From Feynman Proof of Maxwell Equations to Noncommutative Quantum Mechanics

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 Poincaré'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.

Presented at the 3rd Feynman Festival (Collage Park, Maryland, U.S.A., August 2006)

References in corpus (33)

• String Theory and Noncommutative Geometry
• Anomalous Hall Effect and Magnetic Monopoles in Momentum-Space
• Berry phase, hyperorbits, and the Hofstadter spectrum: semiclassical dynamics in magnetic Bloch bands
• Quantum mechanics on the noncommutative plane and sphere
• Topological Nature of Anomalous Hall Effect in Ferromagnet
• Berry phase, hyperorbits, and the Hofstadter spectrum
• Introduction to M(atrix) theory and noncommutative geometry
• Introduction to M(atrix) theory and noncommutative geometry, Part II
• Exotic galilean symmetry in the non-commutative plane, and the Hall effect
• Topological spin transport of photons: the optical Magnus Effect and Berry Phase
• The Landau problem and noncommutative quantum mechanics
• Spin Hall effect and Berry phase of spinning particles
• Spin Hall effect of Photons in a Static Gravitational Field
• Fermat Principle for spinning light
• Electron spin operation by electric fields: spin dynamics and spin injection
• Relativistic Generalization and Extension to the Non-Abelian Gauge Theory of Feynman's Proof of the Maxwell Equations
• Monopole and Berry Phase in Momentum Space in Noncommutative Quantum Mechanics
• Semiclassical Dynamics of Electrons in Magnetic Bloch Bands: a Hamiltonian Approach
• Semiclassical Dynamics of Dirac particles interacting with a Static Gravitational Field
• Minimal coupling and Feynman's proof
• Dirac monopole with Feynman brackets
• Feynman problem in the noncommutative case
• Noncommutative Geometry and Geometric Phases
• Gauge Theories and Fibre Bundles - Applications to Particle Dynamics
• Field Theories from the Relativistic Law of Motion
• Dynamical Equations of Spinning Particles: Feynman's Proof
• Non-constant Non-commutativity in 2d Field Theories and a New Look at Fuzzy Monopoles
• Localization of Massless Spinning Particles and the Berry Phase
• Restoration of Angular Lie Algebra Symmetries from a Covariant Hamiltonian
• Angular Symmetry Breaking Induced by Electromagnetic Field
• Classical spinning particle assuming a covariant Hamiltonian
• Dirac Monopole from Lorentz Symmetry in N-Dimensions: I. The Generator Extension
• Dirac Monopole from Lorentz Symmetry in N-Dimensions: II. The Generalized Monopole

Cited by in corpus (6)

• Electrodynamics on $\kappa$-Minkowski space-time
• Geodesic equation in $k$-Minkowski spacetime
• Maxwell's equations on the $\kappa$-Minkowski spacetime and Electric-Magnetic duality
• On the Plethora of Representations Arising in Noncommutative Quantum Mechanics and An Explicit Construction of Noncommutative 4-tori
• Maxwell's equations and Lorentz force in doubly special relativity
• Noncommutative Classical Dynamics on Velocity Phase Space and Souriau Formalism