Boundary Liouville Theory: Hamiltonian Description and Quantization

TL;DR

This paper develops a Hamiltonian framework for classical and quantum Liouville field theory on a timelike strip, deriving classical solutions, their quantization, and constructing quantum operators, aligning with previous spectral data.

Contribution

It provides a comprehensive Hamiltonian and quantization approach for Liouville theory on a timelike strip, including classical solutions and quantum operator construction.

Findings

Classical solutions with boundary conditions and monodromy properties identified.

Quasiclassical energy spectrum matches previous conformal bootstrap results.

Quantum vertex operator constructed explicitly in the hyperbolic sector.

Abstract

The paper is devoted to the Hamiltonian treatment of classical and quantum properties of Liouville field theory on a timelike strip in 2d Minkowski space. We give a complete description of classical solutions regular in the interior of the strip and obeying constant conformally invariant conditions on both boundaries. Depending on the values of the two boundary parameters these solutions may have different monodromy properties and are related to bound or scattering states. By Bohr-Sommerfeld quantization we find the quasiclassical discrete energy spectrum for the bound states in agreement with the corresponding limit of spectral data obtained previously by conformal bootstrap methods in Euclidean space. The full quantum version of the special vertex operator $e^{-\phi}$ in terms of free field exponentials is constructed in the hyperbolic sector.