Zero Mode Problem of Liouville Field Theory

TL;DR

This paper investigates the quantization of zero modes in Liouville field theory and its reduced particle dynamics, establishing self-adjointness of vertex operators and analyzing their properties on a half-plane.

Contribution

It provides a detailed analysis of zero mode quantization, computes one-point functions, and demonstrates self-adjointness of vertex operators using the reflection amplitude.

Findings

Self-adjointness of particle vertex operators established.

Zero mode realisation on the half-plane derived.

Liouville reflection amplitude computed via operator method.

Abstract

We quantise canonical free-field zero modes $p$, $q$ on a half-plane $p>0$ both, for the 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 realisation on the half-plane, and prove that the particle vertex operators act self-adjointly on a Hilbert space $L^2(\rr_+)$ on account of symmetries generated by the $S$-matrix. Similarly, self-adjointness of the corresponding vertex operator of Liouville field theory in the zero mode sector is obtained by applying the Liouville reflection amplitude, which is derived by the operator method.