Discrete Scalar Quantum Field Theory

TL;DR

This paper develops a discrete scalar quantum field theory on a spacetime lattice, exploring particle interactions, scattering, and decay processes, while addressing the challenges of unitarity and approximation methods.

Contribution

It introduces a novel discrete spacetime framework for scalar quantum fields, defining interaction operators and scattering processes with new approximation techniques.

Findings

Discrete set of particle masses derived from lattice structure

Approximate scattering operators for fundamental processes

Defined cross-sections, decay rates, and lifetimes in the discrete model

Abstract

We begin with a description of spacetime by a 4-dimensional cubic lattice $\sscript$. It follows from this framework that the the speed of light is the only nonzero instantaneous speed for a particle. The dual space $\sscripthat$ corresponds to a cubic lattice of energy-momentum. This description implies that there is a discrete set of possible particle masses. We then define discrete scalar quantum fields on $\sscript$. These fields are employed to define interaction Hamiltonians and scattering operators. Although the scattering operator $S$ cannot be computed exactly, approximations are possible. Whether $S$ is unitary is an unsolved problem. Besides the definitions of these operators, our main assumption is conservation of energy-momentum for a scattering process. This article concludes with various examples of perturbation approximations. These include simplified versions of electron-electron and electron-proton scattering as well as simple decay processes. We also define scattering cross-sections, decay rates and lifetimes within this formalism.