A matrix formulation of quantum stochastic calculus

Abstract

We develop the theory of chaos spaces and chaos matrices. A chaos space is a Hilbert space with a fixed, countably-infinite, direct-sum decomposition. A chaos matrix between two chaos spaces is a doubly-infinite matrix of bounded operators which respects this decomposition. We study operators represented by such matrices, particularly with respect to self-adjointness.

This theory is used to re-formulate the quantum stochastic calculus of Hudson and Parthasarathy. Integrals of chaos-matrix processes are defined using the Hitsuda-Skorokhod integral and Malliavin gradient,following Lindsay and Belavkin. A new way of defining adaptedness is developed and the consequent quantum product Ito formula is used to provide a genuine functional Ito formula for polynomials in a large class of unbounded processes, which include the Poisson process and Brownian motion.

A new type of adaptedness, known as -adaptedness, is defined. We show that quantum stochastic integrals of -adapted processes are well-behaved; for instance, bounded processes have bounded integrals. We solve the appropriate modification of the evolution equation of Hudson and Parthasarathy:



where the coefficients are time-dependent, bounded, -adapted processes acting on the whole Fock space. We show that the usual conditions on the coefficients, viz.



where is unitary and self-adjoint, are necessary and
sufficient conditions for the solution to be unitary. This is a very striking result when compared to the adapted case.