Dipolar quantization and the infinite circumference limit of two-dimensional conformal field theories

TL;DR

This paper introduces dipolar quantization in two-dimensional conformal field theories, achieved by a novel Hamiltonian, leading to an infinite circumference limit and a continuous spectrum, with implications for sine-square deformations.

Contribution

It proposes a new Hamiltonian for 2D CFTs that results in dipolar quantization and an infinite circumference limit, expanding understanding of spectral properties and boundary conditions.

Findings

Continuous and degenerated spectrum in the new theory

Relation between the new Hamiltonian and sine-square deformation

Vacuum states coincide with closed-boundary systems

Abstract

Elaborating on our previous presentation, where the term {\it dipolar quantization} was introduced, we argue here that adopting $L_0-(L_1+L_{-1})/2+{\bar L}_0-({\bar L}_1+{\bar L}_{-1})/2$ as the Hamiltonian instead of $L_0+{\bar L}_0$ yields an infinite circumference limit in two-dimensional conformal field theory. The new Hamiltonian leads to dipolar quantization instead of radial quantization. As a result, the new theory exhibits a continuous and strongly degenerated spectrum in addition to the Virasoro algebra with a continuous index. Its Hilbert space exhibits a different inner product than that obtained in the original theory. The idiosyncrasy of this particular Hamiltonian is its relation to the so-called sine-square deformation, which is found in the study of a certain class of quantum statistical systems. The appearance of the infinite circumference explains why the vacuum states of sine-square deformed systems are coincident with those of the respective closed-boundary systems.