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The two spin-half fermion sector: the spin-singlet and spin-triplet parts and their orbital symmetry

Physical scope

The conditional model computation for two declared spin-half fermions with a finite orbital factor: the decomposition of the antisymmetric sector into spin-singlet and spin-triplet parts, the opposite orbital symmetry in each part, and the explicit components of an antisymmetrised product of two declared one-particle states. The computation is conditional on the adopted antisymmetric sector of P6 and on the declared finite orbital factor; it asserts no spin-statistics theorem, no interaction, no dynamics, no measurement and no empirical claim.

Statement

Adopt the state and composition postulate P1 (State and composition postulate P1: Hilbert spaces, preparations and composite systems) and the identical-particle and spin postulate P6 (Identical-particle and spin assignments P6: permutation sectors and finite-spin representations) in the readings of Declared species sectors and their preparations: the framework's reading of P6, and assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Declare a spin-half fermionic species with an arbitrary model-assigned mass parameter $M_{s p} > 0 k g$ , unused in this calculation and not assigned an empirical value here, whose one-particle space is the completed product $K \hat{\otimes} V$ of a nonzero finite-dimensional complex Hilbert space $K$ (the orbital factor) with $V = C^{2}$ (the internal factor of P6 clause (ii) for $s = \frac{1}{2}$ ), with $V$ carrying the orthonormal basis $(e_{+}, e_{-})$ of Composition of two spin labels: total spin operators, the Clebsch-Gordan block decomposition and the spin-half singlet, and declare $n = 2$ particles in the adopted antisymmetric sector $H_{2}^{-} = ran P_{-}^{(K \otimes V)} \subseteq (K \otimes V) \hat{\otimes} (K \otimes V)$ of the declared species, with the regrouping unitary $Θ$ of Regrouping two factors: the symmetry decomposition of a two-particle sector over an orbital and an internal factor. Write $u \land v : = \sqrt{2} P_{-}^{(K)} (u \otimes v)$ and $u ⊙ v : = \sqrt{2} P_{+}^{(K)} (u \otimes v)$ for $u, v \in K$ , and $σ_{\pm} : = (e_{+} \otimes e_{-} \pm e_{-} \otimes e_{+}) / \sqrt{2}$ for the spin combinations. Then, as conditional statements of the declared model:

(a) sector decomposition: $Θ$ carries $H_{2}^{-}$ onto the orthogonal direct sum $(ran P_{-}^{(K)} \otimes ran P_{+}^{(V)}) + (ran P_{+}^{(K)} \otimes ran P_{-}^{(V)})$ , where $ran P_{-}^{(V)} = C Ψ$ is the one-dimensional spin-singlet line spanned by the singlet $Ψ$ of Composition of two spin labels: total spin operators, the Clebsch-Gordan block decomposition and the spin-half singlet clause (d) and $ran P_{+}^{(V)} = W_{1}$ is the three-dimensional spin-triplet space of that theorem; consequently the antisymmetric two-particle sector is the sum of a spin-singlet part $ran P_{+}^{(K)} \otimes C Ψ$ , of dimension $(\binom{m + 1}{2})$ for $m = \dim K$ , with symmetric orbital factor, and a spin-triplet part $ran P_{-}^{(K)} \otimes W_{1}$ , of dimension $3 (\binom{m}{2})$ , with antisymmetric orbital factor;

(b) opposite orbital symmetry: for all $u, v \in K$ and $χ, χ^{'} \in V$ , $Θ (P_{-}^{(K \otimes V)} ((u \otimes χ) \otimes (v \otimes χ^{'}))) = P_{+}^{(K)} (u \otimes v) \otimes P_{-}^{(V)} (χ \otimes χ^{'}) + P_{-}^{(K)} (u \otimes v) \otimes P_{+}^{(V)} (χ \otimes χ^{'});$ the spin-singlet component therefore multiplies the symmetric orbital component and the spin-triplet component multiplies the antisymmetric orbital component, so within each component the orbital exchange symmetry is opposite to the spin exchange symmetry;

(c) parallel spins: if $u, v \in K$ are orthonormal and $χ \in V$ is a unit vector, then $Θ$ carries the ray of $P_{-}^{(K \otimes V)} ((u \otimes χ) \otimes (v \otimes χ))$ to the ray of $P_{-}^{(K)} (u \otimes v) \otimes (χ \otimes χ)$ , whose unit representative is $(u \land v) \otimes (χ \otimes χ)$ ; for $χ = e_{+}$ this is the $S_{z}^{t o t} = ℏ$ triplet component with an antisymmetric orbital factor;

(d) antiparallel spins: if $u, v \in K$ are orthonormal, then $Θ$ carries the ray of $P_{-}^{(K \otimes V)} ((u \otimes e_{+}) \otimes (v \otimes e_{-}))$ to the ray of $\frac{1}{\sqrt{2}} ((u ⊙ v) \otimes σ_{-} + (u \land v) \otimes σ_{+})$ , an equal-amplitude superposition of the singlet component with symmetric orbital factor and the $S_{z}^{t o t} = 0$ triplet component with antisymmetric orbital factor; in particular such an antisymmetrised product of two orthonormal one-particle states has no definite orbital exchange symmetry.

The model assignment of the antisymmetric sector is that of P6; the statements above are conditional on it and assert no spin-statistics theorem, no interaction, no dynamics and no measurement.

Facts & Assumptions

Given: The declared spin-half fermionic species with orbital factor $K$ of dimension $m$ , internal factor $V = C^{2}$ with basis $(e_{+}, e_{-})$ , the two-particle antisymmetric sector, the regrouping unitary $Θ$ , and vectors $u, v \in K$ , $χ, χ^{'} \in V$ .

[F1]

Sector vocabulary. The sector space is $ran P_{-}^{(K \otimes V)}$ with the inner product of the completed tensor square, the rays of its nonzero vectors are the declared pure preparations of the two particles, and the exchange operator and projections act as in the cited lemmas (Declared species sectors and their preparations: the framework's reading of P6 clauses (c)-(d), The two-particle exchange operator and the symmetric and antisymmetric combinations clauses (a),(b)).

[F2]

Regrouping and its identities. The regrouping unitary $Θ$ is unitary with $Θ ((u \otimes χ) \otimes (v \otimes χ^{'})) = (u \otimes v) \otimes (χ \otimes χ^{'})$ , maps $ran P_{-}^{(K \otimes V)}$ onto $(ran P_{-}^{(K)} \otimes ran P_{+}^{(V)}) + (ran P_{+}^{(K)} \otimes ran P_{-}^{(V)})$ , and satisfies the vector identity $Θ P_{-}^{(K \otimes V)} ((u \otimes χ) \otimes (v \otimes χ^{'})) = P_{+}^{(K)} (u \otimes v) \otimes P_{-}^{(V)} (χ \otimes χ^{'}) + P_{-}^{(K)} (u \otimes v) \otimes P_{+}^{(V)} (χ \otimes χ^{'})$ (Regrouping two factors: the symmetry decomposition of a two-particle sector over an orbital and an internal factor clauses (b),(d),(e)).

[F3]

Spin-half internal factor. For $V = C^{2}$ the total spin operators of Composition of two spin labels: total spin operators, the Clebsch-Gordan block decomposition and the spin-half singlet decompose $V \otimes V = W_{0} \oplus W_{1}$ with $\dim W_{0} = 1$ , $\dim W_{1} = 3$ ; the vector $Ψ = (e_{+} \otimes e_{-} - e_{-} \otimes e_{+}) / \sqrt{2}$ is a unit vector with $S_{i}^{t o t} Ψ = 0$ for all $i$ and is, up to a phase, the unique unit vector in the kernel of $S^{t o t 2}$ ; the products $e_{m} \otimes e_{n}$ are $S_{z}^{t o t}$ -eigenvectors with weight $ℏ (m + n)$ , so $e_{+} \otimes e_{+}$ has weight $ℏ$ and the combinations $σ_{+} = (e_{+} \otimes e_{-} + e_{-} \otimes e_{+}) / \sqrt{2}$ and $σ_{-} = (e_{+} \otimes e_{-} - e_{-} \otimes e_{+}) / \sqrt{2}$ are unit vectors in $W_{1}$ and $W_{0}$ with $S_{z}^{t o t}$ -weight $0$ (Composition of two spin labels: total spin operators, the Clebsch-Gordan block decomposition and the spin-half singlet clauses (b),(c),(d)).

[F4]

Orbital symmetric and antisymmetric combinations. For a Hilbert space $K$ and orthonormal $u, v \in K$ the vectors $u ⊙ v = \sqrt{2} P_{+}^{(K)} (u \otimes v)$ and $u \land v = \sqrt{2} P_{-}^{(K)} (u \otimes v)$ are orthonormal, and $P_{-}^{(K)} (u \otimes v)$ and $P_{+}^{(K)} (u \otimes v)$ are orthogonal eigenvectors of the exchange operator; moreover $P_{-}^{(V)} (χ \otimes χ) = 0$ for every $χ \in V$ (The two-particle exchange operator and the symmetric and antisymmetric combinations clauses (b),(c),(d)).

[F5]

Dimensions. For a $d$ -dimensional space, $\dim ran P_{+}^{(2)} = (\binom{d + 1}{2})$ and $\dim ran P_{-}^{(2)} = (\binom{d}{2})$ (Slater and symmetric occupation-number bases of the finite-dimensional permutation sectors clauses (b),(c)); the dimensions of the two orthogonal summands add (Regrouping two factors: the symmetry decomposition of a two-particle sector over an orbital and an internal factor clause (f)).

Proof

technique · direct; identify the two spin combinations of the internal factor with the antisymmetric and symmetric summands, transfer the regrouping vector identity to rays and read the compositions off the identity of the symmetry-decomposition theorem

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

1.1F3F4algebra

[Decomposition of the internal factor] By [F3] the antisymmetric combination $σ_{-}$ spans $W_{0}$ and $σ_{+}, e_{+} \otimes e_{+}, e_{-} \otimes e_{-}$ span the three-dimensional space $W_{1}$ ; since $e_{+} \otimes e_{-} - e_{-} \otimes e_{+}$ is antisymmetric under the exchange and $e_{+} \otimes e_{+}, e_{-} \otimes e_{-}, e_{+} \otimes e_{-} + e_{-} \otimes e_{+}$ are symmetric, $ran P_{-}^{(V)} = C (e_{+} \otimes e_{-} - e_{-} \otimes e_{+}) = C Ψ = W_{0}$ and $ran P_{+}^{(V)} = {Sym}^{2} V = W_{1}$ ; the two subspaces are orthogonal complements of each other because $V \otimes V = W_{0} \oplus W_{1}$ .

2.1F2F3F5step 1.1algebra

[Sector decomposition] By [F2] the unitary image of the sector is $(ran P_{-}^{(K)} \otimes ran P_{+}^{(V)}) + (ran P_{+}^{(K)} \otimes ran P_{-}^{(V)})$ , an orthogonal direct sum, and step 1.1 identifies its summands as the spin-triplet part $ran P_{-}^{(K)} \otimes W_{1}$ and the spin-singlet part $ran P_{+}^{(K)} \otimes W_{0}$ ; by [F5] the singlet part has dimension $\dim ran P_{+}^{(K)} = (\binom{m + 1}{2})$ , and the triplet part has dimension $3 \dim ran P_{-}^{(K)} = 3 (\binom{m}{2})$ ; this is (a) up to the assignment of the names singlet and triplet, which is the naming of $W_{0}$ and $W_{1}$ in [F3].

3.1F2step 1.1step 2.1algebra

[Opposite symmetry] The identity of (b) is the vector identity of [F2] under the identifications of steps 1.1 and 2.1: the first summand carries the internal factor in $ran P_{-}^{(V)} = W_{0}$ , the singlet, and the orbital factor in $ran P_{+}^{(K)}$ , the symmetric orbital component, while the second summand carries the internal factor in $ran P_{+}^{(V)} = W_{1}$ , the triplet, and the orbital factor in $ran P_{-}^{(K)}$ , the antisymmetric orbital component. This proves (b).

4.1F1F4step 3.1algebra

[Parallel spins] Let $u, v$ be orthonormal and $χ$ a unit vector. By [F4] one has $P_{-}^{(V)} (χ \otimes χ) = 0$ and $P_{+}^{(V)} (χ \otimes χ) = χ \otimes χ$ ; hence the identity of (b) gives $Θ P_{-}^{(K \otimes V)} ((u \otimes χ) \otimes (v \otimes χ)) = P_{-}^{(K)} (u \otimes v) \otimes (χ \otimes χ)$ , and multiplying by $\sqrt{2}$ gives the unit vector $(u \land v) \otimes (χ \otimes χ)$ because $∥ u \land v ∥ = 1$ and $∥ χ \otimes χ ∥ = 1$ by [F4]; the nonzero original vector lies in the declared sector, and its image lies in the regrouped sector of [F2], so the two displayed image rays coincide, which is (c).

4.2F2F3F4step 3.1algebra

[Antiparallel spins] Let $u, v$ be orthonormal. The orbital projections are $P_{+}^{(K)} (u \otimes v) = (u ⊙ v) / \sqrt{2}$ and $P_{-}^{(K)} (u \otimes v) = (u \land v) / \sqrt{2}$ by their definitions. In the internal tensor square, $P_{-}^{(V)} (e_{+} \otimes e_{-}) = σ_{-} / \sqrt{2}$ and $P_{+}^{(V)} (e_{+} \otimes e_{-}) = σ_{+} / \sqrt{2}$ by [F2] and the exchange formulas in [F4]. Substituting these four identities into (b) gives $Θ P_{-}^{(K \otimes V)} ((u \otimes e_{+}) \otimes (v \otimes e_{-})) = \frac{1}{2} ((u ⊙ v) \otimes σ_{-} + (u \land v) \otimes σ_{+})$ . The two tensor summands inside the parentheses are orthogonal unit vectors, by [F3] and [F4], so this complete projected vector has norm $1 / \sqrt{2}$ . Multiplying it by $\sqrt{2}$ gives the unit representative $\frac{1}{\sqrt{2}} ((u ⊙ v) \otimes σ_{-} + (u \land v) \otimes σ_{+})$ of the image ray, as in (d). The orbital exchange $τ_{K} \otimes I$ fixes the first orthogonal nonzero component and negates the second, so their sum cannot be an eigenvector: comparison of the two components would require its eigenvalue to be both $1$ and $- 1$ . Thus the ray has no definite orbital exchange symmetry.

5.1step 1.1step 2.1step 3.1step 4.1step 4.2∎

[Assembly and scope] Steps 1.1 and 2.1 prove (a), step 3.1 proves (b), step 4.1 proves (c) and step 4.2 proves (d). Every statement is conditional on the adopted antisymmetric sector of P6 and on the declared finite orbital factor; the internal factor is the bookkeeping factor of P6 and no spatial position, interaction, dynamics or measurement is introduced.

Remarks

The singlet and triplet names. The names refer to the total spin of the pair in the sense of the internal-factor composition theorem: $W_{0}$ is the kernel of $S^{t o t 2}$ and $W_{1}$ is the triplet block. The orbital factor carries no spin and the theorem attributes no spin to the orbital degrees of freedom.
A Slater determinant is a superposition. As clause (d) shows, the antisymmetrised product of two orthonormal one-particle states with opposite spin orientations contains both the singlet and the triplet $S_{z} = 0$ components with equal amplitudes; a definite orbital exchange symmetry belongs to the components, not to every vector of the sector.

The two spin-half fermion sector: the spin-singlet and spin-triplet parts and their orbital symmetry

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

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