Brownian motion on Lie groups and open quantum systems

TL;DR

This paper explores the connection between quantum dynamical semigroups called twirling semigroups and classical Brownian motion on Lie groups, providing a complete characterization of their generators.

Contribution

It establishes a precise link between twirling semigroups and Brownian motion, and characterizes their generators for finite-dimensional Lie group representations.

Findings

Twirling semigroups associated with finite-dimensional representations are equivalent to random unitary semigroups.

A complete characterization of the generators of these semigroups is provided.

The connection between quantum and classical stochastic processes on Lie groups is clarified.

Abstract

We study the twirling semigroups of (super)operators, namely, certain quantum dynamical semigroups that are associated, in a natural way, with the pairs formed by a projective representation of a locally compact group and a convolution semigroup of probability measures on this group. The link connecting this class of semigroups of operators with (classical) Brownian motion is clarified. It turns out that every twirling semigroup associated with a finite-dimensional representation is a random unitary semigroup, and, conversely, every random unitary semigroup arises as a twirling semigroup. Using standard tools of the theory of convolution semigroups of measures and of convex analysis, we provide a complete characterization of the infinitesimal generator of a twirling semigroup associated with a finite-dimensional unitary representation of a Lie group.