Criteria for Exact Solubility of Relativistic Field Theories by Scattering Transform

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.