Polymer Quantization of a Self-Gravitating Thin Shell

TL;DR

This paper explores polymer quantization of a self-gravitating thin shell, revealing unique features like consistent quantization for super-Planckian black holes, negative energy states, and probability density behavior near collapse.

Contribution

It introduces a polymer quantization approach to self-gravitating shells, enabling analysis of super-Planckian black holes and uncovering novel quantum states and boundary conditions.

Findings

Consistent quantization for super-Planckian black holes.

Existence of negative energy stationary states.

Probability density develops negative regions near collapse.

Abstract

We study the quantum mechanics of self-gravitating thin shell collapse by solving the polymerized Wheeler-DeWitt equation. We obtain the energy spectrum and solve the time dependent equation using numerics. In contradistinction to the continuum theory, we are able to consistently quantize the theory for super-Planckian black holes, and find two choices of boundary conditions which conserve energy and probability, as opposed to one in the continuum theory. Another feature unique to the polymer theory is the existence of negative energy stationary states that disappear from the spectrum as the polymer scale goes to zero. In both theories the probability density is positive semi-definite only for the space of positive energy stationary states. Dynamically, we find that an initial Gaussian probability density develops regions of negative probability as the wavepacket approaches $R=0$ and bounces. This implies that the bouncing state is a sum of both positive and negative eigenstates.