[Paper Review] Group Theoretical Description of Space Inversion, Time Reversal and Charge Conjugation

В. В. Варламов|ArXiv.org|Mar 28, 2002

Algebraic and Geometric Analysis100 references4 citations

TL;DR

This paper presents a group-theoretical framework using Clifford algebras and Pin groups to consistently describe space inversion (P), time reversal (T), and charge conjugation (C) symmetries across all spin fields. By employing automorphisms of the Lorentz and Poincaré groups and analyzing quotient structures in Clifford algebras, it shows that the Maxwell field arises naturally as a composite of Weyl spinors, supporting a unified description of electromagnetic fields via the de Broglie–Jordan neutrino theory of light and Majorana–Oppenheimer electrodynamics.

ABSTRACT

A group theoretical description of basic discrete symmetries (space inversion P, time reversal T and charge conjugation C) is given. Discrete subgroups of orthogonal groups of multidimensional spaces over the fields of real and complex numbers are considered in terms of fundamental automorphisms of Clifford algebras. In accordance with a division ring structure, a complete classification of automorphism groups is established for the Clifford algebras over the fields of real and complex numbers. Finite-dimensional representations of the proper orthochronous Lorentz group are studied in terms of spinor representations of the Clifford algebras. Real, complex, quaternionic and octonionic representations of the Lorentz group are considered. The Atiayh-Bott-Shapiro periodicity is defined on the Lorentz group. Quotient representations of the Lorentz group are introduced. It is shown that quotient representations are the most suitable for description of massless physical fields. An algebraic construction of basic physical fields is presented.

Motivation & Objective

• To develop a consistent, unified group-theoretical description of discrete symmetries (P, T, C) across all spin fields, overcoming limitations in standard relativistic wave equation approaches.
• To resolve ambiguities in higher-spin field descriptions by using automorphisms of the Lorentz and Poincaré groups, particularly through the use of Clifford–Lipschitz and Da̧browski groups.
• To demonstrate that the electromagnetic field (Maxwell field) has a composite structure built from Weyl spinors, supporting the de Broglie–Jordan neutrino theory of light.
• To show that transversely polarized photons emerge naturally from composite neutrino fields, satisfying Lipkin statistics rather than Bose statistics, and to reconcile this with experimental observations.
• To unify Majorana–Oppenheimer electrodynamics and the neutrino theory of light through a consistent group-theoretical framework based on Clifford algebra automorphisms.

Proposed method

• Utilizes Clifford algebras $\mathbb{C}\ell_{p,q}$ and their associated Clifford–Lipschitz groups $\mathbf{Pin}(p,q)$ to construct double coverings of orthogonal groups $O(p,q)$, enabling a systematic treatment of discrete symmetries.
• Applies fundamental automorphisms of Clifford algebras, particularly the pseudoautomorphism $\mathcal{A} \mapsto \widetilde{\mathcal{A}}$, to model charge conjugation and relate it to complex conjugation in spinor representations.
• Employs the semidirect product structure $G_0 \odot \{1, P, T, PT\ olimits$ to describe the full Lorentz group, with discrete transformations realized as outer automorphisms.
• Constructs quotient representations via the homomorphism $\epsilon: \mathbb{C}_{n+1} \to \mathbb{C}_n$ to analyze discrete symmetries in odd-dimensional Clifford algebras and many-body systems.
• Uses the Atiyah–Bott–Shapiro periodicity theorem to relate representations of the Lorentz group to spinor fields and to derive the structure of physical fields in terms of irreducible representations.
• Analyzes the transition from spinor fields to 3D vector fields via symmetric tensor products $\operatorname{Sym}_{(2,0)} \cup \operatorname{Sym}_{(0,2)}$, linking them to the Helmholtz–Silberstein representation $\mathbf{F} = \mathbf{E} + i\mathbf{H}$.

Experimental results

Research questions

• RQ1 How can discrete symmetries P, T, and C be consistently described across all spin fields using group-theoretical methods, especially beyond the $j=1/2$ case?
• RQ2 What is the role of automorphisms of the Lorentz and Poincaré groups in realizing discrete symmetries, and how do they unify different field types?
• RQ3 Can the electromagnetic field be derived as a composite of Weyl spinors through group-theoretical means, supporting the neutrino theory of light?
• RQ4 How do the properties of transversely polarized photons emerge from composite neutrino fields, and what statistics do they obey?
• RQ5 What is the mathematical and physical significance of the $\mathbf{Pin}(p,q)$ groups with different signatures $(a,b,c)$ in describing discrete symmetries?

Key findings

• The full representation space $\mathbb{S}_4 \cup \dot{\mathbb{S}}_4$ reduces to a 3-dimensional symmetric space $\operatorname{Sym}_{(2,0)} \cup \operatorname{Sym}_{(0,2)}$ under the action of the pseudoautomorphism $\mathcal{A} \mapsto \widetilde{\mathcal{A}}$, enabling a transition from spinors to 3D vectors.
• The Maxwell field $ (1,0) \cup (0,1) $ is shown to be isomorphic to $ (1/2,0) \otimes (1/2,0) \cup (0,1/2) \otimes (0,1/2) $, demonstrating its composite structure from Weyl spinors.
• The field $\widetilde{\psi}$ derived from $\widetilde{\phi}$ via projection maps to a 3D vector $\begin{pmatrix} 0 \\ \overset{*}{F}_1 \\ \overset{*}{F}_2 \\ \overset{*}{F}_3 \end{pmatrix}$, confirming the emergence of electromagnetic fields from spinor components.
• Transversely polarized photons arise naturally from composite neutrino fields, satisfying Lipkin statistics rather than Bose statistics, and are identified as quasi-bosons.
• The Pryce Theorem’s assumption of incompatibility between neutrino-based photons and Bose statistics is shown to be unsupported; the group-theoretical framework allows consistent construction of such fields.
• A synthesis of Majorana–Oppenheimer electrodynamics and the de Broglie–Jordan neutrino theory of light is achieved, where the electromagnetic field is fundamentally constructed from spinor fields via tensor products and direct sums.