Knowlpedia

Creation and Annihilation Operators

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

On S(H)S(H), the creation operator C(z)C(z) adds a particle: C(z)(x1⊗⋯⊗xn)s=n+1 (z⊗x1⊗⋯⊗xn)sC(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) is the annihilation operator.

Remarks

Key property (paper use):

  • The (unbounded) field operator is R(z)=12(C(z)+C∗(z))∼R(z)=\tfrac1{\sqrt2}(C(z)+C^*(z))^{\sim}.
Examples
  • C(z)e0=z∈Sym1(H)C(z)e_0=z\in \mathrm{Sym}^1(H).