P(4) affine and superhamiltonian formulations of charged particle dynamics

Abstract

We establish a correspondence between the recently proposedP(4) affine and the standard superhamiltonian descriptions of the electrodynamics of classical charged particles. TheP(4) theory uses a generalized affine connection on the affine frame bundleA(M) over spacetime, and an affine connection is induced on phase space thought of as the vector bundleT ^* M. On the phase space manifoldT ^* M this affine structure defines a covariant canonical symplectic form, which, when coupled with the canonical free-particle superhamiltonian, reproduces the Lorentz force law for classical charged particles. Conversely, one may “split” the noncanonical symplectic form onT ^* M to define an affine connection onA(M) and thus return to theP(4) theory from symplectic geometry. The correspondence also allows a geometrization of superhamiltonian dynamics. Roughly speaking, the symplectic form onT ^* M is geometrized as anR ⁴-affine connection onA(M), and the superhamiltonian is geometrized as an affine difference function on the local momentum-energy tangent affine spaces.