THE ACTION OPTION AND A FEYNMAN QUANTIZATION OF SPINOR FIELDS IN TERMS OF ORDINARY C-NUMBERS
The Feynman sum represents a convenient formulation of quantum mechanics for Bose fields; but to secure a similar formulation applicable to Fermion fields, it was necessary to use anticommuting c-number field histories to insure the anticommutivity of the quantum field opera tors. A method is presented to sum over histories for spinor fields that employs the familiar classical c-number expression for the action, predicts anticommutation rules and Fermi statistics, and retains the invariance of the theory under a change in phase of the complex psi field. The Feynman procedure demands a numerical action value for histories outside the domain for which the action integral was intended (e.g., for histories that 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 readilty 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 chosen 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 Schroedinger field is presented which includes a detailed account of the necessary properties of the over- complete 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. (auth)
- Research Organization:
- Princeton Univ., N.J.
- NSA Number:
- NSA-15-001969
- OSTI ID:
- 4121087
- Journal Information:
- Annals of Physics (New York) (U.S.), Journal Name: Annals of Physics (New York) (U.S.) Vol. Vol: 11; ISSN APNYA
- Country of Publication:
- Country unknown/Code not available
- Language:
- English
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