Quantization of Scalar Field Theory with Internal Symmetry

TL;DR

This paper develops a perturbation theory for quantized scalar fields with internal symmetries, analyzing classical configurations and symmetry effects on bound state energies in a one-dimensional model.

Contribution

It introduces a novel perturbation approach using collective coordinates to study quantum properties of scalar fields with internal symmetries.

Findings

Bound state energies are discrete due to symmetry.

The method accurately captures symmetry properties in quantum scalar field models.

Examples include $U(1)$ and $SU(2)$ symmetries.

Abstract

A simple theoretical model of scalar fields in one spatial dimension with internal symmetry is considered. Assuming the existence of localized classical field configurations, the Schr\"{o}dinger picture is used to describe their quantum properties. Using the collective coordinates method for the Schr\"{o}dinger equation allows the development of a perturbation theory that accurately describes the symmetry properties of the theory. Examples of $U(1)$ and $SU (2)$ symmetries are analyzed and the discreteness of the energy of bound states is shown as a result of the symmetry of the theory.