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Note di matematica/Note di matematica - Università degli studi di Lecce

A quantization of diffusion processes on manifolds

Lu, Y.G.

Published Aug 4, 2009• 0 citations
Publisher

Abstract

Via the well known chaos decomposition with respect to the multidimensional Brownian motion, we give a possible quantization of a diffusion process (in particular, Brownian motion) on a Riemannian manifold. The quantized diffusion process on Riemannian manifold is a quantum Brownian motion on a Holbert space determined by the momentum algebra.

DOI
10.1285/i15900932v14n1p57

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Via the well known chaos decomposition with respect to the multidimensional Brownian motion, we give a possible quantization of a diffusion process (in particular, Brownian motion) on a Riemannian manifold. The quantized diffusion process on Riemannian manifold is a quantum Brownian motion on a Holbert space determined by the momentum algebra.
We consider Brownian motion on quantized non-commutative Riemannian manifolds and show how a set of stochastic processes on sets of complex numbers can be devised. This class of stochastic processes are shown to yield at the outset a Chapman-Kolmogorov equation with a complex diffusion coefficient that can be straightforwardly reduced to the Schrodinger equation. The existence of these processes has been recently shown numerically. In this work we provide an analogous support for the existence of the Chapman-Kolmogorov-Schrodinger equation for them. Besides, it is numerically seen as a Wick rotation can change the heat kernel into the Schrodinger one.
Quantum diffusions are non-commutative generalisations of classical diffusions on manifolds, in which the manifold i s ‘quantised’ by deforming the algebra of smooth functions thereon to a non-commutative algebra, and in which quantum stochastic calculus replaces the Itô calculus of Brownian motion. Quantum diffusions fall into two categories; inner diffusions, which are in a certain sense cohomologically trivial, and which act by conjugating by a unitary process, and outer diffusions, which do not. A natural perturbation procedure is described for linking two cohomologically equivalent quantum diffusions; it involves the construction of a new type of intertwining unitary process.
In the frame of Nelson stochastic quantization for dynamical systems on a manifold, we consider diffusion processes with Brownian covariance given by a Riemannian metric on the manifold. The dynamics is specified through a stochastic variational principle for a generalization of the classical action, with a given kinetic metric. The resulting programming equation, of the Hamilton-Jacobi type, depends on both metrics, the Brownian one and the kinetic one. We introduce a simple notion of compatibility between the two metrics, such that the programming equation and the continuity equation lead to the Schr\"odinger equation on the manifold.
We unify Brownian motion and quantum mechanics in a single mathematical framework. In particular, we show that non-relativistic quantum mechanics of a single spinless particle on a flat space can be described by a Wiener process that is rotated in the complex plane. We then extend this theory to relativistic stochastic theories on manifolds using the framework of second order geometry. As a byproduct, our results suggest that a consistent path integral based formulation of a quantum theory on a Lorentzian (Riemannian) manifold requires an Ito deformation of the Poincare (Galilean) symmetry, arising due to the coupling of the quadratic variation to the affine connection.