Quantum Hamiltonian Reduction of the Schwinger Model

TL;DR

This paper investigates the quantum Hamiltonian reduction method in gauge theories, focusing on the Schwinger model, and highlights how regularization affects the equivalence with classical reduction, which can be restored through redefinitions.

Contribution

It demonstrates the impact of regularization on quantum Hamiltonian reduction and shows how to reconcile quantum and classical results via redefinitions in the Schwinger model.

Findings

Regularization alters the quantum reduction Hamiltonian.

Redefinitions of fermion currents restore agreement with classical reduction.

Quantum and classical reductions are consistent after proper adjustments.

Abstract

We reexamine a unitary-transformation method of extracting a physical Hamiltonian from a gauge field theory after quantizing all degrees of freedom including redundant variables. We show that this {\it quantum Hamiltonian reduction} method suffers from crucial modifications arising from regularization of composite operators. We assess the effects of regularization in the simplest gauge field theory, the Schwinger model. Without regularization, the quantum reduction yields the identical Hamiltonian with the classically reduced one. On the other hand, with regularization incorporated, the resulting Hamiltonian of the quantum reduction disagrees with that of the classical reduction. However, we find that the discrepancy is resolved by redefinitions of fermion currents and that the results are again consistent with those of the classical reduction.