Loop groups and quantum fields

TL;DR

This paper reviews how loop group representation theory aids in constructing quantum fields in various models, highlighting new results on the elliptic Calogero-Sutherland system and emphasizing vertex operators in 1+1D quantum theories.

Contribution

It provides a comprehensive survey of applying loop group representations to quantum field models, including new insights into the elliptic Calogero-Sutherland system.

Findings

Construction of quantum fields via vertex operators

Solution of the Luttinger model and related theories

New results on the elliptic Calogero-Sutherland model

Abstract

This article surveys the application of the representation theory of loop groups to simple models in quantum field theory and to certain integrable systems. The common thread in the discussion is the construction of quantum fields using vertex operators. These examples include the construction and solution of the Luttinger model and other 1+1 dimensional interacting quantum field theories, the construction of anyon field operators on the circle, the `2nd quantization' of the Calogero-Sutherland model using anyons and the geometric construction of quantum fields on Riemann surfaces. We describe some new results on the elliptic Calogero-Sutherland model. (This paper was written in 2001, and we want to make a version of it more accessible because it is a reference for us for subsequent work.)