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Symmetric Fock Space S(H)

The bosonic Fock space ⊕_{n≥0} Sym^n(H) with vacuum vector
Symmetric Fock Space S(H)

The symmetric (bosonic) Fock space over HH is

S(H)=⨁n=0∞Symn(H), S(H)=\bigoplus_{n=0}^\infty \mathrm{Sym}^n(H),

with vacuum vector e0∈Sym0(H)≅Ce_0\in \mathrm{Sym}^0(H)\cong \mathbb C.

Key properties (paper use):

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

Example: If H=CH=\mathbb C, then S(H)≅ℓ2(N0)S(H)\cong \ell^2(\mathbb N_0).