Constrained Hamiltonian Formulation of Kinetic Theory for Relativistic Spin-$1/2$ Particles

Abstract

Describing spin transport away from equilibrium requires a kinetic theory that consistently incorporates spin dynamics and phase space structure. We construct such a collisionless kinetic theory for relativistic spin-$1/2$ particles in external electromagnetic fields using the constrained Hamiltonian formalism. The formulation yields both a covariant kinetic equation and its form in independent physical phase space variables, together with the associated noncanonical Poisson brackets and invariant measure. For massive particles, the interplay between the spin constraints and the electromagnetic field makes the particle velocity generally nonparallel to its kinetic momentum and modifies the momentum and spin evolution. The invariant measure in the reduced phase space obeys an exact Liouville identity even for time-dependent fields. This identity and the equations of motion give local conservation of charge and Boltzmann entropy, as well as balance equations for energy-momentum and total angular momentum. The global equilibrium conditions are consistent with covariant statistical mechanics and the Wigner function approach. The massless construction recovers the standard chiral kinetic equations and phase space measure. This framework provides a consistent Hamiltonian description of spin transport beyond equilibrium.