Real Structures in Clifford Algebras and Majorana Conditions in Any Space-time

TL;DR

This paper systematically analyzes Clifford algebras and Majorana conditions across all spacetime dimensions, classifying admissible free spinor Lagrangians and introducing new structures and invariants relevant to theoretical physics.

Contribution

It introduces an index for inequivalent $ ext{Gamma}$-structures, classifies charge-operators invariant under Wick rotations, and fully characterizes free Majorana-Weyl spinor Lagrangians in various spacetimes.

Findings

Tables of spacetime conditions for non-vanishing kinetic and mass terms

Classification of admissible free Majorana-Weyl spinor Lagrangians

Identification of inequivalent charge-operators in even dimensions

Abstract

Clifford algebras and Majorana conditions are analyzed in any spacetime. An index labeling inequivalent $\Gamma$-structures up to orthogonal conjugations is introduced. Inequivalent charge-operators in even-dimensions, invariant under Wick rotations, are considered. The hermiticity condition on free-spinors lagrangians is presented. The constraints put by the Majorana condition on the free-spinors dynamics are analyzed. Tables specifying which spacetimes admit lagrangians with non-vanishing kinetic, massive or pseudomassive terms (for both charge-operators in even dimensions) are given. The admissible free lagrangians for free Majorana-Weyl spinors are fully classified.