Canonical Quantization Inside the Schwarzschild Black Hole

TL;DR

This paper develops a method for quantizing a scalar field within the Schwarzschild black hole, including the interior, by leveraging symmetries and CPT invariance to define positive frequency modes and propagators.

Contribution

It introduces a novel quantization scheme for the black hole interior that ensures positive definiteness and constructs the propagator as a mode sum, extending previous exterior-focused approaches.

Findings

Quantization scheme valid inside and outside the horizon

Explicit construction of the propagator as a mode sum

Use of CPT invariance to define positive frequencies inside the horizon

Abstract

We propose a scheme for quantizing a scalar field over the Schwarzschild manifold including the interior of the horizon. On the exterior, the timelike Killing vector and on the horizon the isometry corresponding to restricted Lorentz boosts can be used to enforce the spectral condition. For the interior we appeal to the need for CPT invariance to construct an explicitly positive definite operator which allows identification of positive and negative frequencies. This operator is the translation operator corresponding to the inexorable propagation to smaller radii as expected from the classical metric. We also propose an expression for the propagator in the interior and express it as a mode sum.