Stochastic calculus for quantum Brownian motion of non-minimal variance - an approch using integral-sum kernel operators

J. Martin Lindsay

Kalyanapuram Rangachari Parthasarathy

Revista, Zen, MATH, 10

We present a novel regularization scheme for path integrals applied to quantum mechanical systems with singular potentials. Our approach focuses on the heat kernel asymptotics and their connection to Brownian motion. We introduce a short-time expansion for the heat kernel, incorporating the effects of singular potentials through a carefully constructed effective potential. This effective potential is derived using a combination of distributional calculus and stochastic calculus, ensuring that the resulting path integral is well-defined and produces physically meaningful results. The regularization method is then applied to specific examples of singular potentials, such as the inverse square potential, demonstrating its effectiveness in resolving the ambiguities associated with these potentials. We further explore the implications of our regularization scheme for the behavior of Brownian motion in the presence of singular potentials, connecting the quantum mechanical description to the classical stochastic process.

Andrzej Łuczak

R. L. Hudson

We replace the independent one-dimensional Brownian motions in the de…nition of classical Levy 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 Levy area can then be constructed in a way similar to Levy’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. Levy’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 Levy 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)

R. L. Hudson