Creation and Annihilation Operators

On symmetric Fock space $S(H)$, the creation operator $C(z)$ adds a particle: $C(z)(x_1\otimes\cdots\otimes x_n)_s=\sqrt{n+1}\,(z\otimes x_1\otimes\cdots\otimes x_n)_s$.

Its adjoint $C^*(z)$ is the annihilation operator.

Key property (paper use):

• The (unbounded) field operator is $R(z)=\tfrac1{\sqrt2}(C(z)+C^*(z))^{\sim}$.

Example: $C(z)e_0=z\in \mathrm{Sym}^1(H)$.