Gaussian quantum stochastic processes on the CCR algebra

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Abstract
We define a stationary Gaussian quantum stochastic process (GQSP) on the C*‐algebra of the canonical commutation relations over a real symplectic Hilbert space. Physically GQSPs can describe diffusion with non‐Markovian memory effects in quantum harmonic oscillators of arbitrary dimension. We find that in analogy with the commutative case a GQSP is completely specified by a operator‐valued autocovariance function satisfying certain positive definiteness and reality conditions. The autocovariance function also determines the response of the system to a class of time‐dependent generalized forces, and it has a spectral representation in terms of a positive operator‐valued measure.

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