Criteria for Exact Solubility of Relativistic Field Theories by Scattering Transform

TL;DR

This paper establishes criteria for the exact solvability of relativistic field theories using scattering transforms, extending the method beyond models with Lax pairs by leveraging Lorentz invariance and algebraic properties.

Contribution

It introduces a new framework for constructing scattering transforms directly from relativistic invariance, identifying specific models like sine-Gordon and Massive Thirring as quantisable.

Findings

Scattering transforms can be built from Lorentz invariance and algebraic closure.

Quantisable models are restricted to sine-Gordon and Massive Thirring types under certain assumptions.

Potential extensions involve chirality in the target space of the scattering transform.

Abstract

Scattering transform is a well known powerful tool for quantisation of field theories in (1+1) dimensions. Conventionally only those models whose classical counterparts admit a Lax pair (origin of which is always mysterious) have been quantised in this way. In relativistic quantum field theories we show that the scattering transforms can be constructed ab initio from its invariance under Lorentz transformation (both proper and improper), irreducible transformation nature of scalar and Dirac fields, the existence of a momentum scale associated with asymptotic nature of the scattering transform and the closure of short distance operator product algebra. For single fields it turns out that theories quantisable by scattering transforms are restricted to sine-Gordon type for spin-0 and Massive Thirring type for spin-1/2 if the target space of the scattering transform matrix is assumed to be parity invariant. There are interesting unexplored extensions if the target space is given chirality.