Poisson structure and Moyal quantisation of the Liouville theory Original Research Article

The symplectic and Poisson structures of the Liouville theory are derived from the SL(2,R) WZNW theory by gauge invariant Hamiltonian reduction. Causal non-equal time Poisson brackets for a Liouville field are presented. Using the symmetries of the Liouville theory, symbols of local fields are constructed and their ∗-products calculated. Quantum deformations consistent with the canonical quantisation result, and a non-equal time commutator is given.