Knowlpedia

Creation and Annihilation Operators

Operators adding/removing one symmetric tensor factor in bosonic Fock space

For z∈Hz\in H, the creation operator on the algebraic finite-particle subspace of S(H)S(H) is defined by

C(z)(x1⊗⋯⊗xn)s=n+1 (z⊗x1⊗⋯⊗xn)s.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)∗C(z)^*, on its natural domain, is the annihilation operator. These operators are generally unbounded.

Field operator

In the paper's notation, the associated field operator is the closure

R(z)=12(C(z)+C(z)∗)∼.R(z)=\frac1{\sqrt2}\bigl(C(z)+C(z)^*\bigr)^{\sim}.

In particular, C(z)e0=z∈Sym⁡1(H)C(z)e_0=z\in\operatorname{Sym}^1(H), where e0e_0 is the vacuum vector.