The action option and a Feynman quantization of spinor fields in terms of ordinary c-numbers

Abstract

The Feynman sum represents a convenient formulation of quantum mechanics for Bose fields, but, to secure a similar formulation applicable to Fermion fields, it has been necessary to use “anticommuting c-number” field histories to insure the anticommutivity of the quantum field operators. Here, a method is presented to sum over histories for spinor fields which (1) employs the familiar classical c-number expression for the action, (2) predicts anti-commutation rules and Fermi statistics, and (3) retains the invariance of the theory under a change in phase of the complex ψ field. The Feynman procedure demands a numerical action value for histories outside the domain for which the action integral was intended, for example, for histories which are discontinuous with respect to space or time. One is therefore presented with an “action option”, i.e., the action value for such “unruly” histories may be defined in various ways. Depending on the choice made, the resulting quantum theory can be made to manifest either Bose or Fermi statistics. This ambiguity is inherent in the formalism itself. However, the proper choice to extend the classical information is most readily determined by constructing the sum over histories by a summation over multiple products of matrix elements of the unitary operator which advances the state an infinitesimal time. This summation need not be limited to the familiar discrete basis vectors; instead a “generalized representation” can be employed which involves, for each Fermion degree of freedom, continuously many, nonindependent vectors. When a suitable parameterization is chose for this “overcomplete family of states” the multiple product of matrix elements for a given history reduces to the exponential of the appropriate action functional evaluated for that history. A unified formulation of both statistics for the Schrödinger field is presented which includes a detailed account of the necessary properties of the overcomplete family of states and a derivation of the functional measure for Fermion fields. The propagator and a functional expression for the ground state of the neutrino field are presented as applications of the method to relativistic spinor fields.