Spinor Structure and Quantum Mechanics from Relativistic Mass-Shell Factorisation in Phase Space

Abstract

Spinor structure is usually introduced as part of relativistic quantum theory. We show that its underlying algebra can instead emerge without quantisation from a covariant statistical description of a massive relativistic particle where we require the theory retain both sheets of the massive mass shell. A finite-dimensional factorisation of the complete mass-shell constraint, linear in all four momentum components, forces a Clifford algebra. In (3+1) dimensions its minimal complex representation is four-dimensional, while each on-shell factor selects a rank-two sector; statistical completeness therefore requires an arbitrary $2\times 2$ matrix distribution within that sector before quantisation. Projecting the Weyl-ordered matrix Liouvillian gives relativistic transport while preserving all internal populations and coherences. For universal deformations on flat canonical phase space satisfying the stated covariance, matrix-neutrality, homogeneity and associativity assumptions, the resulting product is the matrix Weyl--Moyal product. Its parameter has dimensions of action and, when identified with $\hslash$, supplies the scale of physical spin-$\frac{1}{2}$. Requiring all-order constraint preservation gives a vanishing star commutator, while preservation of the selected star-spectral sector yields the two-sided Dirac--Wigner equations. Our central conclusion is therefore that relativistic mass-shell factorisation supplies spinor structure, while completion of the constraint-preservation hierarchy leads to the Weyl--Moyal phase-space formulation of quantum mechanics.

13 pages. Major revision and substantial expansion into a self-contained two-part treatment. Part I derives Clifford algebra and rank-two spinor structure from complete relativistic mass-shell factorisation. Part II completes the constraint-preservation hierarchy, deriving the matrix Moyal product and its star-eigenvalue equations and then recovering the Dirac-Wigner equations