DefinitionDefinition: AI-adaptedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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.

Symmetric and antisymmetric identical-particle Fock spaces

Definition

Assume the Axiom of Countable Choice, written ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $I$ be a specified finite or countably infinite mode set with a fixed enumeration, and let $h = ℓ^{2} (I)$ have its coordinate orthonormal basis $(e_{j})_{j \in I}$ . Put $h^{\otimes 0} = C$ , $h^{\otimes 1} = h$ , and define each completed tensor power recursively by $h^{\otimes (n + 1)} = h^{\otimes n} \hat{\otimes} h$ , using the first-variable-linear product inner product of Hilbert tensor construction: inner product, completion and product bases. For $n \geq 1$ , let $U_{π}$ permute the tensor factors for $π \in S_{n}$ , and let

P_{\pm}^{(n)} = \frac{1}{n!} \sum_{π \in S_{n}} χ_{\pm} (π) U_{π}, χ_{+} (π) = 1, χ_{-} (π) = sgn (π),

with $P_{\pm}^{(0)} = I_{C}$ . The permutation-sector theorem Permutation sector projections on the n-fold tensor power makes these orthogonal projections. Define the symmetric and antisymmetric sectors $H_{\pm}^{(n)} : = ran P_{\pm}^{(n)}$ . Define

F_{\pm} (h) : = {ψ = (ψ_{n})_{n \geq 0} : ψ_{n} \in H_{\pm}^{(n)}, \sum_{n \geq 0} ∥ ψ_{n} ∥^{2} < \infty}, ⟨ ψ, φ ⟩ : = \sum_{n \geq 0} ⟨ ψ_{n}, φ_{n} ⟩ .

The vector $Ω = (1, 0, 0, \dots)$ is the vacuum, and the finite-particle subspace consists of vectors with only finitely many nonzero components.

This is a mathematical construction for a specified mode basis. Choosing its symmetric or antisymmetric sector for a physical species is a separate model postulate. No spin-statistics theorem or physical species assignment is part of the definition.

Facts & Assumptions

Given: A finite or countable enumerated mode set $I$ , the coordinate Hilbert space $ℓ^{2} (I)$ , its completed tensor powers, and the symmetric-group action.

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): every at most countable family of nonempty sets has a choice function.

[F1]

Hilbert tensor construction: inner product, completion and product bases, Statement (c)–(d): the completed tensor product of complex Hilbert spaces has the product inner product, is complete, and the products of complete orthonormal families form a complete orthonormal family.

[F2]

Permutation sector projections on the n-fold tensor power, Statement (a)–(c): the factor permutations are unitary, and their symmetric and antisymmetric averages are orthogonal projections whose ranges are the fixed and sign-transforming subspaces.

[F3]

Cauchy–Schwarz: $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥ ∥ y ∥$ , with equality exactly for dependent pairs: the inner product pairing satisfies the Cauchy–Schwarz inequality.

Proof

1.1givenA1F1F2algebra

The tensor factors and product pairing in the Definition are supplied by [F1]. For every $n \geq 1$ , [F2] makes $P_{\pm}^{(n)}$ an orthogonal projection. The range of an orthogonal projection is closed: if $P x_{k} = x_{k}$ and $x_{k} \to x$ , boundedness of $P$ gives $P x = \lim_{k} P x_{k} = \lim_{k} x_{k} = x$ . Therefore each $H_{\pm}^{(n)}$ is a closed subspace of the complete tensor Hilbert space and is itself complete.

1.2givenF1F3algebra

For $ψ, φ \in F_{\pm} (h)$ , Cauchy–Schwarz for the square-summable nonnegative sequences $(∥ ψ_{n} ∥)_{n}$ and $(∥ φ_{n} ∥)_{n}$ gives $\sum_{n} ∣ ⟨ ψ_{n}, φ_{n} ⟩ ∣ \leq (\sum_{n} ∥ ψ_{n} ∥^{2})^{1 / 2} (\sum_{n} ∥ φ_{n} ∥^{2})^{1 / 2} < \infty$ . Hence the displayed pairing converges absolutely, is sesquilinear, and is positive definite because $⟨ ψ, ψ ⟩ = \sum_{n} ∥ ψ_{n} ∥^{2} = 0$ forces every coordinate to vanish.

2.1step 1.1step 1.2algebra

Let $(ψ^{(k)})_{k}$ be Cauchy in this norm. For each fixed $n$ , $(ψ_{n}^{(k)})_{k}$ is Cauchy in the complete sector $H_{\pm}^{(n)}$ , so it converges to some $ψ_{n} \in H_{\pm}^{(n)}$ . If $k \geq K$ and $∥ ψ^{(k)} - ψ^{(m)} ∥ \leq ε$ for all $m \geq K$ , then for every finite $N$ , passing $m \to \infty$ in the finite sum yields $\sum_{n \leq N} ∥ ψ_{n} - ψ_{n}^{(k)} ∥^{2} \leq ε^{2}$ . Also $\sum_{n \leq N} ∥ ψ_{n} ∥^{2} \leq 2 ∥ ψ^{(k)} ∥^{2} + 2 ε^{2}$ . Taking suprema over finite $N$ proves $\sum_{n} ∥ ψ_{n} ∥^{2} < \infty$ , so $ψ \in F_{\pm} (h)$ , and then $∥ ψ^{(k)} - ψ ∥ \leq ε$ for all $k \geq K$ . Thus the direct sum is complete.

3.1step 1.2algebra∎

For $ψ \in F_{\pm} (h)$ , truncating after sector $N$ changes its squared norm by $\sum_{n > N} ∥ ψ_{n} ∥^{2} \to 0$ . Therefore the finite-particle subspace is dense and $Ω$ is a unit vector.

Symmetric and antisymmetric identical-particle Fock spaces

Definition

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources