Title:

A matrix formulation of quantum stochastic calculus

We develop the theory of chaos spaces and chaos matrices. A chaos space is a Hilbert space with a fixed, countablyinfinite, directsum decomposition. A chaos matrix between two chaos spaces is a doublyinfinite matrix of bounded operators which respects this decomposition. We study operators represented by such matrices, particularly with respect to selfadjointness. This theory is used to reformulate the quantum stochastic calculus of Hudson and Parthasarathy. Integrals of chaosmatrix processes are defined using the HitsudaSkorokhod 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 $\Omega$adaptedness, is defined. We show that quantum stochastic integrals of $\Omega$adapted processes are wellbehaved; for instance, bounded processes have bounded integrals. We solve the appropriate modification of the evolution equation of Hudson and Parthasarathy: $U(t)=I+\int_{0}^{t}E(s)\mathrm{d}\Lambda(s)+F(s)\mathrm{d} A(s)+ G(s)U(s)\mathrm{d} A^{\dagger}(s)+H(s)U(s)\mathrm{d} s, $ where the coefficients are timedependent, bounded, $\Omega$adapted processes acting on the whole Fock space. We show that the usual conditions on the coefficients, viz. $(E,F,G,H)=(WI,L,WL^{*},iK+\mbox{$\frac{1}{2}$}LL^{*})$ where $W$ is unitary and $K$ selfadjoint, are necessary and sufficient conditions for the solution to be unitary. This is a very striking result when compared to the adapted case.
