Erlangen 2022 – wissenschaftliches Programm

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A: Fachverband Atomphysik

A 32: Precision spectroscopy of atoms and ions IV (joint session A/Q)

A 32.3: Vortrag

Freitag, 18. März 2022, 11:15–11:30, A-H2

Path integral formalism of Dirac propagators for atomic physics — •Sreya Banerjee and Zoltán Harman — Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany

The very basic building blocks of perturbative calculations in atomic structure and collision theory are Green’s functions. We extend this study of Green’s functions, in the nonperturbative regime, using Feynman’s path integral approach. As a first step, we derive the free Dirac propagator followed by the derivation of the Dirac-Coulomb Green’s function (DCGF) in spherical coordinates, using this formalism.

For the free relativistic Dirac particle, the effective Hamiltonian for the iterated Dirac equation is constructed. The corresponding Green’s function is expanded into partial waves in spherical coordinates. The radial part of this Green’s function is then converted into a path integral, through reparametrisation of the paths by local time rescaling, followed by a one-to-one mapping of the radial variable with the local time parameter. This yields a closed form of the Green’s function. Following the same procedure, the DCGF is diagonalised in Biedenharn’s basis into a radial path integral, the effective action of which is similar to that of the non-relativistic hydrogen atom. We convert the radial path integral from Coulomb type to that of an isotropic harmonic oscillator through coordinate transformation along with local time rescaling. As such, an explicit path integral representation of the DCGF is obtained, along with the energy spectrum of the bound states.