LemmaStatement: AI-adaptedProof: AI-adapted

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Finite spin components of an N-fold spatial tensor product

Statement

Slot convention. For $ℓ \geq 1$ , $S_{ℓ}$ acts on ${0, \dots, ℓ - 1}$ as in Permutation sector projections on the n-fold tensor power. On one-based tuple positions use $\hat{π} (k) : = 1 + π (k - 1)$ for $1 \leq k \leq ℓ$ . Conjugation by $k \mapsto k - 1$ gives $\hat{π σ} = \hat{π} \circ \hat{σ}$ and $\hat{π^{- 1}} = {\hat{π}}^{- 1}$ ; thus the supplier's tensor action becomes $U_{π} (v_{1} \otimes \dots \otimes v_{ℓ}) = v_{{\hat{π}}^{- 1} (1)} \otimes \dots \otimes v_{{\hat{π}}^{- 1} (ℓ)}$ . The parameters of $U_{π}$ and $sgn (π)$ remain canonical zero-based permutations.

Assume Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $N, r \geq 1$ be integers, $H = L^{2} (R^{3}; C^{r})$ , $S_{N} = (C^{r})^{\otimes N}$ , and $M = r^{N}$ . In the product basis $e_{a} = e_{a_{1}} \otimes \dots \otimes e_{a_{N}}$ indexed by $a \in {1, \dots, r}^{N}$ , identify $S_{N}$ with $C^{M}$ and write $L_{N} = L^{2} (R^{3 N}; C^{M})$ . Then:

(a) There is a unique unitary $Θ_{N} : H^{\otimes N} \to L_{N}$ such that $[Θ_{N} (f_{1} \otimes \dots \otimes f_{N})]_{a} (x_{1}, \dots, x_{N}) = \prod_{j = 1}^{N} [f_{j} (x_{j})]_{a_{j}} .$

(b) For $π \in S_{N}$ , let $U_{π}$ be the permutation of the complete one-particle factors and $R_{π} e_{a} = e_{a_{{\hat{π}}^{- 1} (1)}} \otimes \dots \otimes e_{a_{{\hat{π}}^{- 1} (N)}}$ the permutation of spin factors. With $X^{π} = (x_{\hat{π} (1)}, \dots, x_{\hat{π} (N)})$ , the conjugate permutation is $U_{π} : = Θ_{N} U_{π} Θ_{N}^{- 1}, (U_{π} Ψ) (X) = R_{π} Ψ (X^{π}) .$ In particular $[U_{π} Ψ]_{a} (X) = Ψ_{(a_{\hat{π} (1)}, \dots, a_{\hat{π} (N)})} (X^{π})$ ; averaging this combined action gives $P_{\pm} = Θ_{N} P_{\pm}^{(N)} Θ_{N}^{- 1}$ .

(c) The component domain $D_{N} = W^{2, 2} (R^{3 N}; C^{M})$ is the set of tuples with every scalar component in $W^{2, 2} (R^{3 N}; C)$ . If a scalar self-adjoint $T$ has this scalar domain and its domain is preserved by every spatial block permutation, commuting with that permutation on the domain, its finite direct sum $T_{M} Ψ = (T Ψ_{a})_{a}$ on $D_{N}$ is self-adjoint, retains every scalar lower bound $T \geq c I$ , and commutes with the combined permutations on that domain. The restrictions of $T_{M}$ to $ran P_{\pm}$ , with domain $D_{N} \cap ran P_{\pm}$ , are self-adjoint when the sector is nonzero; on the zero sector the unique zero operator has domain ${0}$ .

(d) For matrices $A_{j} \in B (C^{r})$ , the constant spin action $A_{1} \otimes \dots \otimes A_{N}$ preserves $D_{N}$ and commutes with $T_{M}$ . Its permutation covariance is $R_{π} (A_{1} \otimes \dots \otimes A_{N}) R_{π}^{- 1} = A_{{\hat{π}}^{- 1} (1)} \otimes \dots \otimes A_{{\hat{π}}^{- 1} (N)} .$ Thus a common internal unitary $A^{\otimes N}$ commutes with $U_{π}$ , preserves both sectors and commutes with their restricted operators. This is a tensor identity and asserts no physical spin-statistics assignment. For $r = 1$ all spin actions are scalar and $Θ_{N}$ is the iterated scalar product unitary; for $N = 1$ it is the identity component identification.

Facts & Assumptions

Given: $N, r, H, S_{N}, M, L_{N}$ , the product spin basis, and in (c) the scalar self-adjoint $T$ with domain $W^{2, 2}$ and spatial permutation invariance.

[F1]

The scalar Euclidean product map is unitary and permits finite component sums (Euclidean L² product unitary: L²(R^a)⊗̂L²(R^b) ≅ L²(R^{a+b}) Statement); iterating it gives the scalar N-factor product map. The component identification is unitary for every Euclidean dimension and finite component count (Unitary identifications of the component-valued wave-function space (a), (b)).

[F2]

Tensor pairings multiply, elementary tensors span densely, and spin product bases are orthonormal; bounded factor maps extend with their tensor action and adjoints (Hilbert tensor construction: inner product, completion and product bases, Bounded tensor maps: extension, norm estimate, adjoint and commutation). Factor permutations and their symmetric/sign averages are unitary and orthogonal projections respectively (Permutation sector projections on the n-fold tensor power (a)-(c)).

[F3]

A self-adjoint operator invariant on its domain under a unitary finite-group permutation representation restricts self-adjointly to each nonzero symmetric/sign sector (Permutation-restricted self-adjointness (a)-(d)); the scalar-only application (e) is not needed for a spin sector.

Proof

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

1.1F1F2givenalgebra

[Finite component product unitary] Resolve each one-particle vector into its $r$ components by [F1]. On the dense span of vectors $(g_{1} e_{a_{1}}) \otimes \dots \otimes (g_{N} e_{a_{N}})$ , send this vector to the function $(\prod_{j} g_{j} (x_{j})) e_{a}$ . Inner products of such vectors are the product of the scalar pairings times $\prod_{j} δ_{a_{j} b_{j}}$ by [F2]; their image pairings have exactly the same value by the iterated scalar unitary [F1]. Linearity therefore gives a well-defined isometry on this dense span. Its image is dense: in each of the finitely many components the scalar products span densely by [F1]. The isometry extends to the completion; its range is closed by completeness and dense, hence all of $L_{N}$ . Expansion of the finite components gives the formula in (a), and equality on a dense span gives uniqueness.

2.1step 1.1F2algebra

[Combined permutation] On the component product vectors of step 1.1, applying $U_{π}$ replaces the factor in slot $j$ by the former factor in slot ${\hat{π}}^{- 1} (j)$ . Evaluating the product formula gives $(U_{π} Ψ) (X) = R_{π} Ψ (X^{π})$ , including the displayed component-index formula. Both operators are bounded unitaries, so the equality extends by density. Conjugating the finite averages of [F2] gives $P_{\pm}$ ; this simultaneously permutes spatial blocks and spin slots.

3.1givenstep 2.1algebra

[Finite component domain and self-adjointness] Since there are finitely many components, the definition of weak derivatives shows $D_{N} = (W^{2, 2} (R^{3 N}; C))^{\oplus M}$ : derivative testing and integrability hold componentwise in both directions, and all norms are finite sums. The direct sum $T_{M}$ is dense and symmetric. A vector $η$ lies in $D (T_{M}^{*})$ precisely when testing separately in each component puts $η_{a} \in D (T^{*}) = D (T)$ for every $a$ ; conversely these finitely many adjoint values give a vector in $L_{N}$ . Thus $D (T_{M}^{*}) = D_{N}$ and $T_{M}^{*} = T_{M}$ . If $T \geq c I$ , summing the scalar quadratic inequalities gives $T_{M} \geq c I$ . The hypothesis on scalar spatial permutations gives domain preservation and commutation for their componentwise action; $R_{π}$ is a constant permutation of finitely many components, so it preserves the domain and commutes with $T_{M}$ too. Their combined action in step 2.1 has both properties.

4.1step 2.1step 3.1F3algebra

[Sector restrictions] Apply [F3] to the self-adjoint $T_{M}$ of step 3.1 and the combined representation from step 2.1. It gives reduction and self-adjointness on every nonzero $ran P_{\pm}$ with exactly the stated intersection domain. If a sector is zero, its Hilbert space and dense operator domain are both ${0}$ , so its unique operator is its own adjoint.

5.1step 1.1step 2.1step 3.1step 4.1F2algebra∎

[Spin covariance and edge cases] A constant spin matrix makes finite linear combinations of the scalar components, so it preserves $D_{N}$ and commutes with the componentwise scalar $T_{M}$ . On elementary spin tensors the conjugation by $R_{π}$ reorders the matrices to $A_{{\hat{π}}^{- 1} (j)}$ in slot $j$ ; by linearity this is the displayed covariance on all of $S_{N}$ . For a common $A$ , the reordering leaves $A^{\otimes N}$ unchanged, so it commutes with each combined permutation and its averages; if $A$ is unitary the inverse has the same properties, proving onto preservation of both sectors and their domains. At $r = 1$ the sole component is scalar, and at $N = 1$ the product formula is the original vector-valued function. These establish (d) and all stated branches.

Depends on

Dependency tree · 2 levels

Sources

G. Teschl, Mathematical Methods in Quantum Mechanics, Section 1.4 (finite tensor products and orthogonal sums); local derivation from the declared suppliers