CANONICAL REPRESENTATION OF STATIONARY QUANTUM GAUSSIAN PROCESSES

Stimulated by the quantum generalization of the canonical representation theory for Gaussian processes in Ref. 1, we first give the representations (not necessarily canonical) of two stationary Gaussian processes X and Y by means of white noises q_t and p_t with no assumptions on their commutator. We then assume that q_t + ip_t annihilates the vacuum state and prove that the representations are the joint Boson–Fock ones if and only if X and Y have a scalar commutator.