Noncommutative Classical and Quantum Mechanics for Quadratic Lagrangians (Hamiltonians)

TL;DR

This paper develops a formalism for classical and quantum quadratic systems with noncommutative coordinates, transforming noncommutativity into Hamiltonian modifications, and explores a charged particle in a noncommutative plane with magnetic field.

Contribution

It introduces a method to handle noncommutativity in quadratic Hamiltonians by coordinate transformation, simplifying analysis of such systems.

Findings

Exact path integral expression for quadratic models with noncommutativity.

Application to a charged particle in a noncommutative plane with magnetic field.

Introduction of an effective Planck constant dependent on noncommutativity.

Abstract

We consider classical and quantum mechanics for an extended Heisenberg algebra with additional canonical commutation relations for position and momentum coordinates. In our approach this additional noncommutativity is removed from the algebra by linear transformation of coordinates and transmitted to the Hamiltonian (Lagrangian). Since linear transformations do not change the quadratic form of Hamiltonian (Lagrangian), and Feynman's path integral has well-known exact expression for quadratic models, we restricted our analysis to this class of physical systems. The compact general formalism presented here can be easily realized in any particular quadratic case. As an important example of phenomenological interest, we explored model of a charged particle in the noncommutative plane with perpendicular magnetic field. We also introduced an effective Planck constant $\hbar_{eff}$ which depends on noncommutativity.