Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

M. Berg, C. DeWitt-Morette, S. Gwo, E. Kramer

TL;DR

The paper elucidates the existence of two inequivalent Pin groups, Pin(1,3) and Pin(3,1), as nontrivial double covers of the full Lorentz group and shows how they refine fermion classification beyond orientation-preserving transformations. It develops both finite- and infinite-dimensional representations (pinors vs spinors), their adjoints, and bundle-theoretic formulations, including Majorana pinors and charge conjugation within the Pin framework. While many observables agree across Pin types, the authors identify concrete contexts—such as neutrinoless double beta decay and vacuum currents on topologically nontrivial spaces—where Pin(1,3) vs Pin(3,1) yield distinct predictions, and they extend the discussion to arbitrary spacetime dimensions and string-theoretic implications. The work provides a practical, structured toolkit (definitions, representations, observables, and topological considerations) for applying Pin-group concepts in high-energy physics, quantum field theory, and beyond.

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.

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

TL;DR

The paper elucidates the existence of two inequivalent Pin groups, Pin(1,3) and Pin(3,1), as nontrivial double covers of the full Lorentz group and shows how they refine fermion classification beyond orientation-preserving transformations. It develops both finite- and infinite-dimensional representations (pinors vs spinors), their adjoints, and bundle-theoretic formulations, including Majorana pinors and charge conjugation within the Pin framework. While many observables agree across Pin types, the authors identify concrete contexts—such as neutrinoless double beta decay and vacuum currents on topologically nontrivial spaces—where Pin(1,3) vs Pin(3,1) yield distinct predictions, and they extend the discussion to arbitrary spacetime dimensions and string-theoretic implications. The work provides a practical, structured toolkit (definitions, representations, observables, and topological considerations) for applying Pin-group concepts in high-energy physics, quantum field theory, and beyond.

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.

Paper Structure

This paper contains 38 sections, 286 equations, 14 figures, 5 tables.

Table of Contents

  1. Dictionary of Notation
  2. Introduction
  3. Background
  4. As seen by physicists
  5. As seen by Wigner
  6. As seen by mathematicians
  7. The Pin groups in 3 space, 1 time dimensions
  8. The Pin groups
  9. A Spin group is a subgroup of a Pin group
  10. Pin group and Spin group representations on finite dimensional spaces. Classical fields
  11. Pin and Spin representations on infinite-dimensional spaces. Quantum fields.
  12. Bundles; Fermi fields on manifolds
  13. Bundle reduction
  14. Search for Observable Differences
  15. Computing observables with Pin(1,3) and Pin(3,1)
  16. ...and 23 more sections

Figures (14)

  • Figure 1: Components of the Lorentz group
  • Figure 2: Double cover of the Lorentz group
  • Figure 3: Spins do not change under reflection.
  • Figure 4: Tree-level QED diagram for $\Sigma^0 \rightarrow\Lambda^0 + e^+ + e^-$.
  • Figure 5: A type of experiment which should give different results for the two types of pinors.
  • ...and 9 more figures