From Mass-Shell Factorisation to Spin: An Attempt at a Matrix-Valued Liouville Framework for Relativistic Classical and Quantum Phase-Spacetime

Abstract: While Liouville's theorem is first-order in time for the phase-space distribution itself, the relativistic mass-shell constraint $p^\mu p_\mu = m^2$ is naively second-order in energy. We argue that it is reasonable to unify both energy branches within a single Hamiltonian by factorizing $(p^2 - m^2)$ in analogy with Dirac's approach in relativistic quantum mechanics. We show the resulting matrix-based Liouville equation remains first order and naturally yields a $4\times4$ matrix-valued probability density function in phase space as a classical analogue of a relativistic spin-half Wigner function. We investigate its classical physics and deformation quantisation.