On Hamiltonian and Quantum Dynamics of Massless Particles

TL;DR

This paper reviews the relativistic dynamics of massless particles, explores Hamiltonian flows on twistor space, and introduces a covariant quantization method, including an example of a quantum harmonic oscillator with spin.

Contribution

It presents a Hamiltonian framework on twistor space for massless particles with arbitrary helicity and proposes a covariant quantization procedure, extending classical dynamics to quantum models.

Findings

Equations of motion are integrable for zero helicity.

Hamiltonian flows reproduce classical massless particle dynamics.

A covariant quantization method is proposed and exemplified.

Abstract

A short review of special relativistic dynamics describing a particle acted upon by an arbitrary conservative external force is presented. If the mass of the particle is zero and the force is central then the equations of motion turn out to be completely integrable. A well-known result. Hamiltonian flows on the twistor phase space T are constructed which, for conservative forces and value of the helicity equal to zero, reproduce equations of motion of the classical massless particle. For helicities different from zero the same hamiltonian flows produce equations of motion showing a curious "Zitterbewegung" like behaviour. A canonical Poincare covariant quantization procedure on T is suggested. One simple example describing a spinning and massless 3-D quantum mechanical harmonic oscillator is analysed in some detail.