Quasi-free quantum stochastic integrals for the CAR and CCR

Abstract

A theory of quantum martingales and quantum stochastic integrals in quasi-free representations of the CAR and CCR is presented. For the CAR, the results generalize some of those developed in Barnett, Streater, and Wilde ( J. Funct. Anal. 48 (1982), 172–212, J. London Math. Soc. 27 (1983), 373–384) and for the CCR, the results contain the standard Itô theory of stochastic integration with respect to Brownian motion as a special case.