© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

Phase Space Path Integral for the Dirac Equation

János Polonyi

Published Sep 16, 1998• 0 citations

Abstract

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

Related papers

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 spin and the chirality flips.

arXiv:hep-th/9809116•Sep 16, 1998•0 citations

Spinor chain path integral for the Dirac equation

Ted Jacobson A

A path integral that reduces to Feynman's checkerboard rule in one space dimension is found for the retarded Dirac propagator in three space dimensions. The only variable are two-component spinors and a binary chirality variable. No action functional is employed. Each spinor together with a chirality corresponds to a spacetime displacement during a time epsilon . A sequence of spinors and chiralities determines a polygonal spacetime path, the first and last spinors specify the initial and final spin states. The transition amplitude for a sequence is given by ( nu N mod nu N-1 )( nu N-1 mod . . . mod nu 2 )( nu 2 mod nu 1 )(i epsilon m) R where nu i are the spinors, < mod > is the ordinary inner product in spin space, m is the electron mass and R is the number of times the chirality switches. Integrating over all sequences corresponding to a given displacement yields the Dirac propagator in the limit epsilon to 0. With epsilon >0 this formulation provides an alternative to the point particle model of the electron. In an external electromagnetic potential A mu the amplitude for a path C is multiplied by exp(-ie integral c A mu dx mu ), requiring spacetime coordinates to specify A mu (x). The usual perturbation expansion is derived from this rule. These results are extended to nonAbelian gauge potentials. Quantised interaction of particles is not treated.

Journal of Physics A Mathematical and General•Aug 21, 1984•47 citations

Path integral representation of the evolution operator for the Dirac equation

Alexander S. Lukyanenko, Inna Lukyanenko

A path integral representation of the evolution operator for the four-dimensional Dirac equation is proposed. A quadratic form of the canonical momenta regularizes the original representation of the path integral in the electron phase space. This regularization allows one to obtain a representation of the path integral over trajectories in the configuration space. This form of the path integral is useful for the formulation of perturbation theory in an external electromagnetic field.

arXiv:gr-qc/0605129•Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE•Feb 23, 2006•0 citations

Propagation of a Dirac particle. A path integral approach

Takashi Ichinose, Hiroshi Tamura

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.

Journal of Mathematical Physics•Jun 1, 1984•32 citations

Trajectory construction of Dirac evolution

Peter Holland

We extend our programme of representing the quantum state through exact stand-alone trajectory models to the Dirac equation. We show that the free Dirac equation in the angular coordinate representation is a continuity equation for which the real and imaginary parts of the wave function, angular versions of Majorana spinors, define conserved densities. We hence deduce an exact formula for the propagation of the Dirac spinor derived from the self-contained first-order dynamics of two sets of trajectories in 3-space together with a mass-dependent evolution operator. The Lorentz covariance of the trajectory equations is established by invoking the ‘relativity of the trajectory label'. We show how these results extend to the inclusion of external potentials. We further show that the angular version of Dirac's equation implies continuity equations for currents with non-negative densities, for which the Dirac current defines the mean flow. This provides an alternative trajectory construction of free evolution. Finally, we examine the polar representation of the Dirac equation, which also implies a non-negative conserved density but does not map into a stand-alone trajectory theory. It reveals how the quantum potential is tacit in the Dirac equation.

arXiv:1910.05829•Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences•Apr 1, 2020•2 citations

Path integral for the radial Dirac equation

Takashi Ichinose

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.

Journal of Mathematical Physics•Jan 29, 2005•4 citations

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

A path integral that reduces to Feynman's checkerboard rule in one space dimension is found for the retarded Dirac propagator in three space dimensions. The only variable are two-component spinors and a binary chirality variable. No action functional is employed. Each spinor together with a chirality corresponds to a spacetime displacement during a time epsilon . A sequence of spinors and chiralities determines a polygonal spacetime path, the first and last spinors specify the initial and final spin states. The transition amplitude for a sequence is given by ( nu N mod nu N-1 )( nu N-1 mod . . . mod nu 2 )( nu 2 mod nu 1 )(i epsilon m) R where nu i are the spinors, < mod > is the ordinary inner product in spin space, m is the electron mass and R is the number of times the chirality switches. Integrating over all sequences corresponding to a given displacement yields the Dirac propagator in the limit epsilon to 0. With epsilon >0 this formulation provides an alternative to the point particle model of the electron. In an external electromagnetic potential A mu the amplitude for a path C is multiplied by exp(-ie integral c A mu dx mu ), requiring spacetime coordinates to specify A mu (x). The usual perturbation expansion is derived from this rule. These results are extended to nonAbelian gauge potentials. Quantised interaction of particles is not treated.

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.

We extend our programme of representing the quantum state through exact stand-alone trajectory models to the Dirac equation. We show that the free Dirac equation in the angular coordinate representation is a continuity equation for which the real and imaginary parts of the wave function, angular versions of Majorana spinors, define conserved densities. We hence deduce an exact formula for the propagation of the Dirac spinor derived from the self-contained first-order dynamics of two sets of trajectories in 3-space together with a mass-dependent evolution operator. The Lorentz covariance of the trajectory equations is established by invoking the ‘relativity of the trajectory label'. We show how these results extend to the inclusion of external potentials. We further show that the angular version of Dirac's equation implies continuity equations for currents with non-negative densities, for which the Dirac current defines the mean flow. This provides an alternative trajectory construction of free evolution. Finally, we examine the polar representation of the Dirac equation, which also implies a non-negative conserved density but does not map into a stand-alone trajectory theory. It reveals how the quantum potential is tacit in the Dirac equation.