Journal of Mathematical Physics

Path integral for the radial Dirac equation

Takashi Ichinose

Published Jan 29, 2005• 4 citations

Abstract

For the radial Dirac equation, a countably additive path space measure on the space of continuous paths living on the real half-line is constructed to give a path integral representation of its Green function.

DOI: 10.1063/1.1829374

Related papers

Path Integral for the Dirac Equation

János Polonyi

arXiv:hep-th/9809116Sep 16, 19980 citations

Path Integral Approach to Relativistic Quantum Mechanics

Takashi Ichinose, Hiroshi Tamura

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.

Progress of Theoretical Physics SupplementJan 1, 198713 citations

Path space measure for the 3+1-dimensional Dirac equation in momentum space

Tõru Nakamura

Nonstandard analysis is used to construct a measure over paths for the path integral solution to the Dirac equation in 3+1 dimension. Paths are considered in momentum space, because the Green function in the configuration space contains a derivative of δ function which keeps us from assigning a measure over paths. The solution is obtained not only as the standard part of a nonstandard path sum with respect to a nonstandard measure, but also as a standard path integral with respect to a standard measure extracted from the nonstandard one. The result is an extension of Gaveau’s work [B. Gaveau, J. Funct. Anal. 58, 310–319 (1984)].

Journal of Mathematical PhysicsAug 1, 20006 citations

A functional integral with respect to a countably additive measure representing a solution of the Dirac equation

Nikolaj Nikolaevich Shamarov

SHAMAROV measures of Poisson type constructed in [1] to represent solutions of Schrödinger's equation and the properties of probability distributions on function spaces, generated by homogeneous Markov processes.1

Transactions of the Moscow Mathematical SocietyNov 16, 20052 citations

Path space measures for Dirac and Schrödinger equations: Nonstandard analytical approach

Tõru Nakamura

A nonstandard path space * -measure is constructed to justify the path integral formula for the Dirac equation in two-dimensional space–time. A standard measure as well as a standard path integral is obtained from it. We also show that, even for the Schrödinger equation, for which there is no standard measure appropriate for a path integral, there exists a nonstandard measure to define a * -path integral whose standard part agrees with the ordinary path integral as defined by a limit from time-slice approximant.

Journal of Mathematical PhysicsAug 1, 19977 citations

Phase Space Path Integral for the Dirac Equation

János Polonyi

A c-number path integral representation is constructed for the solution of the Dirac equation. The integration is over the real trajectories in the continuous three-space and other two canonical pairs of compact variables controlling the dynamics of the spin and the chirality flips. The problems to overcome in order to find a c-number path integral representation for the solution of the Dirac equation i¯h∂tΨ = HΨ = [ (−ic¯h ∂ + eA) · α + mc 2 β − eV] Ψ (1) is (i) to generate the first order derivatives in the space coordinates and (ii) to treat the spinor indices. The traditional method to tackle these problems is to use space-time lattice and sum over the lattice trajectories [1], and the application of Grassman variables [2]. The inherent limitation of this solution lies in the difficulty in arriving at the propagation in continuous space and in developing analytic non-perturbative approximations. We present

Sep 16, 19980 citations