Introduction to a Quantum Theory over a Galois Field

TL;DR

This paper introduces a quantum theory based on Galois fields that eliminates infinities and offers a new perspective on particles, antiparticles, and conservation laws, with implications for fundamental physics.

Contribution

It proposes a novel quantum framework over Galois fields, addressing issues like infinities and the cosmological constant problem, and reinterprets particle symmetries and conservation laws.

Findings

Infinities are absent in the Galois field quantum theory.

Particle-antiparticle splitting occurs only approximately at low energies.

Standard quantum properties like spin-statistics are consistent within this framework.

Abstract

We consider a quantum theory based on a Galois field. In this approach infinities cannot exist, the cosmological constant problem does not arise, and one irreducible representation (IR) of the symmetry algebra splits into independent IRs describing a particle an its antiparticle only in the approximation when de Sitter energies are much less than the characteristic of the field. As a consequence, the very notions of particles and antiparticles are only approximate and such additive quantum numbers as the electric, baryon and lepton charges are conserved only in this approximation. There can be no neutral elementary particles and the spin-statistics theorem can be treated simply as a requirement that standard quantum theory should be based on complex numbers.