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Composition of two spin labels: total spin operators, the Clebsch-Gordan block decomposition and the spin-half singlet

Physical scope

The internal-factor reading of the composition of two declared spin labels s and t under the composition postulate P1, the spin assignment P6 and the locally adopted diagonal two-factor rotation action: the total spin operators on the tensor product of the two internal factors, their commutation relations and Casimir, the Clebsch-Gordan decomposition into total-spin blocks with multiplicities, and the explicit spin-half singlet. The item concerns the internal factors only; the full two-particle spatial model, statistics, interactions and any measured correlation belong to the composites pages and are not asserted here.

Statement

Adopt the composition postulate P1 and the internal-factor assignment of the spin postulate P6 in the readings of State and composition postulate P1: Hilbert spaces, preparations and composite systems and Identical-particle and spin assignments P6: permutation sectors and finite-spin representations (The state space of the framework: preparations, pure rays and density operators, Hilbert tensor construction: inner product, completion and product bases), and assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The Axiom of Choice). Fix two declared spin labels $s, t \in {0, \frac{1}{2}, 1, \frac{3}{2}, \dots}$ with internal factors $V_{s} = C^{2 s + 1}$ , $V_{t} = C^{2 t + 1}$ and spin matrices $J_{i}^{(s)}$ , $J_{i}^{(t)}$ of Finite spin matrices, and consider the composite internal factor $V_{s} \otimes V_{t}$ . On this composite, explicitly adopt the diagonal internal rotation assignment $D_{s, t} (W) : = D_{s} (W) \otimes D_{t} (W)$ for $W \in S U (2)$ , where each $D_{s}, D_{t}$ is the representation assigned by P6 clause (ii). This two-factor action is a local composite model assignment, not an additional conclusion of P6. Put $S_{i}^{t o t} : = ℏ (J_{i}^{(s)} \otimes I_{V_{t}} + I_{V_{s}} \otimes J_{i}^{(t)}), S^{t o t 2} : = \sum_{i = 1}^{3} (S_{i}^{t o t})^{2}, S_{z}^{t o t} : = S_{3}^{t o t} .$ Write $S_{i}^{(s)} : = ℏ J_{i}^{(s)}$ and $S^{(s) 2} : = \sum_{i} (S_{i}^{(s)})^{2}$ , and similarly for $t$ ; define $S_{\pm}^{t o t} : = S_{1}^{t o t} \pm i S_{2}^{t o t}$ . Then:

(a) each $S_{i}^{t o t}$ is a self-adjoint endomorphism of $V_{s} \otimes V_{t}$ satisfying $[S_{i}^{t o t}, S_{j}^{t o t}] = i ℏ \sum_{k} ε_{i j k} S_{k}^{t o t}$ , $S^{t o t 2}$ is self-adjoint and commutes with every $S_{i}^{t o t}$ , and $S^{t o t 2} = S^{(s) 2} \otimes I_{V_{t}} + I_{V_{s}} \otimes S^{(t) 2} + 2 \sum_{i = 1}^{3} S_{i}^{(s)} \otimes S_{i}^{(t)};$

(b) there is an orthogonal decomposition $V_{s} \otimes V_{t} = ⨁_{j = ∣ s - t ∣}^{s + t} W_{j}$ into $S^{t o t}$ -invariant subspaces with $\dim W_{j} = 2 j + 1$ , on each of which the triple $(S_{1}^{t o t}, S_{2}^{t o t}, S_{3}^{t o t})$ is unitarily equivalent to the spin- $j$ matrices of Finite spin matrices scaled by $ℏ$ ; in particular $S^{t o t 2} = ℏ^{2} j (j + 1) I$ on $W_{j}$ , each $j$ occurs exactly once, and there is an orthonormal basis $(v_{j, k})$ , $0 \leq k \leq 2 j$ , of $W_{j}$ with $S_{z}^{t o t} v_{j, k} = ℏ (j - k) v_{j, k}, S_{+}^{t o t} v_{j, k} = ℏ \sqrt{k (2 j - k + 1)} v_{j, k - 1}, S_{-}^{t o t} v_{j, k} = ℏ \sqrt{(k + 1) (2 j - k)} v_{j, k + 1};$ the coefficients of the $v_{j, k}$ in the product basis $(e_{m}^{(s)} \otimes e_{n}^{(t)})$ are the Clebsch-Gordan coefficients and are determined by the phase of the top vector $u_{j}$ (a unit vector of $S_{z}^{t o t}$ -weight $ℏ j$ orthogonal to all $W_{j^{'}}$ with $j^{'} > j$ ), with $u_{s + t} = e_{s}^{(s)} \otimes e_{t}^{(t)}$ admissible;

(c) for every $M$ the $S_{z}^{t o t}$ -eigenspace with weight $ℏ M$ has dimension $N (M) = # {m \in {- s, \dots, s} : M - m \in {- t, \dots, t}}$ , and the identity $(e_{m}^{(s)} \otimes e_{n}^{(t)})_{m, n}$ is an orthonormal basis of $V_{s} \otimes V_{t}$ of $S_{z}^{t o t}$ -eigenvectors with weights $ℏ (m + n)$ ;

(d) for $s = t = \frac{1}{2}$ one has $V_{1 / 2} \otimes V_{1 / 2} = W_{0} \oplus W_{1}$ with $\dim W_{0} = 1$ and $\dim W_{1} = 3$ ; the vector $Ψ : = \frac{1}{\sqrt{2}} (e_{1 / 2}^{(1 / 2)} \otimes e_{- 1 / 2}^{(1 / 2)} - e_{- 1 / 2}^{(1 / 2)} \otimes e_{1 / 2}^{(1 / 2)})$ is a unit vector with $S_{i}^{t o t} Ψ = 0$ for every $i$ , and it is the unique unit vector in the kernel of $S^{t o t 2}$ up to a phase;

(e) Scope. The statement concerns the internal factors of a declared composite, the assignments of P1 and P6 and the declared diagonal two-factor action. It asserts no spatial two-particle model, no statistics or permutation symmetry, no interaction, no dynamics, no measurement and no empirical correlation; in particular it does not by itself imply the singlet correlations of a separated pair, which belong to the composites page.

Facts & Assumptions

Given: The postulates P1 and P6 in the stated readings; the Axiom of Countable Choice; the local diagonal rotation assignment $D_{s, t}$ ; the spin labels $s, t$ ; the internal factors with their bases $(e_{m}^{(s)})$ , $(e_{n}^{(t)})$ ; the total spin operators; a real number $M$ .

[F1]

Finite spin matrices. On $V_{s}$ the matrices $J_{i}^{(s)}$ are self-adjoint with $[J_{i}^{(s)}, J_{j}^{(s)}] = i \sum_{k} ε_{i j k} J_{k}^{(s)}$ , $J^{(s) 2} = s (s + 1) I$ , $J_{+}^{(s)} e_{m}^{(s)} = \sqrt{(s - m) (s + m + 1)} e_{m + 1}^{(s)}$ and $J_{z}^{(s)} e_{m}^{(s)} = m e_{m}^{(s)}$ (and analogously for $t$ ) (Finite spin matrices clauses (a),(b),(c),(d)).

[F2]

Tensor spin composition. On $V_{s} \otimes V_{t}$ the operators $J_{i}^{t o t} : = J_{i}^{(s)} \otimes I + I \otimes J_{i}^{(t)}$ are self-adjoint with the angular momentum commutators, $J^{t o t 2} = J^{(s) 2} \otimes I + I \otimes J^{(t) 2} + 2 \sum_{i} J_{i}^{(s)} \otimes J_{i}^{(t)}$ , there is an orthogonal decomposition $V_{s} \otimes V_{t} = ⨁_{j = ∣ s - t ∣}^{s + t} W_{j}$ with $\dim W_{j} = 2 j + 1$ , each $j$ occurring once, on each block the triple is unitarily equivalent to the spin- $j$ matrices, the weight multiplicity is $N (M) = # {m \in L_{s} : M - m \in L_{t}}$ with the identity basis $(e_{m}^{(s)} \otimes e_{n}^{(t)})$ , and the block bases $v_{j, k} = (k! (2 j) \dots (2 j - k + 1))^{- 1 / 2} (J_{-}^{t o t})^{k} u_{j}$ satisfy the displayed ladder actions, with the phase of $u_{j}$ fixing the Clebsch-Gordan coefficients and $u_{s + t} = e_{s}^{(s)} \otimes e_{t}^{(t)}$ admissible (Tensor spin composition clauses (a),(b),(c),(d),(e), Hilbert tensor construction: inner product, completion and product bases).

[F3]

Internal-factor reading. P1 supplies the composite internal factor $V_{s} \otimes V_{t}$ , and P6 clause (ii) assigns $D_{s}, D_{t}$ separately. The diagonal composite action $D_{s, t} (W) = D_{s} (W) \otimes D_{t} (W)$ is adopted in the Statement. Its representation law and unitarity follow from tensor composition and adjoints, and its continuity from the finite-dimensional continuity of each factor. Along $A_{i} (u)$ its derivative at zero is $- i (J_{i}^{(s)} \otimes I + I \otimes J_{i}^{(t)})$ by the product rule and the P6 coordinate generators. Thus $S_{i}^{t o t}$ are the scaled generators of this assigned action. The model's spin observables are $S = ℏ J$ on each factor (Identical-particle and spin assignments P6: permutation sectors and finite-spin representations clause (ii), State and composition postulate P1: Hilbert spaces, preparations and composite systems, The massive spin-s particle model of the framework: state space, operators, units and frame readings, Orbital, spin and total angular momentum of the massive spin-s model: assigned sharp observables, units and readings).

Proof

Proof technique: direct; rescale the matrix composition theorem by $ℏ$ , transfer its decomposition and ladder statements to the total spin operators, and evaluate the spin-half singlet explicitly.

Given: The data of the statement and [F1]-[F3].

1.1F2algebra

[Operators and algebra] By [F2] the operators $J_{i}^{t o t}$ are self-adjoint with the angular momentum commutators and the displayed Casimir identity; multiplying by $ℏ$ and using $S_{i}^{t o t} = ℏ J_{i}^{t o t}$ , $S_{i}^{(s)} = ℏ J_{i}^{(s)}$ , $S_{i}^{(t)} = ℏ J_{i}^{(t)}$ and $S^{(s) 2} = ℏ^{2} J^{(s) 2}$ gives $[S_{i}^{t o t}, S_{j}^{t o t}] = i ℏ \sum_{k} ε_{i j k} S_{k}^{t o t}$ and the identity of (a). Each $S_{i}^{t o t}$ is self-adjoint, so its square is self-adjoint; because the space is finite-dimensional, their sum $S^{t o t 2}$ is self-adjoint. Its commutation with every $S_{i}^{t o t}$ follows from the Casimir commutation in [F2], which completes (a).

1.2F1F2algebra

[Singlet computation] Let $s = t = \frac{1}{2}$ and $Ψ = \frac{1}{\sqrt{2}} (e_{1 / 2}^{(s)} \otimes e_{- 1 / 2}^{(t)} - e_{- 1 / 2}^{(s)} \otimes e_{1 / 2}^{(t)})$ ; by [F1] $J_{+}^{(s)} e_{1 / 2}^{(s)} = 0$ , $J_{+}^{(s)} e_{- 1 / 2}^{(s)} = e_{1 / 2}^{(s)}$ and $J_{-}^{(s)} e_{1 / 2}^{(s)} = e_{- 1 / 2}^{(s)}$ , $J_{-}^{(s)} e_{- 1 / 2}^{(s)} = 0$ , and analogously for $t$ ; hence $J_{+}^{t o t} (e_{1 / 2}^{(s)} \otimes e_{- 1 / 2}^{(t)}) = e_{1 / 2}^{(s)} \otimes e_{1 / 2}^{(t)} = J_{+}^{t o t} (e_{- 1 / 2}^{(s)} \otimes e_{1 / 2}^{(t)})$ and $J_{-}^{t o t} (e_{1 / 2}^{(s)} \otimes e_{- 1 / 2}^{(t)}) = e_{- 1 / 2}^{(s)} \otimes e_{- 1 / 2}^{(t)} = J_{-}^{t o t} (e_{- 1 / 2}^{(s)} \otimes e_{1 / 2}^{(t)})$ , so $J_{\pm}^{t o t} Ψ = 0$ , giving $J_{1}^{t o t} Ψ = J_{2}^{t o t} Ψ = 0$ by $J_{\pm}^{t o t} = J_{1}^{t o t} \pm i J_{2}^{t o t}$ . Separately, [F1] gives $J_{3}^{t o t} (e_{1 / 2}^{(s)} \otimes e_{- 1 / 2}^{(t)}) = (\frac{1}{2} - \frac{1}{2}) e_{1 / 2}^{(s)} \otimes e_{- 1 / 2}^{(t)} = 0$ and $J_{3}^{t o t} (e_{- 1 / 2}^{(s)} \otimes e_{1 / 2}^{(t)}) = (- \frac{1}{2} + \frac{1}{2}) e_{- 1 / 2}^{(s)} \otimes e_{1 / 2}^{(t)} = 0$ , hence $J_{3}^{t o t} Ψ = 0$ as well; $Ψ$ is unit because the product basis is orthonormal.

2.1step 1.1F2algebra

[Decomposition and Clebsch-Gordan data] Transferring [F2] to the total spin operators by the rescaling of step 1.1 gives the orthogonal decomposition $V_{s} \otimes V_{t} = ⨁_{j = ∣ s - t ∣}^{s + t} W_{j}$ with $\dim W_{j} = 2 j + 1$ , the unitary equivalence with the spin- $j$ matrices on each block and hence $S^{t o t 2} = ℏ^{2} j (j + 1) I$ on $W_{j}$ , the weight multiplicity of (c) and the identity basis; the block bases $v_{j, k}$ have the displayed ladder actions with the stated phase convention, which are (b) and (c).

3.1step 1.2step 2.1F2algebra

[Uniqueness of the singlet] For $s = t = \frac{1}{2}$ the decomposition of step 2.1 has the two blocks $j = 0$ and $j = 1$ with dimensions $1$ and $3$ , and $S^{t o t 2} = 0$ on the one-dimensional block $W_{0}$ and $S^{t o t 2} = ℏ^{2} \cdot 2 I$ on $W_{1}$ ; hence $\ker S^{t o t 2} = W_{0}$ is one-dimensional, and since $Ψ \neq 0$ satisfies $S^{t o t 2} Ψ = 0$ by step 1.2, it spans $W_{0}$ : the kernel is the line $C Ψ$ . This is (d).

4.1step 1.1step 2.1step 3.1F3∎

[Assembly and scope] Step 1.1 proves (a), step 2.1 proves (b),(c), and step 3.1 proves (d); only the internal factors, the assignments of [F3] and the finite matrix composition theorem were used, no spatial two-particle factor, statistics, dynamics, measurement or empirical input entered, which is the scope statement (e).

Remarks

Role of the page. This theorem is the internal-factor supplier for the composites page, where the two-particle spatial factors, the preparation and the correlation analysis are added; no measured correlation or Bell statement is made here.
Phases. The block vectors are determined by the phase of the top vector; different phases multiply all Clebsch-Gordan coefficients of a block by the same phase, so all outcome probabilities built from one block are unaffected.

Composition of two spin labels: total spin operators, the Clebsch-Gordan block decomposition and the spin-half singlet

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources