Conformally Invariant Quantization of the Liouville Theory

The Liouville theory is quantized with use of Fock-space methods, an infinite set of charges $L_n$, $n=0, \pm 1, \dots$, is constructed which represents the conformal algebra in two dimensions, and consequences of this algebra are discussed. It is then argued, with use of variational methods in Fock space, that the spectrum of the Liouville Hamiltonian is continuous, and that there exist energy eigenstates obeying the constraints $L_n|E〉=0\mathrm{}$, $n>\mathrm{}0\mathrm{}$.