Quantum Lévy area as a quantum martingale limit

We replace the independent one-dimensional Brownian motions in the de…nition of classical Lévy area by the mutually noncommuting "momentum" and "position" Brownian motions of non-Fock quantum stochastic calculus, which are independent in the sense of factorisation of joint characteristic functions. The corresponding quantum Lévy area can then be constructed in a way similar to Lévy’s original construction as a quantum martingale limit. Acknowledgement 1 The author thanks the referee for drawing his attention to a number of mistypes. 1 Introduction. Lévy’s stochastic area [10], [11], as well as being of great intrinsic interest and possessing numerous connections with classical mathematical analysis and mathematical physics [7], has recently come to prominence in the theory of rough paths [12], [13], [5]. Here we present a quantum or noncommutative version of Lévy area in which the two independent one-dimensional Brownian motions used to de…ne it are replaced by the pair of quantum Brownian motions (P;Q) ; forming the quantum Wiener process of [2]. Each of P and Q consists of a process of mutually commuting self-adjoint operators which can be simultaneously diagonalised as multiplication by a classical Brownian motion of appropriate variance in the Hilbert space of complex-valued functions square-integrable with respect to Wiener measure, But P and Q together satisfy the commutation relation P (s)Q(t) Q(t)P (s) = imin fs; tg I; (1)