Discrete phase space - II: The second quantization of free relativistic wave fields

TL;DR

This paper develops a second quantization framework for free relativistic wave fields within a discrete phase space, deriving key equations and quantization relations, and confirming their consistency with continuous space-time formulations.

Contribution

It introduces a second quantization approach for relativistic fields in discrete phase space, extending traditional methods to a new discrete setting.

Findings

Exact computation of total momentum, energy, and charge matches continuous space-time results.

Formulation of Klein-Gordon, Maxwell, and Dirac equations as difference equations in discrete phase space.

Quantization relations are established for scalar, electromagnetic, and spin-1/2 fields.

Abstract

The Klein-Gordon equation, the Maxwell equation, and the Dirac equation are presented as partial difference equations in the eight-dimensional covariant discrete phase space. These equations are also furnished as difference-differential equations in the arena of discrete phase space and continuous time. The scalar field and electromagnetic fields are quantized with commutation relations. The spin-1/2 field is quantized with anti-commutation relations. Moreover, the total momentum, energy and charge of these free relativisitic quantized fields in the discrete phase space and continuous time are computed exactly. The results agree completely with those computed from the relativisitic fields defned on the space-time continuum.