From Feynman Proof of Maxwell Equations to Noncommutative Quantum Mechanics

TL;DR

This paper reviews Feynman's proof of Maxwell's equations using minimal assumptions, explores the structure of relativistic particles with Feynman brackets, and extends to noncommutative quantum mechanics revealing new effects like the spin Hall effect and Berry phase implications.

Contribution

It introduces a novel connection between Feynman brackets, noncommutative quantum mechanics, and physical phenomena such as the spin Hall effect and Berry phase, with potential experimental tests.

Findings

Magnetic angular momentum and monopoles arise from Lorentz Lie algebra structure.

Noncommutative coordinates induce a spin Hall effect.

Berry phase influences light propagation in media.

Abstract

In 1990, Dyson published a proof due to Feynman of the Maxwell equations assuming only the commutation relations between position and velocity. With this minimal assumption, Feynman never supposed the existence of Hamiltonian or Lagrangian formalism. In the present communication, we review the study of a relativistic particle using ``Feynman brackets.'' We show that Poincar\'e's magnetic angular momentum and Dirac magnetic monopole are the consequences of the structure of the Lorentz Lie algebra defined by the Feynman's brackets. Then, we extend these ideas to the dual momentum space by considering noncommutative quantum mechanics. In this context, we show that the noncommutativity of the coordinates is responsible for a new effect called the spin Hall effect. We also show its relation with the Berry phase notion. As a practical application, we found an unusual spin-orbit contribution of a nonrelativistic particle that could be experimentally tested. Another practical application is the Berry phase effect on the propagation of light in inhomogeneous media.