Over the last few years a non-commutative stochastic calculus has been developed [e.g. R. L. Hudson and K. R. Parthasarathy, Commun. Math. Phys. 93, 301-323 (1984; Zbl 0546.60058)] in which the Wiener process of classical probability is replaced by a pair of ’operator-valued processes’ \((Q,P)\) satisfying: \[ \begin{alignedat}{2} &\text{I}\quad&Q(s)P(t)-P(t)Q(s)&=2is\wedge t \\ &&Q(s)Q(t)-Q(t)Q(s)&=0 \\ &&P(s)P(t)-P(t)P(s)&=0 \\ &&E[Q(s)P(t)]&=is\wedge t \\ &&E[Q(s)Q(t)]&=E[P(s)P(t)] \\ &&&=(\cosh2\psi)s\wedge t \end{alignedat} \text{ OR } \begin{alignedat}{2} &\text{II}\quad&Q(s)P(t)+P(t)Q(s)&=0 \\ &&Q(s)Q(t)-Q(t)Q(s)&=2s\wedge t \\ &&P(s)P(t)-P(t)P(s)&=2s\wedge t \\ &&E[Q(s)P(t)]&=(\cos 2\theta)s\wedge t \\ &&E[Q(s)Q(t)]&=E[Q(t)Q(s)] \\ &&&=s\wedge t \end{alignedat} \] The two correspond to Bose-Einstein and Fermi-Dirac statistics, respectively. In the former case the individual processes \(Q\) and \(P\) are each essentially Brownian motions (of variance \(\cosh2\psi\)) whereas in the latter case \(Q\) fails to commute at different times, as does \(P\); they are called Clifford processes.
The main problem considered is the representability of processes which are martingales with respect to the (operator algebra) filtration determined by \((Q,P)\) (case II \(0<\theta<\pi/2)\) in terms of stochastic integrals. Thus a non-commutative analogue of the Kunita-Watanabe theorem is sought. Clifford martingales were shown to be expressible as integrals with respect to the Clifford process in C. Barnett, R. F. Streater and I. F. Wilde, J. Funct. Anal. 48, 172-212 (1982; Zbl 0492.46051); in case I (\(\psi>0\)) martingales were shown to be given by a sum of integrals w.r.t. \(Q\) and \(P\) in R. L. Hudson and the author, A non-commutative martingale representation theorem for non-Fock quantum Brownian motion. ibid. 61, 202-221 (1985), and similarly in the present case: for \(\theta\in (0,\pi /2)\), for any martingale \(M\) there are adapted \(L^2\)-processes \(J,K\) s.t. for \(t\geq 0,\) \[ M(t) = M(0) + \int^{t}_{0} J(s) dQ(s) + \int^{t}_{0} K(s) dP(s) \tag{\(*\)} \] [In the paper, the process \(A:=2^{-1/2}(Q+iP)\) and its adjoint are used]. The representation (\(*\)) fails when \(\theta =0,\pi /2\) (and when \(\psi =0)\), however, it has recently been discovered that, by addition of an integral with respect to a third process (gauge/preservation process) representability is achieved for so called ’regular’ martingales [K. R. Parthasarathy et al., Delhi preprint (1985)].