Quantization of Carrollian conformal scalar theories

TL;DR

This paper investigates the quantization of Carrollian conformal scalar theories, comparing two schemes, and explores their symmetry properties, correlation functions, and the applicability of the state-operator correspondence.

Contribution

It provides a detailed analysis of two quantization schemes for Carrollian conformal scalars, highlighting their differences in unitarity, symmetry realization, and correlation functions.

Findings

Canonical quantization yields a unitary Hilbert space and matching correlators with path integral results.

Highest-weight vacuum quantization is non-unitary and breaks rotational symmetry in 3D.

The usual state-operator correspondence does not hold in Carrollian conformal theories.

Abstract

In this work, we study the quantization of Carrollian conformal scalar theories, including two-dimensional(2D) magnetic scalar and three-dimensional(3D) electric and magnetic scalars. We discuss two different quantization schemes, depending on the choice of the vacuum. We show that the standard canonical quantization corresponding to the induced vacuum yields a unitary Hilbert space and the 2-point correlation functions in this scheme match exactly with the ones computed from the path integral. In the canonical quantization, the BMS symmetry can be realized without anomaly. On the other hand, for the quantization based on the highest-weight vacuum, it does not have a unitary Hilbert space. In 2D, the correlators in the highest-weight vacuum agree with the ones obtained by taking the $c\to 0$ limit of the 2D CFT, and there is an anomalous term in the commutation relations between the Virasoso generators, whose form is similar to the one in 2D CFT. In 3D, there is no good definition of the highest-weight vacuum without breaking the rotational symmetry. In our study, we find that the usual state-operator correspondence in CFT does not hold in the Carrollian case.