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Spin one half: spread identity, incompatible component outcomes and saturated Robertson bounds

Physical scope

The spin-one-half internal factor of the massive particle model: the exact identity for the sum of the three component spreads in every unit state, the resulting lower bound hbar^2/2 and the impossibility of a state with all three component spreads small, the failure of spectral compatibility for every pair of spin components with the explicit projection witness, and the saturation of the Robertson bound in the S_z eigenstates. The statement is a conditional consequence of P1, P2, P5 and P6, under the Axiom of Choice and Countable Choice required by the cited spectral and moment suppliers, and the cited finite-dimensional mathematics; it asserts no measurement, apparatus, dynamics, empirical value or claim about angular momentum in space.

Statement

Adopt the state and composition postulate P1, the observable and probability postulate P2, the kinematics and massive-particle model postulate P5 and the internal-factor assignment P6, read for the spin- $\frac{1}{2}$ 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 $s = \frac{1}{2}$ and $r = 2$ (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, 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). Assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The Axiom of Choice), the hypotheses of the cited spectral, moment and Robertson suppliers. Write $S_{i} = ℏ J_{i}$ for the model's internal-factor operators of The massive spin-s particle model of the framework: state space, operators, units and frame readings clause (c), so that $J_{i} = \frac{1}{2} σ_{i}$ in the Pauli matrices of Finite spin matrices clause (e), and let $Δ_{ψ} S_{i}$ be the preparation spread of Preparation variance and spread of an assigned sharp observable. Then, for every unit $ψ \in H_{1 / 2}$ :

(a) with $⟨ S ⟩_{ψ} : = (⟨ S_{x} ψ, ψ ⟩, ⟨ S_{y} ψ, ψ ⟩, ⟨ S_{z} ψ, ψ ⟩) \in R^{3}$ , $\sum_{i \in {x, y, z}} (Δ_{ψ} S_{i})^{2} = \frac{3 ℏ^{2}}{4} - ∣ ⟨ S ⟩_{ψ} ∣^{2} \geq \frac{ℏ^{2}}{2},$ so at least one component satisfies $Δ_{ψ} S_{i} \geq ℏ / \sqrt{6}$ ; in particular no unit state has all three component spreads equal to zero;

(b) for every pair of distinct components the spectral projection-valued measures fail to commute; explicitly, with $P_{i}^{\pm} : = \frac{1}{2} (I \pm σ_{i})$ the spectral projections of $S_{i}$ at $\pm ℏ / 2$ , $P_{x}^{+} P_{y}^{+} - P_{y}^{+} P_{x}^{+} = \frac{i}{2} σ_{z} \neq 0,$ and cyclically; consequently no pair of the three components possesses a joint sharp observable in the sense of Compatible sharp observables, their joint law and the ideal joint instrument;

(c) in the state $ψ = f e_{1 / 2}$ with any unit spatial factor $f \in L^{2} (R^{3}; C)$ and the basis vector $e_{1 / 2}$ of the internal factor, $⟨ S_{z} ⟩_{ψ} = \frac{ℏ}{2}, ⟨ S_{x} ⟩_{ψ} = ⟨ S_{y} ⟩_{ψ} = 0, Δ_{ψ} S_{x} = Δ_{ψ} S_{y} = \frac{ℏ}{2}, Δ_{ψ} S_{z} = 0,$ and $Δ_{ψ} S_{x} Δ_{ψ} S_{y} = ℏ^{2} / 4 = \frac{1}{2} ∣ ⟨ [S_{x}, S_{y}] ψ, ψ ⟩ ∣$ , so the Robertson bound of The Robertson relation for preparation spreads of assigned sharp observables is attained in this state;

(d) Scope. Clauses (a)-(c) are conditional consequences of P1, P2, P5, P6, the stated choice hypotheses and the cited mathematics of the finite internal matrices acting on $H_{1 / 2}$ . They concern the internal-factor coupling constants of the model; they assert no laboratory measurement of a spin component, no Stern-Gerlach apparatus model, no dynamics, no empirical value and no interpretation of $S_{i}$ as a rotating body or as orbital angular momentum.

Facts & Assumptions

Given: The postulates P1, P2, P5, P6 read through The massive spin-s particle model of the framework: state space, operators, units and frame readings with $s = \frac{1}{2}$ ; the Axiom of Countable Choice and the Axiom of Choice; the internal operators $S_{i}$ ; a unit vector $ψ \in H_{1 / 2}$ ; a unit spatial factor $f$ ; the basis vector $e_{1 / 2}$ .

[F1]

Internal operators. $S_{i} = ℏ J_{i}$ acts on the internal factor as the matrix action, $J_{i} = \frac{1}{2} σ_{i}$ , and the Pauli matrices satisfy $σ_{i}^{2} = I$ and $σ_{i} σ_{j} = δ_{i j} I + i \sum_{k} ε_{i j k} σ_{k}$ , so $S_{i}^{2} = \frac{ℏ^{2}}{4} I$ , $S_{x}^{2} + S_{y}^{2} + S_{z}^{2} = \frac{3 ℏ^{2}}{4} I$ and $[S_{x}, S_{y}] = i ℏ S_{z} = i \frac{ℏ^{2}}{2} σ_{z}$ (Finite spin matrices clause (e), The massive spin-s particle model of the framework: state space, operators, units and frame readings clause (c)).

[F2]

Spreads in the finite factor. $S_{i}$ is bounded and self-adjoint, so for every unit $ψ$ the mean $⟨ S_{i} ψ, ψ ⟩$ and the variance $(Δ_{ψ} S_{i})^{2} = ⟨ S_{i}^{2} ψ, ψ ⟩ - ⟨ S_{i} ψ, ψ ⟩^{2}$ exist (Mean, second moment and variance of an assigned observable in a preparation clauses (a) and (b), Preparation variance and spread of an assigned sharp observable clauses (b) and (c), Assigned sharp observables and the outcome law of the framework clause (a)).

[F3]

Spectral projections of the components. For $λ = \pm ℏ / 2$ the spectral projection $E_{S_{i}} ({λ})$ is the orthogonal projection onto the eigenspace $\ker (S_{i} - λ)$ (Atoms of the spectral measure and eigenvalues of a self-adjoint operator clause (a), Projection valued measure); the matrices $P_{i}^{\pm} = \frac{1}{2} (I \pm σ_{i})$ satisfy $P_{i}^{\pm} = P_{i}^{\pm 2} = P_{i}^{\pm *}$ , $P_{i}^{+} + P_{i}^{-} = I$ and $S_{i} P_{i}^{\pm} = \pm \frac{ℏ}{2} P_{i}^{\pm}$ , so $P_{i}^{\pm} = E_{S_{i}} ({\pm ℏ / 2})$ (Finite spin matrices clause (e), The Hilbert orthogonal projection onto a closed subspace).

[F4]

Cauchy-Schwarz. In a real or complex inner-product space, $∣ ⟨ u, v ⟩ ∣ \leq ∥ u ∥ ∥ v ∥$ (Cauchy–Schwarz: $∣ ⟨ u, v ⟩ ∣ \leq ∥ u ∥ ∥ v ∥$ , with equality exactly for linearly dependent vectors). Thus in $C^{2}$ , $∣ ⟨ σ_{i} z, z ⟩ ∣ \leq ∥ z ∥^{2}$ because $σ_{i}$ is unitary by [F1].

[F5]

Robertson relation. For two assigned sharp observables and a unit vector in the intersection of the two operator domains and the two product domains, the spread product is at least half the absolute value of the commutator pairing; bounded observables satisfy the hypotheses automatically (The Robertson relation for preparation spreads of assigned sharp observables clauses (b) and (c)).

[F6]

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

Proof

technique · evaluate the finite spin matrices pointwise on the internal factor, sum the three component variances, bound the vector mean by the Bloch identity, and read off the noncommutation of the spectral projections and the saturated Robertson bound in the $S_{z}$ eigenstate

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

1.1givenF1F2algebra

[Spread identity] By [F1], $\sum_{i} S_{i}^{2} = \frac{3 ℏ^{2}}{4} I$ , so for a unit $ψ$ , $\sum_{i} (Δ_{ψ} S_{i})^{2} = \sum_{i} ⟨ S_{i}^{2} ψ, ψ ⟩ - \sum_{i} ⟨ S_{i} ψ, ψ ⟩^{2} = \frac{3 ℏ^{2}}{4} - ∣ ⟨ S ⟩_{ψ} ∣^{2}$ by [F2]. This is the identity of (a).

1.2givenF1F3F6algebra

[Noncommutation of the component projections] By [F3] the projections $P_{i}^{\pm} = E_{S_{i}} ({\pm ℏ / 2})$ are the spectral projections of the components. Using $σ_{i}^{2} = I$ and $σ_{x} σ_{y} = i σ_{z}$ , $σ_{y} σ_{x} = - i σ_{z}$ from [F1], $P_{x}^{+} P_{y}^{+} - P_{y}^{+} P_{x}^{+} = \frac{1}{4} (σ_{x} σ_{y} - σ_{y} σ_{x}) = \frac{i}{2} σ_{z} \neq 0,$ with nonzero action on any unit spatial factor times $e_{1 / 2}$ . Cyclic permutations give nonvanishing differences for the other two pairs, so every two distinct components fail to strongly commute. If a joint sharp observable $H$ for components $i, j$ existed, [F6] applied to $Λ \times R$ and $R \times Σ$ would give $E_{S_{i}} (Λ) E_{S_{j}} (Σ) = H (Λ \times Σ) = E_{S_{j}} (Σ) E_{S_{i}} (Λ)$ for all Borel sets, contradicting those projection witnesses. This proves (b) without an ideal-instrument assumption.

2.1step 1.1F1F2F4algebra

[Bound on the vector mean] Choose a representative of the class $ψ$ and write $g (x) : = (⟨ σ_{x} ψ (x), ψ (x) ⟩, ⟨ σ_{y} ψ (x), ψ (x) ⟩, ⟨ σ_{z} ψ (x), ψ (x) ⟩) \in R^{3}$ , which is defined for almost every $x$ and measurable. For $z = (z_{1}, z_{2}) \in C^{2}$ the Pauli relations of [F1] give $⟨ σ_{x} z, z ⟩ = 2 Re (z_{1} \overline{z_{2}})$ , $⟨ σ_{y} z, z ⟩ = - 2 Im (z_{1} \overline{z_{2}})$ and $⟨ σ_{z} z, z ⟩ = ∣ z_{1} ∣^{2} - ∣ z_{2} ∣^{2}$ , hence the Bloch identity $\sum_{i} ⟨ σ_{i} z, z ⟩^{2} = 4 ∣ z_{1} z_{2} ∣^{2} + (∣ z_{1} ∣^{2} - ∣ z_{2} ∣^{2})^{2} = ∥ z ∥^{4}$ ; so $∣ g (x) ∣ = ∥ ψ (x) ∥_{C^{2}}^{2}$ for almost every $x$ , and $g \in L^{1} (R^{3}; R^{3})$ with $∥ g ∥_{1} = ∥ ψ ∥^{2} = 1$ ; and $⟨ S_{i} ψ, ψ ⟩ = \frac{ℏ}{2} \int_{R^{3}} g_{i} (x) d x$ because $S_{i}$ acts on the internal factor. Put $M : = \int_{R^{3}} g (x) d x$ . If $M \neq 0$ , set $v : = M / ∣ M ∣$ ; the real Cauchy-Schwarz inequality [F4] gives $∣ M ∣ = v \cdot M = \int v \cdot g \leq \int ∣ g ∣$ . If $M = 0$ , the same bound is immediate. Hence $∣ ⟨ S ⟩_{ψ} ∣ = \frac{ℏ}{2} ∣ \int_{R^{3}} g (x) d x ∣ \leq \frac{ℏ}{2} \int_{R^{3}} ∣ g (x) ∣ d x = \frac{ℏ}{2} .$ Substituting into step 1.1 gives $\sum_{i} (Δ_{ψ} S_{i})^{2} \geq \frac{3 ℏ^{2}}{4} - \frac{ℏ^{2}}{4} = \frac{ℏ^{2}}{2}$ ; since the sum of the three nonnegative numbers is at least $ℏ^{2} / 2$ , their maximum is at least $ℏ^{2} / 6$ , so some component satisfies $Δ_{ψ} S_{i} \geq ℏ / \sqrt{6}$ . This completes (a).

2.2step 1.2F1F2F5algebra

[The $S_{z}$ eigenstate] Let $ψ = f e_{1 / 2}$ with $∥ f ∥ = 1$ . Since $S_{i}$ acts on the internal factor, the pointwise expectations are constants: $σ_{z} e_{1 / 2} = e_{1 / 2}$ gives $⟨ S_{z} ⟩_{ψ} = \frac{ℏ}{2}$ , and $σ_{x} e_{1 / 2} = e_{- 1 / 2}$ , $σ_{y} e_{1 / 2} = i e_{- 1 / 2}$ with the basis orthonormal give $⟨ S_{x} ⟩_{ψ} = ⟨ S_{y} ⟩_{ψ} = 0$ . By [F1], $⟨ S_{x}^{2} ⟩_{ψ} = ⟨ S_{y}^{2} ⟩_{ψ} = \frac{ℏ^{2}}{4}$ , so $Δ_{ψ} S_{x} = Δ_{ψ} S_{y} = ℏ / 2$ ; and $S_{z} ψ = \frac{ℏ}{2} ψ$ by the same eigenrelation, so $Δ_{ψ} S_{z} = 0$ . Finally $[S_{x}, S_{y}] = i ℏ S_{z}$ from [F1] gives $\frac{1}{2} ∣ ⟨ [S_{x}, S_{y}] ψ, ψ ⟩ ∣ = \frac{ℏ}{2} ⟨ S_{z} ψ, ψ ⟩ = \frac{ℏ^{2}}{4} = Δ_{ψ} S_{x} Δ_{ψ} S_{y}$ , so the Robertson bound of [F5] is attained. This is (c).

3.1step 1.1step 2.1step 1.2step 2.2∎

[Scope] Clauses (a)-(c) used the internal matrices, the spread definition, the spectral-projection criterion and the Robertson relation; no measurement procedure, apparatus, dynamics or empirical input entered, which is the scope statement (d).

Spin one half: spread identity, incompatible component outcomes and saturated Robertson bounds

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources