A quantization procedure based on completely positive maps and Markov operators

TL;DR

This paper explores the behavior of completely positive maps and Markov operators to develop a quantization method linking classical dynamics with quantum-like state spaces, revealing isometric properties and commutativity in limit sets.

Contribution

It introduces a novel quantization procedure based on CP maps induced by Markov operators, connecting classical and quantum dynamics through a new mathematical framework.

Findings

$oldsymbol{ ext{Limit sets of CP maps exhibit isometric behavior.}}$

$oldsymbol{ ext{States in the convex hulls of these sets commute.}}$

$oldsymbol{ ext{A non-expansive linear map links classical functions and matrix spaces.}}$

Abstract

We describe $\omega$-limit sets of completely positive (CP) maps over finite-dimensional spaces. In such sets and in its corresponding convex hulls, CP maps present isometric behavior and the states contained in it commute with each other. Motivated by these facts, we describe a quantization procedure based on CP maps which are induced by Markov (transfer) operators. Classical dynamics are described by an action over essentially bounded functions. A non-expansive linear map, which depends on a choice of a probability measure, is the centerpiece connecting phenomena over function and matrix spaces.