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arXiv:2309.13688

Path integral formalism for the free Dirac propagator in spherical coordinates

Sreya Banerjee, Z. Harman

Published Sep 24, 2023• 0 citations
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Abstract

The relativistic Green's function of a free spin-1/2 fermion is derived using the Feynman path integral formalism in spherical coordinates. The Green's function is reduced to an exactly solvable path integral by an appropriate coordinate transformation. The result is given in terms of spherical Bessel functions and spherical spinors, and agrees with previous solutions of the problem.

DOI
10.48550/arxiv.2309.13688

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The relativistic Green's function of a free spin-1/2 fermion is derived using the Feynman path integral formalism in spherical coordinates. The Green's function is reduced to an exactly solvable path integral by an appropriate coordinate transformation. The result is given in terms of spherical Bessel functions and spherical spinors, and agrees with previous solutions of the problem.
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An exact path-integral solution of the Dirac equation with a Coulomb potential is found. The energy-dependent Green's function of the second-order Dirac-Coulomb equation, expressed as a polar-coordinate path integral, is reduced to the exactly solvable path integral for an isotropic harmonic oscillator by a coordinate transformation combined with a local time rescaling. The Green's function of the linear Dirac equation is evaluated with the help of the Biedenharn transformation. The energy spectrum for the bound states is also obtained.
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In this paper, we investigate a central coordinate system within the framework of the Wigner–Dunkl deformation of quantum mechanics in d spatial dimensions, focusing on the dynamics of a relativistic spinless particle described by the Klein–Gordon equation. The analysis is carried out using Feynman’s path integral formalism in hyperspherical coordinates, where the Lagrangian includes an effective potential depending explicitly only on radial variables. This approach allows for the exact separation of variables in d-dimensional Dunkl space, enabling an analytical evaluation of the path integral. Particular emphasis is placed on the attractive Coulomb-type potential, for which the exact propagator is obtained in closed form. As a result, we derive the energy spectrum and corresponding wave functions of the Dunkl–Klein–Gordon system, expressed in terms of special functions that reflect the modified symmetries induced by the Dunkl operators.
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