The Pin Groups in Physics: C, P, and T

TL;DR

This paper reviews the mathematical structure of Pin groups, their role in classifying fermions under space and time reversal, and discusses experimental implications and theoretical advantages in physics.

Contribution

It introduces the significance of two distinct Pin groups in physics, providing a framework for fermion classification and clarifying properties under parity and time reversal.

Findings

Pin(3,1) is relevant to neutrinoless double beta decay.

Vacuum expectation values differ for Pin(1,3) and Pin(3,1) on topologically nontrivial spaces.

Pin groups offer a simple framework for studying fermions and defining intrinsic parities.

Abstract

We review the role in physics of the Pin groups, double covers of the full Lorentz group. Pin(1,3) is to O(1,3) what Spin(1,3) is to SO(1,3). The existence of two Pin groups offers a classification of fermions based on their properties under space or time reversal finer than the classification based on their properties under orientation preserving Lorentz transformations -- provided one can design experiments that distinguish the two types of fermions. Many promising experimental setups give, for one reason or another, identical results for both types of fermions. Two notable positive results show that the existence of two Pin groups is relevant to physics: 1) In a neutrinoless double beta decay, the neutrino emitted and reabsorbed in the course of the interaction can only be described in terms of Pin(3,1). 2) If a space is topologically nontrivial, the vacuum expectation values of Fermi currents defined on this space can be totally different when described in terms of Pin(1,3) and Pin(3,1). Possibly more important than the two above predictions, the Pin groups provide a simple framework for the study of fermions; they make possible clear definitions of intrinsic parities and time reversal. A section on Pin groups in arbitrary spacetime dimensions is included.