Canonical Quantization of Teukolsky fields on Kerr Background

TL;DR

This paper develops a canonical quantization framework for Teukolsky fields on Kerr spacetime, establishing conditions for consistent commutation relations and deriving the Hadamard parametrix for renormalization.

Contribution

It introduces a novel quantization scheme for Teukolsky fields, linking the Hertz potential and Newman-Penrose scalars via Teukolsky-Starobinsky identities, and provides tools for local renormalization.

Findings

Canonical commutation relations hold when Teukolsky-Starobinsky constants are positive.

Derived the Hadamard parametrix for Teukolsky equations.

Established the relation between Teukolsky field energy and gravitational perturbation energy.

Abstract

Electromagnetic and gravitational perturbations on Kerr spacetime can be reconstructed from solutions to the Teukolsky equations. We study the canonical quantization of solutions to these equations for any integer spin. Our quantization scheme involves the analysis of the Hertz potential and one of the Newman-Penrose scalars, which must be related via the Teukolsky-Starobinsky identities. We show that the canonical commutation relations between the fields can be implemented if and only if the Teukolsky-Starobinsky constants are positive, which is the case both for gravitational perturbations and Maxwell fields. We also obtain the Hadamard parametrix of the Teukolsky equation, which is the basic ingredient for a local and covariant renormalization scheme for non-linear observables. We also discuss the relation of the canonical energy of Teukolsky fields to that of gravitational perturbations.