On the quantum dynamics of non-commutative systems

TL;DR

This review explores the quantum dynamics of non-commutative systems, focusing on their consistency, the conditions for unitarity, and the functional quantization approach, including explicit path integral calculations.

Contribution

It provides a unified framework for classical and quantum dynamics of non-commutative systems using constrained system theory and establishes the compatibility of operator and functional quantization methods.

Findings

Derived conditions for the existence of the Born series and unitarity.

Established compatibility between operator and functional quantization.

Calculated the Feynman kernel for a non-commutative 2D harmonic oscillator.

Abstract

This is a review paper concerned with the global consistency of the quantum dynamics of non-commutative systems. Our point of departure is the theory of constrained systems, since it provides a unified description of the classical and quantum dynamics for the models under investigation. We then elaborate on recently reported results concerned with the sufficient conditions for the existence of the Born series and unitarity and turn, afterwards, into analyzing the functional quantization of non-commutative systems. The compatibility between the operator and the functional approaches is established in full generality. The intricacies arising in connection with the explicit computation of path integrals, for the systems under scrutiny, is illustrated by presenting the detailed calculation of the Feynman kernel for the non-commutative two dimensional harmonic oscillator.