Hidden symmetries in two dimensional field theory

TL;DR

This paper reveals hidden $SU(2) imes SU(2)$ chiral symmetry in a simple two-dimensional massless scalar field, clarifying its relation to fermionic theories and Goldstone bosons using elementary quantum field theory.

Contribution

It simplifies the understanding of symmetries in two-dimensional bosonization, explicitly exposing hidden symmetries with elementary methods.

Findings

Identifies a hidden $SU(2) imes SU(2)$ symmetry in a trivial scalar field theory.

Clarifies the interpretation of the scalar field as a Goldstone boson.

Uses elementary quantum field theory to analyze complex symmetry structures.

Abstract

The bosonization process elegantly shows the equivalence of massless scalar and fermion fields in two space-time dimensions. However, with multiple fermions the technique often obscures global symmetries. Witten's non-Abelian bosonization makes these symmetries explicit, but at the expense of a somewhat complicated bosonic action. Frenkel and Kac have presented an intricate mathematical formalism relating the various approaches. Here I reduce these arguments to the simplest case of a single massless scalar field. In particular, using only elementary quantum field theory concepts, I expose a hidden $SU(2)\times SU(2)$ chiral symmetry in this trivial theory. I then discuss in what sense this field should be interpreted as a Goldstone boson.