# Discrete Spacetime Quantum Field Theory

## TL;DR

This paper develops a discrete spacetime quantum field theory, deriving elementary length, symmetry groups, and discrete equations, providing a new framework for understanding particle physics at the Planck scale.

## Contribution

It introduces a discrete spacetime model with a corresponding symmetry group and derives discrete versions of fundamental quantum field equations and interactions.

## Key findings

- Existence of an elementary length close to Planck's length.
- A simplified discrete Lorentz-like symmetry group.
- Discrete Klein-Gordon and Dirac equations derived.

## Abstract

This paper begins with a theoretical explanation of why spacetime is discrete. The derivation shows that there exists an elementary length which is essentially Planck's length. We then show how the existence of this length affects time dilation in special relativity. We next consider the symmetry group for discrete spacetime. This symmetry group gives a discrete version of the usual Lorentz group. However, it is much simpler and is actually a discrete version of the rotation group. From the form of the symmetry group we deduce a possible explanation for the structure of elementary particle classes. Energy-momentum space is introduced and mass operators are defined. Discrete versions of the Klein-Gordon and Dirac equations are derived. The final section concerns discrete quantum field theory. Interaction Hamiltonians and scattering operators are considered. In particular, we study the scalar spin~0 and spin~1 bosons as well as the spin~$1/2$ fermion cases