Worldline Supersymmetry and Dimensional Reduction

For any worldline reformulation of a quantum field theory for Dirac fermions, this paper shows that worldline supersymmetry may generally be enforced by the vanishing of the commutator of the Dirac operator with the worldline Hamiltonian. The action of supersymmetry on the worldline Lagrangian may not, however, be written in terms of the variations on the fields in the usual way, except when the spinning particle couples just to a one-form. By reduction from six to four dimensions of the worldline reformulation for a spinning particle coupled to a three-form, corrections to the superworldline Lagrangian are presented which are needed in order to reproduce correct field theory results from worldline perturbation theory in an unambiguous way.