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Progress of Theoretical Physics Supplement

Path Integral Approach to Relativistic Quantum Mechanics

Takashi Ichinose, Hiroshi Tamura

Published Jan 1, 1987• 13 citations
Publisher

Abstract

Countably additive path space measures are constructed, in two space-time dimensions, to give rigorous path integral formulas representing the fundamental solution of the Cauchy problem for the Dirac equation as well as the retarded and advanced propagator for the Dirac particle. The theory also applies to a free particle, a particle in a central electric field and a particle in parallel electric and uniform magnetic fields in four-dimensional space-time.

DOI
10.1143/ptps.92.144

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Proceedings of the Japan Academy Series A Mathematical Sciences•Jan 1, 1982•28 citations
Countably additive path space measures are constructed, in two space-time dimensions, to give rigorous path integral formulas representing the fundamental solution of the Cauchy problem for the Dirac equation as well as the retarded and advanced propagator for the Dirac particle. The theory also applies to a free particle, a particle in a central electric field and a particle in parallel electric and uniform magnetic fields in four-dimensional space-time.
Rigorous path integral formulas are given which represent, in two space-time dimensions, the fundamental solution of the Cauchy problem for the Dirac equation as well as the retarded and advanced propagators for the Dirac particle. It is also shown that the theory can be applied to a free particle and a particle in a central electric field in four space-time dimensions and reveals some aspects of the path integral. Heuristically discussed is the connection with the phase space path integral or Hamiltonian path integral.
The feasibility of applying the path-integral formalism for solving the Dirac equation is shown in the case of a free particle for which the Dirac propagator is obtained by evaluating an appropriate path integral, directly constructed from the Dirac equation. Furthermore, the propagator for a Dirac electron in a constant magnetic field is indirectly obtained from the propagator of an auxiliary wave equation by evaluating a world-line (space-time path) integral. The spectrum of the Dirac equation is also, in this case, extracted from an auxiliary propagator.
A nonstandard path space is constructed that gives the mathematically rigorous formulation for the path integral representation of the fundamental solution to the Cauchy problem for the Dirac equation in (1+1)-dimensional space-time. Nonstandard analysis makes the mathematical concepts elementary, consequently, the procedures to prove theorems are considerably simplified. A difference scheme with infinitesimal spacing is available in determining the probability distribution over the path-space and this method is also available for the heat equation and for the free Schrödinger equation.
We construct generalized quantum Cauchy pre-measures that correspond to the analytic continuation of the transition probability of the Cauchy process to imaginary time. We show that these complex pre-measures of time translations extend to a measure on the space dual to a real Hilbert space whose support is locally compact in the uniform convergence topology and with velocities in the Hilbert space. At that the quantum Cauchy-Dirac and Cauchy-Maxwell pre-measures of time translations of electrons and photons, that correspond to the retarded Green's functions of the Dirac and Maxwell equations with no sources viewed as generalized functions on bump functions, are unitary equivalent to quantum Cauchy pre-measures Therefore these pre-measures on the space dual to a real Hilbert space are σσσ-additive as well, but their support on the electron (respectively, photon) trajectories are compact in the uniform convergence topology with velocities in the Hilbert space. We study the way the classical relativistic mechanics of particle comes from the quantum mechanics of the free Dirac particle.