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arXiv:0709.0816 · Beyond the Quantum

NONCOMMUTATIVE QUANTUM MECHANICS VIEWED FROM FEYNMAN FORMALISM

José Lages, Alain Bérard, Hervé Mohrbach, Yves Grandati, P. GOSSELIN

Published Sep 1, 2007• 2 citations
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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 Poincaré'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.

DOI
10.1142/9789812771186_0019

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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 Poincaré'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.
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.
We consider Feynman's path integral approach to quantum mechanics with a noncommutativity in position and momentum sectors of the phase space. We show that a quantum-mechanical system with this kind of noncommutativity is equivalent to the another one with usual commutative coordinates and momenta. We found connection between quadratic classical Hamiltonians, as well as Lagrangians, in their commutative and noncommutative regimes. The general procedure to compute Feynman's path integral on this noncommutative phase space with quadratic Lagrangians (Hamiltonians) is presented. Using this approach, a particle in a constant field, ordinary and inverted harmonic oscillators are elaborated in detail.
Classical and quantum mechanics based on an extended Heisenberg algebra with additional canonical commutation relations for position and momentum coordinates are considered. In this approach additional noncommutativity is removed from the algebra by a linear transformation of coordinates and transferred to the Hamiltonian (Lagrangian). This linear transformation does not change the quadratic form of the Hamiltonian (Lagrangian), and Feynman’s path integral preserves its exact expression for quadratic models. The compact general formalism presented here can be easily illustrated in any particular quadratic case. As an important result of phenomenological interest, we give the path integral for a charged particle in the noncommutative plane with a perpendicular magnetic field. We also present an effective Planck constant ħ eff which depends on additional noncommutativity.
In order to evaluate the Feynman path integral in noncommutative quantum mechanics, we consider properties of a Lagrangian related to a quadratic Hamiltonian with noncommutative spatial coordinates. A quantum-mechanical system with noncommutative coordinates is equivalent to another one with commutative coordinates. We found connection between quadratic classical Lagrangians of these two systems. We also shown that there is a subclass of quadratic Lagrangians, which includes harmonic oscillator and particle in a constant field, whose connection between ordinary and noncommutative regimes can be expressed as a linear change of position in terms of a new position and velocity.
A paper by Dyson, published nearly two decades ago, describing Feynman’s proof of Maxwell equations, has generated many comments, analysis, discussions and generalizations of the proof. Feynman’s derivation is assumed to be based on two main sets of equations. One is supposed to be the second law of Newton and the other a set of basic commutation relations from quantum physics.Here we present a new comment on this paper, focusing mainly on the initial arguments and applying a new method of analysis and interpretation of physics, named observational realism. The present discussion does not alter the technical steps of Feynman, but do clarify their basis. We show that Newton’s physics is not a starting point in Feynman’s derivation, neither is quantum physics involved in it, but the foundations of relativity only.
Starting from square-integrable wave functions on a Lie group, we build an invertible Fourier transform mapping them on wave functions on the dual of the Lie algebra. This is a group-theoretic version of the map from position space to momentum space, with generally noncommuting momenta owing to the group structure. As a result, the multiplication of momentum-dependent functions involves star products, which makes the construction of noncommutative Fourier series much more involved than that of their commutative cousin. This is especially true when compact subgroups are present, in which case we carefully take into account quotients of the operator algebra, and the resulting normalization issues. We show that our formalism provides an isometry of Hilbert spaces, and use it to derive a noncommutative Poisson summation formula for any compact Lie group. This is a key preliminary for the computation of Wigner functions and path integrals for quantum systems on group manifolds.