Angular Quantization in CFT

TL;DR

This paper introduces an angular quantization scheme for 2D conformal field theories, offering an alternative to radial quantization, with explicit constructions and potential applications in string theory and entanglement entropy analysis.

Contribution

It develops the theory of angular quantization for 2D CFTs, including explicit Fock space constructions and discusses its relation to boundary conditions and entanglement entropy.

Findings

Provides explicit Fock space constructions for free 2D CFTs.

Shows angular quantization as an alternative to radial quantization.

Discusses applications in string theory and entanglement entropy.

Abstract

The most common quantization scheme in which to study a conformal field theory is radial quantization, wherein a Hilbert space of states is defined on a sphere, whose Hamiltonian when mapped to the plane is the dilatation generator and which boasts a state/operator correspondence. In this paper, we consider an alternative quantization scheme for 2d CFTs in which the plane is foliated by constant-angle slices, as opposed to concentric circles, whose Hamiltonian is the rotation generator. In this angular quantization, there is no state/operator correspondence but instead an "asymptotics/operator correspondence". A central feature is that the quantization slice ends on two operators, and a regulator must be chosen by excising holes around each operator and imposing suitable boundary conditions such that the holes shrink to the desired local operators. This angular quantization may be viewed as constructing CFTs in Minkowski space, or separately as studying non-conformal (but local) boundary conditions in CFTs. We provide explicit Fock space constructions for various free 2d CFTs. In addition to the motivation of applying angular quantization in string theory situations where traditional radial quantization is insufficient, we comment on its relation to modular Hamiltonian approaches to entanglement entropy.