Let $H$ be a complex [Hilbert space](/page/Hilbert%20Space) whose [inner product](/page/Inner%20Product) $(\cdot,\cdot)_H$ is linear in the first variable, and let
be the full Fock space over $H$, with vacuum vector $\Omega$. For $f \in H$, let $\ell(f): \mathcal{F}(H) \to \mathcal{F}(H)$ be the left creation operator, defined by $\ell(f)\Omega = f$ and