Quasi-Linear Quantum Field Theories for Maps to Groups and Their Quotients

Abstract

I describe a functional integral for maps from \(\mathbb{R}\times \mathbb{R}^{\rm n}\) to a Lie group or its quotient which has a simple renormalization that leads to a quantum field theory for maps from \(\mathbb{R}^{\rm n}\) into the Lie group or its quotient whose Hamiltonian is the time translation generator for a unitary action of the n+1 dimensional Poincare group on the quantum Hilbert space. I also explain how the renormalization provides a functional integral for maps from a Riemann surface to a compact Lie group or its quotient that exhibits many conformal field theoretic properties.