Dirac, Majorana, and Weyl Spinors in Arbitrary Dimension and Signature

Abstract

Fermionic field types in arbitrary signature are often inferred from complex Spin representations alone, which obscures distinctions associated with real structures, disconnected spacetime reflections, and action-level bilinear constraints. We give a convention-explicit classification that treats the full real Clifford algebra and its even subalgebra separately, thereby distinguishing pinors from spinors and Pin-equivariant Majorana-type structures from structures that are only Spin-equivariant. Adjoint, complex-conjugation, and transposition intertwiners are combined with elementary and collective reflection lifts into a unified sign calculus. The resulting local Clifford classification is organised into thirty-two periodicity sectors determined by the Bott class and the total dimension modulo eight, without including theory-dependent gauge, anomaly, or global constraints. For each sector, we identify the available Dirac, Weyl, Majorana, symplectic Majorana, Majorana-Weyl, and symplectic Majorana-Weyl fields. The existence of an invariant pairing is distinguished from intertwiner compatibility, Grassmann and chiral selection rules, covariant index contraction, and Hermiticity of local bilinear operators.