Schwartz/distribution spaces and Hilbert Fock conventions

For an integer $d\ge 1$, let $\mathcal{S}({\mathbb{R}}^d)$ be the complex Schwartz space with seminorms $p_{\alpha \beta }(f)={\sup }_{x\in {\mathbb{R}}^d}\mathrm{\mid }{x}^{\alpha }{\mathrm{\partial }}^{\beta }f(x)\mathrm{\mid }$ for all multi-indices $\alpha ,\beta$; this is exactly Schwartz space and its seminorms. Let $\mathcal{D}({\mathbb{R}}^d)=C_c^{\mathrm{\infty }}({\mathbb{R}}^d;\mathbb{C})$ with the support convention of Test function space d of an open set and the locally convex LF topology of Test function topology. The tempered-distribution space ${\mathcal{S}}^{\mathrm{\prime }}({\mathbb{R}}^d)$ is the space of continuous complex-linear functionals on this topological $\mathcal{S}({\mathbb{R}}^d)$, paired bilinearly with test functions, as in Tempered distribution.

For every complex Hilbert space $H$, use the second-variable-linear pairing $⟨u,v{⟩}_2:=(v,u{)}_1$, where $(\cdot ,\cdot {)}_1$ is the first-variable-linear Hilbert pairing of Real and complex inner-product spaces and their induced length. Thus $⟨u,v{⟩}_2$ is linear in $v$ and conjugate-linear in $u$. For each tensor product, use the same norm completion as Hilbert tensor construction: inner product, completion and product bases, but equip it with the second-variable-linear pairing obtained by conjugating that supplier's first-variable-linear pairing. This leaves the norm and completion unchanged; on elementary tensors it gives the product of the second-variable pairings. Let $H^{\widehat{\otimes }n}$ be the iterated Hilbert tensor completion, with $H^{\widehat{\otimes }0}:=\mathbb{C}$; the tensor inner product is the product of the factor pairings. Identify parenthesizations by the isometry fixing elementary tensors: the product pairing is the same on their dense span, so the map and its inverse extend to the completions. For $n\ge 1$, let $P_{s,n}=n{!}^{-1}{\sum }_{\pi \in S_n}{U}_{\pi }$ and $P_{a,n}=n{!}^{-1}{\sum }_{\pi \in S_n}\mathrm{sgn}(\pi )U_{\pi }$, using the permutation action and the orthogonal projections of Permutation sector projections on the n-fold tensor power. Define $H_s^{(n)}=\mathrm{ran}{P}_{s,n}$ and $H_a^{(n)}=\mathrm{ran}{P}_{a,n}$, with $H_s^{(0)}=H_a^{(0)}=\mathbb{C}$. These are the symmetric and antisymmetric $n$-particle sectors.

The bosonic and fermionic Fock Hilbert spaces are the square-summable direct sums ${\mathcal{F}}_s(H)=\{({\mathrm{\Psi }}_n{)}_{n\ge 0}:{\mathrm{\Psi }}_n\in H_s^{(n)},{\sum }_n\mathrm{\parallel }{\mathrm{\Psi }}_n{\mathrm{\parallel }}^2<\mathrm{\infty }\}$ and ${\mathcal{F}}_a(H)=\{({\mathrm{\Psi }}_n{)}_{n\ge 0}:{\mathrm{\Psi }}_n\in H_a^{(n)},{\sum }_n\mathrm{\parallel }{\mathrm{\Psi }}_n{\mathrm{\parallel }}^2<\mathrm{\infty }\}$, with the componentwise inner product. Their vacuum is $\mathrm{\Omega }=(1,0,0,\dots )$. The finite-particle subspace ${\mathcal{D}}_{\mathrm{f}\mathrm{i}\mathrm{n}}$ consists of sequences with only finitely many nonzero components.

For $h\in H$, the map $C_h:H\to \mathbb{C}$, $C_h(v)=⟨h,v{⟩}_2$, is linear and bounded: $\mathrm{\mid }{C}_h(v)\mathrm{\mid }=\mathrm{\mid }(v,h{)}_1\mathrm{\mid }\le \mathrm{\parallel }h\mathrm{\parallel }\mathrm{\parallel }v\mathrm{\parallel }$ by Cauchy–Schwarz: $\mathrm{\mid }⟨u,v⟩\mathrm{\mid }\le \parallel u\parallel \parallel v\parallel$, with equality exactly for linearly dependent vectors. For any Hilbert space $K$, identify $\mathbb{C}\widehat{\otimes }K$ with $K$ by $J_K(z\otimes v)=zv$. Every algebraic tensor is $1\otimes w$; the tensor norm is then $\mathrm{\parallel }w\mathrm{\parallel }$, so $J_K$ extends to a surjective isometry on the completion (with inverse $w\mapsto 1\otimes w$). Throughout the following formulas, $C_h\otimes I^{\otimes (n-1)}$ means $J_{H^{\widehat{\otimes }(n-1)}}\circ (C_h\widehat{\otimes }I)$, with $I$ the identity on $H^{\widehat{\otimes }(n-1)}$. The bounded tensor-map supplier therefore applies and gives norm at most $\mathrm{\parallel }h\mathrm{\parallel }$. This includes $n=1$, where $H^{\widehat{\otimes }0}=\mathbb{C}$. Contraction commutes with permutations of the remaining $n-1$ factors, so it takes symmetric or antisymmetric tensors to the corresponding $(n-1)$-particle sector (and to $\mathbb{C}$ for $n=1$).

For $n\ge 0$, bosonic creation on sector $n$ is $a_n^{†}(h){\mathrm{\Psi }}_n=\sqrt{n+1}{P}_{s,n+1}(h\otimes {\mathrm{\Psi }}_n)$; for $n\ge 1$, bosonic annihilation is $a_n(h){\mathrm{\Psi }}_n=\sqrt{n}(C_h\otimes I^{\otimes (n-1)}){\mathrm{\Psi }}_n$, where $C_h$ is the bounded functional above, its tensor extension is supplied by Bounded tensor maps: extension, norm estimate, adjoint and commutation, and $a(h)\mathrm{\Omega }=0$. Define fermionic creation by $c_n^{†}(h){\mathrm{\Psi }}_n=\sqrt{n+1}{P}_{a,n+1}(h\otimes {\mathrm{\Psi }}_n)$; for $n\ge 1$, fermionic annihilation is the corresponding first-factor contraction $c_n(h){\mathrm{\Psi }}_n=\sqrt{n}(C_h\otimes I^{\otimes (n-1)}){\mathrm{\Psi }}_n$, and $c(h)\mathrm{\Omega }=0$. The annihilation maps are conjugate-linear in $h$; the creation maps are linear in $h$. These formulas define sector maps on the finite-particle subspace; their maximal closed domains and operator relations are proved in Bosonic and fermionic Fock operators with exact domains. No physical statistics assignment or spin–statistics theorem is part of these mathematical definitions.

Foundational assumption for this construction. Assume the Axiom of Countable Choice The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$). The cited Hilbert tensor construction, bounded tensor-map extension, and permutation-sector projection results each state this hypothesis. Accordingly, the tensor completions, sector projections, and Fock construction above are taken in this context. This records the choice premise already listed among the dependencies; it does not assign physical statistics to a species.