Fermionic Fock Space over a Complex Hilbert Space

Assume Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)), and let $H$ be a complex Hilbert space. For each $k\ge 0$, let ${\mathrm{\Lambda }}^{k}H$ be the Hilbert exterior power with the Gram-determinant inner product and wedge convention of Hilbert exterior powers and induced operators, with ${\mathrm{\Lambda }}^{0}H=\mathbb{C}$. The algebraic direct sum $X_0={⨁}_{k\ge 0}^{\mathrm{a}\mathrm{l}\mathrm{g}}{\mathrm{\Lambda }}^{k}H$, where distinct particle-number summands are orthogonal and the inner product on a finite sum is the sum of the component inner products, is an inner-product space. Its norm completion exists under Countable Choice by Every metric space has a completion, constructed as the equivalence classes of its Cauchy sequences and has its compatible Banach-space and Hilbert-space structures by The metric completion of a normed space carries a unique compatible Banach-space structure and The norm completion of an inner-product space is a Hilbert space. The fermionic Fock space over $H$ is this Hilbert completion. The vacuum vector $\mathrm{\Omega }$ is the image of $1\in {\mathrm{\Lambda }}^{0}H$ in the completion.

On finite sums of decomposable wedges define the creation rule $C(f)\xi =f\wedge \xi$ for $f\in H$. This rule is initially defined on a dense subspace. Its bounded extension and the CAR relations for $C(f)$ and its Hilbert-space adjoint are proved in Fock Creation and Annihilation Operators Satisfy the CAR.