Nonperturbative quantization approach for QED on the Hopf bundle

TL;DR

This paper develops a nonperturbative quantization method for Dirac and Maxwell fields on a curved spacetime with a Hopf bundle topology, deriving solutions, spectra, and quantum state properties.

Contribution

It introduces a novel nonperturbative quantization scheme for interacting fields on a curved manifold with explicit solutions and operator equations.

Findings

Discrete spectra of classical solutions on the Hopf bundle

Quantization scheme for free Dirac and Maxwell fields

A simplified model illustrating the quantization approach

Abstract

We consider the Dirac equation and Maxwell's electrodynamics in $\mathbb{R} \times S^3$ spacetime, where a three-dimensional sphere is the Hopf bundle $S^3 \rightarrow S^2$. In both cases, discrete spectra of classical solutions are obtained. Based on the solutions obtained, the quantization of free, noninteracting Dirac and Maxwell fields is carried out. The method of nonperturbative quantization of interacting Dirac and Maxwell fields is suggested. The corresponding operator equations and the infinite set of the Schwinger-Dyson equations for Green's functions is written down. To illustrate the suggested scheme of nonperturbative quantization, we write a simplified set of equations describing some physical situation. Also, we discuss the properties of quantum states and operators of interacting fields.