The Ito algebra of quantum Gaussian fields

Abstract

The notion of mutual quadratic variation (square bracket) is extended to a quantum probabilistic framework. The mutual quadratic variations of the annihilation, creation, and number fields in a Gaussian representation are calculated, in both the Boson and the Fermion case, in the strong topology on a common invariant domain. It is proved that the corresponding Ito table closes at the second order. The Fock representation is characterized, among the Gaussian ones, by the property that its Ito table closes at the first order.