A groupoidal description of elementary particles

TL;DR

This paper extends the classification of elementary particles to curved space-times using groupoids, generalizing Wigner's approach and introducing a new family of representations related to magnetic backgrounds.

Contribution

It introduces a groupoid-based framework for classifying elementary particles in curved space-times, extending Mackey's theory to this setting.

Findings

Classifies elementary particles via irreducible projective representations of kinematical groupoids.

Recovers Wigner's classification in Minkowski space-time.

Identifies a new family of representations for massless particles with magnetic-like backgrounds.

Abstract

In this work, we show that extending the standard description of space-time symmetries from groups of isometries to the more flexible framework of kinematical groupoids allows for the extension of Wigner's program to curved space-times. We propose a new definition of elementary particles as irreducible projective representations of the kinematical groupoids supporting the theory. By choosing a natural kinematical groupoid associated with any space-time, called the \textit{Wigner groupoid}, we demonstrate that such irreducible projective representations are characterized by quantum numbers similar to those characterizing the irreducible projective representations of the Poincar\'e group. Describing the irreducible projective representations of groupoids poses its own difficulties. To address this, we develop a suitable extension of Mackey's theory of induced representations of groups, proving that projective representations of transitive Lie groupoids with connected isotropy groups are in one-to-one correspondence with the projective representations of their isotropy groups. The application of these results provides a classification of elementary particles valid for a large class of space-times. This classification largely reproduces Wigner's standard classification on Minkowski space-time, while a new family of representations emerges, corresponding to massless particles in the presence of a magnetic-like background field.