Structure, classifcation, and conformal symmetry, of elementary particles over non-archimedean space-time

TL;DR

This paper explores how the structure, classification, and conformal symmetry of elementary particles are affected when spacetime is modeled over non-archimedean fields like p-adic numbers, using a new representation theory approach.

Contribution

It introduces a novel variant of the Mackey machine for projective unitary representations of semidirect product groups over non-archimedean fields, analyzing particle symmetries in p-adic spacetime.

Findings

Conformal symmetry cannot be maintained for massive particles over p-adic spacetime.

Constructs conformal spacetime over p-adic fields.

Extends group representation theory to non-archimedean geometries.

Abstract

It is known that no length or time measurements are possible in sub-Planckian regions of spacetime. The Volovich hypothesis postulates that the micro-geometry of spacetime may therefore be assumed to be non-archimedean. In this letter, the consequences of this hypothesis for the structure, classification, and conformal symmetry of elementary particles, when spacetime is a flat space over a non-archimedean field such as the $p$-adic numbers, is explored. Both the Poincar\'e and Galilean groups are treated. The results are based on a new variant of the Mackey machine for projective unitary representations of semidirect product groups which are locally compact and second countable. Conformal spacetime is constructed over $p$-adic fields and the impossibility of conformal symmetry of massive and eventually massive particles is proved.