

Measure toolkit. For $σ$ -finite measure spaces, integrals of nonnegative product-measurable functions may be iterated (Tonelli's theorem for nonnegative measurable functions on a sigma-finite product); for nonnegative measurable $h$ the set function $A \mapsto \int_{A} h d μ$ is a measure (The indefinite integral of a nonnegative measurable function is a measure); finite measures on the $σ$ -algebra generated by a $π$ -system that agree on the $π$ -system and have equal mass are equal (Finite measures agreeing on a generating pi-system and on the whole space are equal); the Euclidean Borel $σ$ -algebra of $R^{3}$ is generated by the rectangles (The Borel product of R^m and R^n is the Borel sigma-algebra of R^{m+n}). Under Countable Choice, every Borel measure finite on compact sets on a second-countable locally compact Hausdorff space is regular (Locally finite Borel measures on second-countable LCH spaces are regular). The usual Euclidean $R^{3}$ is such a space: rational-center, positive rational-radius balls form a countable base; the metric is Hausdorff; and closed bounded balls are compact by Heine-Borel in $R^{n}$ : with the Euclidean metric a subset of $R^{n}$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line, so every point has a compact neighborhood. A coordinate hyperplane in $R^{3}$ has Lebesgue measure zero: its indicator has iterated integral zero by [F5] because the inner integral over a singleton vanishes.

Proof

technique · direct; read the spectral measures of the componentwise operators from the conjugation lemmas, compute the laws by Tonelli, construct the joint position projection-valued measure and prove its uniqueness on rectangles, and transport the momentum measure through the unitary momentum representation

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

1.1F1F2F4F5algebra

[Position spectral measure and law] [F2] gives $E_{Q_{j}} (B) = M_{1_{x_{j}^{- 1} (B)}}$ , so for a unit $ψ$ , $μ_{ψ}^{Q_{j}} (B) = ∥ E_{Q_{j}} (B) ψ ∥^{2} = \int_{x_{j}^{- 1} (B)} ∥ ψ (x) ∥_{C^{r}}^{2} d x$ by [F1] and the definition of the componentwise norm. The measure is the indefinite integral $A \mapsto \int_{A} ∥ ψ ∥^{2} d x$ of a nonnegative integrable function [F5], hence atomless: for every $λ$ the set $x_{j}^{- 1} ({λ})$ is a coordinate hyperplane, whose indicator has iterated integral zero by [F5], so $μ_{ψ}^{Q_{j}} ({λ}) = 0$ . Finally the essential range of $x_{j}$ is $R$ because every interval slab $x_{j}^{- 1} ((λ - ε, λ + ε))$ has positive Lebesgue measure, so [F4] gives $σ (Q_{j}) = R$ . This proves (a).

1.2F1F2algebra

[Internal-factor invariance] Let $V \in U (r)$ . By [F2] the matrix action $V_{\times}$ is unitary and commutes with $E_{Q_{j}} (B)$ and $E_{P_{k}} (B)$ ; hence $μ_{V_{\times} ψ}^{Q_{j}} (B) = ∥ E_{Q_{j}} (B) V_{\times} ψ ∥^{2} = ∥ V_{\times} E_{Q_{j}} (B) ψ ∥^{2} = ∥ E_{Q_{j}} (B) ψ ∥^{2} = μ_{ψ}^{Q_{j}} (B)$ , and the same computation with $P_{k}$ proves (d).

2.1step 1.1F2F4F5algebra

[Joint position law] Define $G (B) : = M_{1_{B}}$ for Borel $B \subseteq R^{3}$ ; each $G (B)$ is a bounded self-adjoint projection because pointwise multiplication by an indicator is one, and $G$ is a PVM: $G (B) G (C) = G (B \cap C)$ , the total mass is $I$ , and strong countable additivity follows by dominated convergence for $∥ G (B) ψ - \sum_{n \leq N} G (B_{n}) ψ ∥^{2} = \int ∣ 1_{B} - \sum_{n \leq N} 1_{B_{n}} ∣ ∥ ψ ∥^{2}$ . For every $h \in H_{s}$ , the scalar measure is $G_{h} (B) = ⟨ G (B) h, h ⟩ = \int_{B} ∥ h (x) ∥_{C^{r}}^{2} d x$ , a Borel measure of finite total mass $∥ h ∥^{2}$ by [F5], hence finite on every compact set. Apply the regularity corollary in [F5] on the usual Euclidean $R^{3}$ , using the Countable Choice already assumed: each $G_{h}$ is regular. In particular the measure for a unit $ψ$ is the regular probability measure $μ_{ψ} (B) = \int_{B} ∥ ψ (x) ∥^{2} d x$ ; the zero vector gives the zero regular measure. This proves regularity of the actual component-valued PVM $G$ in Projection valued measure, including its finite matrix amplification. On rectangles, $G (B_{1} \times B_{2} \times B_{3}) = M_{1_{B_{1}} (x_{1}) 1_{B_{2}} (x_{2}) 1_{B_{3}} (x_{3})} = E_{Q_{1}} (B_{1}) E_{Q_{2}} (B_{2}) E_{Q_{3}} (B_{3})$ by [F2], since the componentwise multipliers commute. For a unit $ψ$ the scalar law is $μ_{ψ}^{G} (B) = ∥ G (B) ψ ∥^{2} = \int_{B} ∥ ψ (x) ∥^{2} d x$ . If $G^{'}$ is another regular PVM with the same rectangle values, then for every unit $φ$ the finite measures $B \mapsto ⟨ G (B) φ, φ ⟩$ and $B \mapsto ⟨ G^{'} (B) φ, φ ⟩$ agree on the rectangles, a $π$ -system generating the Borel $σ$ -algebra of $R^{3}$ [F5], and have equal total mass $∥ φ ∥^{2}$ ; [F5] makes them equal on all Borel sets, and polarization recovers all pairings, so $G^{'} = G$ . This proves (b).

2.2step 1.1F3F4algebra

[Momentum spectral measure and law] By [F3] and [F4], $U^{(s)}$ is unitary, $D (M_{p_{k}}) = U^{(s)} D (P_{k})$ and $M_{p_{k}} = U^{(s)} P_{k} U^{(s) - 1}$ ; the multiplication operator $M_{p_{k}}$ on $L^{2} (d p; C^{r})$ is self-adjoint with spectral PVM $M_{1_{{p_{k} \in B}}}$ [F4], so transporting through the unitary $U^{(s)}$ and using uniqueness of the spectral PVM [F4] gives $E_{P_{k}} (B) = U^{(s) - 1} M_{1_{{p_{k} \in B}}} U^{(s)}$ . For a unit $ψ$ , $μ_{ψ}^{P_{k}} (B) = ∥ E_{P_{k}} (B) ψ ∥^{2} = ∥ M_{1_{{p_{k} \in B}}} U^{(s)} ψ ∥^{2} = \int_{{p_{k} \in B}} ∥ ψ_{p} (p) ∥^{2} d p$ [F3]. The law is the indefinite integral of the integrable function $∥ ψ_{p} ∥^{2}$ over a slice condition, hence atomless by the hyperplane argument of step 1.1; and the essential range of $p_{k}$ is $R$ , so $σ (P_{k}) = R$ [F4]. This proves (c).

3.1givenstep 1.1step 2.1step 2.2step 1.2∎

[Scope] Clauses (a)-(d) are consequences of P2, P5 and P6 as read by The massive spin-s particle model of the framework: state space, operators, units and frame readings together with the displayed mathematics; no dynamics, joint position-momentum statistics, apparatus model or empirical claim is used, and the last sentences of the statement record the scope.

Remarks

Relation to the spinless laws. For $s = 0$ one has $r = 1$ and the statement reduces to the position and momentum outcome laws of the spinless massive model already established on the observables page; the present theorem supplies the component-valued case and the invariance of the spatial laws under the internal factor.
Nondegenerate continuum. Both spectra equal $R$ and both laws are atomless: the model admits no normalizable position or momentum eigenvector, and no exact value is an outcome with positive probability. Finite-resolution readings are the Borel bin probabilities displayed above.

Position and momentum outcome laws of the massive spin-s model

Statement

Facts & Assumptions

Proof

Remarks

Depends on

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