Principles of Discrete Time Mechanics: IV. The Dirac Equation, Particles and Oscillons

TL;DR

This paper extends discrete time mechanics to the Dirac equation, deriving new oscillatory solutions called oscillons, which influence particle interactions but are not observable as particles in continuous time.

Contribution

It introduces oscillons as novel solutions to the discrete time Dirac equation and analyzes their properties and implications for quantum field theory.

Findings

Found new oscillon solutions oscillating with twice the fundamental period.

Oscillons can have definite charge, momentum, and spin but are not observable in continuous time.

Oscillons correspond to states with negative squared norm, affecting particle interactions.

Abstract

We apply the principles of discrete time mechanics discussed in earlier papers to the first and second quantised Dirac equation. We use the Schwinger action principle to find the anticommutation relations of the Dirac field and of the particle creation operators in the theory. We find new solutions to the discrete time Dirac equation, referred to as oscillons on account of their extraordinary behaviour. Their principal characteristic is that they oscillate with a period twice that of the fundamental time interval T of our theory. Although these solutions can be associated with definite charge, linear momentum and spin, such objects should not be observable as particles in the continuous time limit. We find that for non-zero T they correspond to states with negative squared norm in Hilbert space. However they are an integral part of the discrete time Dirac field and should play a role in particle interactions analogous to the role of longitudinal photons in conventional quantum electrodynamics.