Regularization as quantization in reducible representations of CCR

TL;DR

This paper introduces a covariant quantization scheme using reducible CCR representations, which naturally regularizes quantum electrodynamics, avoids infrared divergences, and maintains gauge invariance, offering a new perspective on quantum field quantization.

Contribution

It proposes a novel reducible representation-based quantization scheme for QED that regularizes infrared issues and preserves covariance and gauge invariance.

Findings

Poisson photon statistics naturally emerge

No infrared divergence for pointlike sources

Classical fields obtained via coherent-state averages

Abstract

A covariant quantization scheme employing reducible representations of canonical commutation relations with positive-definite metric and Hermitian four-potentials is tested on the example of quantum electrodynamic fields produced by a classical current. The scheme implies a modified but very physically looking Hamiltonian. We solve Heisenberg equations of motion and compute photon statistics. Poisson statistics naturally occurs and no infrared divergence is found even for pointlike sources. Classical fields produced by classical sources can be obtained if one computes coherent-state averages of Heisenberg-picture operators. It is shown that the new form of representation automatically smears out pointlike currents. We discuss in detail Poincar\'e covariance of the theory and the role of Bogoliubov transformations for the issue of gauge invariance. The representation we employ is parametrized by a number that is related to R\'enyi's $\alpha$. It is shown that the ``Shannon limit" $\alpha\to 1$ plays here a role of correspondence principle with the standard regularized formalism.