A natural extension of the conformal Lorentz group in a field theory context

TL;DR

This paper explores a finite-dimensional algebra related to quantum field theory, revealing an extension of the conformal Lorentz group that could aid non-perturbative quantum field theory construction without supersymmetry.

Contribution

It introduces a new algebraic structure whose automorphism group extends the conformal Lorentz group and models particle-antiparticle creation operators, offering a novel approach to quantum field theory.

Findings

Automorphism group relates to conformal Lorentz group

Non-semisimple automorphisms model state dressing

Potential for non-perturbative QFT construction

Abstract

In this paper a finite dimensional unital associative algebra is presented, and its group of algebra automorphisms is detailed. The studied algebra can physically be understood as the creation operator algebra in a formal quantum field theory at fixed momentum for a spin 1/2 particle along with its antiparticle. It is shown that the essential part of the corresponding automorphism group can naturally be related to the conformal Lorentz group. In addition, the non-semisimple part of the automorphism group can be understood as "dressing" of the pure one-particle states. The studied mathematical structure may help in constructing quantum field theories in a non-perturbative manner. In addition, it provides a simple example of circumventing Coleman-Mandula theorem using non-semisimple groups, without SUSY.