Symmetric Fock Space S(H)

Let $H$ be a complex Hilbert space. Its symmetric (bosonic) Fock space is the Hilbert direct sum

S(H)=\widehat{\bigoplus}_{n=0}^\infty \operatorname{Sym}^n(H),

where $\operatorname{Sym}^n(H)$ is the completed symmetric $n$ -fold tensor power. The unit vector $e_0\in\operatorname{Sym}^0(H)\cong\mathbb C$ is the vacuum vector.

Remarks

Key properties (paper use):

Carries creation/annihilation operators and the Fock–Cook field operators.
The canonical action of $U(H)$ second-quantizes to a unitary action on $S(H)$ .

Examples

If $H=\mathbb C$ , then $S(H)\cong \ell^2(\mathbb N_0)$ .