Fock space

Description

Let $H$ be a complex Hilbert space. For $n \ge 1$, let $H^{\otimes n}$ be the tensor product of $n$ copies of $H$, and let $H^{\otimes 0} = \C$. The full Fock space over $H$ is the Hilbert sum $$\mathcal{F}(H) = \widehat{\bigoplus}_{n \ge 0} H^{\otimes n}.$$ Let $S_+$ and $S_-$ be the orthogonal projections of $H^{\otimes n}$ onto its symmetric and antisymmetric tensors. The bosonic Fock space is $\mathcal{F}_+(H) = \widehat{\bigoplus}_{n \ge 0} S_+ H^{\otimes n}$, and the fermionic Fock space is $\mathcal{F}_-(H) = \widehat{\bigoplus}_{n \ge 0} S_- H^{\otimes n}$. Each of them is a Hilbert space. Indeed, each summand is a closed subspace of $H^{\otimes n}$, and a Hilbert sum of Hilbert spaces is a Hilbert space.