The Liouville Field Theory Zero-Mode Problem

We quantize the canonical free-field zero modes $p$, $q$ on the half-plane $p>0$ for both Liouville field theory and its reduced Liouville particle dynamics. We describe the particle dynamics in detail, calculate one-point functions of particle vertex operators, deduce their zero-mode realization on the half-plane, and prove that the particle vertex operators act self-adjointly on the Hilbert space $L^2(\mathbb{R}_+)$ because of symmetries generated by the $S$-matrix. Similarly, we obtain the self-adjointness of the corresponding Liouville field theory vertex operator in the zero-mode sector by applying the Liouville reflection amplitude, which is derived by the operator method.