Gauge-invariant Hamiltonian formulation of Maxwell-Dirac field equations

Abstract We present a Poisson structure on the basic space of gauge-invariant variables for spinor electrodynamics. The coupling of the matter and radiation appears in the Poisson structure rather than the energy-momentum tensor. Poincare symmetries are realized as canonical transformations. In particular, we obtain a mainfestly gauge-invariant Liouville equation for the density matrix of spinor fields, which exhibits the nonlocal electromagnetic effects explicitly. By letting the Compton wavelength of the matter go to zero, we obtain the Vlasov equation for a polarized relativistic plasma.