Spinor Structure from Relativistic Mass-Shell Factorisation in Phase Space

Abstract

Spin is usually regarded as one of the most intrinsically quantum phenomena, while its lack of a natural analogue in classical physics presents an obstacle to phase-space deformation-quantisation accounts of its origin. We show that this difficulty may arise from imposing an overly restrictive scalar Hamiltonian structure on relativistic phase space. Requiring the complete massive mass-shell constraint to be represented by a single finite-dimensional expression that is linear in all four components of momentum forces its coefficient matrices to satisfy a Clifford algebra. The minimal complex representation of this algebra is four-dimensional. Requiring statistical completeness within the rank-two subspace selected by the mass-shell factor then leads to a four-by-four matrix-valued ensemble distribution. At each on-shell momentum, the linear mass-shell operator selects a two-dimensional subspace, so a general classical ensemble is described by a $2 \times 2$ matrix before quantisation. Projecting the Weyl-ordered Liouville equation into this subspace gives relativistic transport while preserving arbitrary populations and coherences and the remaining projector components determine the first deformation correction. Expansion of the matrix Moyal star commutator yields the symmetrised classical matrix Liouvillian at leading order. If the stronger two-sided star constraints are imposed, they reproduce the left and right Dirac-Wigner equations and introduce $\hbar$ as the scale converting the dimensionless internal algebra into physical angular momentum. These results suggest a non-quantum origin for spinor structure and a route to reconciling relativistic covariance, classical phase-space transport, and quantum spin.