Noncommutative Quantum Mechanics Viewed from Feynman Formalism

TL;DR

This paper explores how noncommutative quantum mechanics can be derived from Feynman's formalism, revealing connections to magnetic monopoles and extending the framework to dual momentum space.

Contribution

It extends Feynman's approach to noncommutative quantum mechanics, linking algebraic structures to physical phenomena like magnetic monopoles.

Findings

Poincaré's magnetic angular momentum arises from sO(3) Lie algebra.

Dirac magnetic monopole emerges naturally in the formalism.

Extension to dual momentum space facilitates noncommutative quantum mechanics.

Abstract

Dyson published in 1990 a proof due to Feynman of the Maxwell equations. This proof is based on the assumption of simple commutation relations between position and velocity. We first study a nonrelativistic particle using Feynman formalism. We show that Poincar\'{e}'s magnetic angular momentum and Dirac magnetic monopole are the direct consequences of the structure of the sO(3) Lie algebra in Feynman formalism. Then we show how to extend this formalism to the dual momentum space with the aim of introducing Noncommutative Quantum Mechanics which was recently the subject of a wide range of works from particle physics to condensed matter physics.