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arXiv:1103.4531 · Stochastics and Dynamics

SYMMETRY REDUCTION OF BROWNIAN MOTION AND QUANTUM CALOGERO–MOSER MODELS

Simon Hochgerner

Published May 30, 2012• 1 citation
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Abstract

Let Q be a Riemannian G-manifold. This paper is concerned with the symmetry reduction of Brownian motion in Q and ramifications thereof in a Hamiltonian context. Specializing to the case of polar actions, we discuss various versions of the stochastic Hamilton–Jacobi equation associated to the symmetry reduction of Brownian motion and observe some similarities to the Schrödinger equation of the quantum–free particle reduction as described by Feher and Pusztai [10]. As an application we use this reduction scheme to derive examples of quantum Calogero–Moser systems from a stochastic setting.

DOI
10.1142/s0219493712500074

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Let Q be a Riemannian G-manifold. This paper is concerned with the symmetry reduction of Brownian motion in Q and ramifications thereof in a Hamiltonian context. Specializing to the case of polar actions, we discuss various versions of the stochastic Hamilton–Jacobi equation associated to the symmetry reduction of Brownian motion and observe some similarities to the Schrödinger equation of the quantum–free particle reduction as described by Feher and Pusztai [10]. As an application we use this reduction scheme to derive examples of quantum Calogero–Moser systems from a stochastic setting.
Relations between free motion on the GL + ( n , ℝ) group manifold and the dynamics of an n -particle system with spin degrees of freedom on a line interacting with a pairwise 1/sinh 2 x “potential” (Euler-Calogero-Sutherland model) are discussed within a Hamiltonian reduction. Two kinds of reductions of the degrees of freedom are considered: that which is due to continuous invariance and that which is due to discrete symmetry. It is shown that, upon projecting onto the corresponding invariant manifolds, the resulting Hamiltonian system represents the Euler-Calogero-Sutherland model in both cases.
Lie symmetry group methods provide a useful tool for the analysis of differential equations in a variety of areas in physics and applied mathematics. The nature of symmetry is that it provides information on properties which remain invariant under transformation. In differential equations this invariance provides a route toward complete integrations, reductions, linearisations and analytical solutions which can evade standard techniques of analysis. In this thesis we study two problems in quantum mechanics from a symmetry perspective: We consider for pedagogical purposes the linear time dependent Schrodinger equation in a potential and provide a symmetry analysis of the resulting equations. Thereafter, as an original contribution, we study the group theoretic properties of the density matrix equation for the quantum Brownian motion of a free particle interacting with a bath of harmonic oscillators. We provide a number of canonical reductions of the system to equations of reduced dimensionality as well as several complete integrations.
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.
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 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.
The energy-based stochastic extension of the Schrödinger equation is a rather special nonlinear stochastic differential equation on Hilbert space, involving a single free parameter, that has been shown to be very useful for modeling the phenomenon of quantum state reduction. Here we construct a general closed form solution to this equation, for any given initial condition, in terms of a random variable representing the terminal value of the energy and an independent Brownian motion. The solution is essentially algebraic in character, involving no integration, and is thus suitable as a basis for efficient simulation studies of state reduction in complex systems.