Form Factors from Vertex Operators and Correlation Functions at q=1

TL;DR

This paper applies affine Lie algebras to 2D quantum field theory at the free fermion point, introducing a new momentum space bosonization approach for form factors and analyzing correlation functions using radial quantization.

Contribution

It develops a novel method for computing form factors via vertex operators and explores the structure of the field space at the free fermion point using affine Lie algebra techniques.

Findings

Form factors computed as vacuum expectation values in momentum space.

Radial quantization reveals a quasi-chiral factorization of fields.

Correlation functions analyzed through space-time translational anomalies.

Abstract

We present a new application of affine Lie algebras to massive quantum field theory in 2 dimensions, by investigating the $q\to 1$ limit of the q-deformed affine $\hat{sl(2)}$ symmetry of the sine-Gordon theory, this limit occurring at the free fermion point. We describe how radial quantization leads to a quasi-chiral factorization of the space of fields. The conserved charges which generate the affine Lie algebra split into two independent affine algebras on this factorized space, each with level 1, in the anti-periodic sector. The space of fields in the anti-periodic sector can be organized using level-$1$ highest weight representations, if one supplements the $\slh$ algebra with the usual local integrals of motion. Using the integrals of motion, a momentum space bosonization involving vertex operators is formulated. This leads to a new way of computing form-factors, as vacuum expectation values in momentum space. The problem of non-trivial correlation functions in this model is also discussed; in particular it is shown how space-time translational anomalies which arise in radial quantization can be used to compute the short distance expansion of some simple correlation functions.