Classical scattering in Liouville field theory

A detailed description of classical scattering in Liouville field theory (LFT) is presented. Contrary to widespread belief, LFT scattering is shown to be non-trivial, although it is finite-dimensional in some sense. In particular, for certain phase spaces, the LFT S-matrix is represented as a transformation of the Poisson group SL(2,R). The completeness and conformal invariance of the scattering are also indicated. Singular fields are treated on an equal footing with regular ones, except that only the latter are given a consistent Hamiltonian interpretation. A number of unexpected peculiarities of LFT scattering are revealed. First, for some exceptional field configurations, the asymptotic fields are not solutions of d'Alembert's equation, rather they are a sum of the d'Alembert and Liouville components. Second, the scattering occurs in 'two or three spaces'. And last, depending on the choice of the algebra of observables, the conventional splitting of the d'Alembert field into leftward and rightward components is either in general impossible or essentially non-unique.