Quantum Stochastic Calculus and Quantum Gaussian Processes

In this lecture we present a brief outline of boson Fock space stochastic calculus based on the creation, conservation and annihilation operators of free field theory, as given in the 1984 paper of Hudson and Parthasarathy. We show how a part of this architecture yields Gaussian fields stationary under a group action. Then we introduce the notion of semigroups of quasifree completely positive maps on the algebra of all bounded operators in the boson Fock space $\Gamma (\mathbb{C}^n)$ over $\mathbb{C}^n.$ These semigroups are not strongly continuous but their preduals map Gaussian states to Gaussian states. They were first introduced and their generators were shown to be of the Lindblad type by Vanheuverzwijn. They were recently investigated in the context of quantum information theory by Heinosaari, Holevo and Wolf. Here we present the exact noisy Schr\"odinger equation which dilates such a semigroup to a quantum Gaussian Markov process.