Light-Cone Quantization a of Scalar Field on Time-Dependent Backgrounds

TL;DR

This paper explores light-cone quantization of scalar fields in curved spacetimes, emphasizing the importance of null hypersurfaces and scalar densities for consistent quantization, and connects it with particle second quantization.

Contribution

It clarifies the proper conditions for light-cone quantization on curved backgrounds and introduces the use of scalar densities instead of scalar fields for consistent formulation.

Findings

Proper light-cone quantization requires null hypersurfaces.

Scalar fields must be treated as scalar densities in this context.

The approach aligns with second quantization of particles on curved backgrounds.

Abstract

We discuss what is light-cone quantization on a curved spacetime also without a null Killing vector. Then we consider as an example the light-cone quantization of a scalar field on a background with a Killing vector and the connection with the second quantization of the particle in the same background. It turns out that the proper way to define the light-cone quantization is to require that the constant light-cone time hypersurface is null or, equivalently, that the particle Hamiltonian is free of square roots. Moreover, in order to quantize the scalar theory it is necessary to use not the original scalar rather a scalar field density, i.e. the Schr\"odinger wave functional depends on a scalar density and not on the original field. Finally we recover this result as the second quantization of a particle on the same background, where it is necessary to add as input the fact that we are dealing with a scalar density.