Spin, Statistics, and Reflections, II. Lorentz Invariance

TL;DR

This paper extends the connection between modular P$_1$CT-symmetry and the spin-statistics relation to Lorentz-invariant models, using reflection groups and modular conjugations to verify the relation in a universal covering of the Lorentz group.

Contribution

It introduces a Lorentz-invariant model of the universal covering group using reflection groups and constructs a representation satisfying the spin-statistics relation.

Findings

Extended the modular P$_1$CT-symmetry analysis to Lorentz-invariant settings.

Constructed a model of the universal covering of the Lorentz group as a reflection group.

Verified the spin-statistics relation within this new Lorentz-invariant framework.

Abstract

The analysis of the relation between modular P$_1$CT-symmetry -- a consequence of the Unruh effect -- and Pauli's spin-statistics relation is continued. The result in the predecessor to this article is extended to the Lorentz symmetric situation. A model $\G_L$ of the universal covering $\widetilde{L_+^\uparrow}\cong SL(2,\complex)$ of the restricted Lorentz group $L_+^\uparrow$ is modelled as a reflection group at the classical level. Based on this picture, a representation of $\G_L$ is constructed from pairs of modular P$_1$CT-conjugations, and this representation can easily be verified to satisfy the spin-statistics relation.