Quasifree quantum stochastic integrals for the CAR and CCR

by C. Barnett, R. F. Streater and I. F. Wilde.

J. Functional Anal., 52, 19-47, 1983.

ABSTRACT

A theory of quantum martingales and quantum stochastic integrals in quasifree representations of the CAR and CCR is presented. For the CAR, the results generalise some of those developed in The Ito-Clifford Integral, and for the CCR, the results contain the standard Ito theory of stochastic integration with respect to Brownian motion as a special case.

NOTES

One of the results of this paper is that the stochastic integral has a unique closure that is affiliated to the von Neumann algebra of the filtration.

CAR stands for "canonical anticommutation relations", and CCR stands for "canonical commutation relations". See the book by G. G. Emch, Algebraic Methods in Statistical Mechanics and Quantum Field Theory, Interscience, New York, 1972.

J. M. Lindsay and I.F. Wilde ("On non-Fock boson stochastic integrals", J. Functional Anal., 65, 77-82 (1986)) have shown that our theory leads to the same definition of stochastic integral as the method of Hudson and Parthasarathy, in the case considered by them (Quantum Ito's formula and stochastic evolutions, Commun. Math. Phys., 93, 301-323, 1984). It is also easy to show that our version of the CAR theory is the same as that worked on by D. Applebaum. A version of the Bose case using ideas from white noise analysis has been given by N. Obata (Publ. RIMS Kyoto, 31, 667-702, 1995).