Abstract
A new analysis of the Poisson structure of the Liouville field theory (LFT) in an infinite volume is presented. The second Poisson structure of the Korteweg-de Vries equation is thoroughly investigated as an essential part of the approach, and a variety of correct Poisson brackets is found. This (along with other causes) leads to a variety of correct LFT Poisson structures. Special attention is paid to the most important LFT property of conformal invariance. In particular, a maximal conformal group suitable for the adopted LFT phase spaces is found, and various Hamiltonian representations of the conformal algebra are described. The properties of local commutativity and canonicity are proved.
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