Proper time Quantization of a Thin Shell

TL;DR

This paper compares quantum theories of a thin dust shell built on proper time versus coordinate time, finding that proper time quantization aligns better with physical expectations and offers a more consistent approach.

Contribution

It demonstrates that proper time quantization of a thin shell yields physically consistent solutions, unlike coordinate time quantization, highlighting its importance in quantum gravity models.

Findings

Proper time quantization admits solutions for shell mass above the Planck mass.

Coordinate time quantization admits no solutions for shell mass above the Planck mass.

Proper time approach aligns better with observational expectations.

Abstract

"Time" has different meanings in classical general relativity and in quantum theory. While all choices of a time function yield the same local classical geometries, quantum theories built on different time functions are not unitarily equivalent. This incompatibility is most vivid in model systems for which exact quantum descriptions in different time variables are available. One such system is a spherically symmetric, thin dust shell. In this essay we will compare the quantum theories of the shell built on proper time and on a particular coordinate time. We find wholly incompatible descriptions: whereas the shell quantum mechanics in coordinate time admits no solutions when the mass is greater than the Planck mass, its proper time quantum mechanics only admits solutions when the mass is greater than the Planck mass. The latter is in better agreement with what is expected from observation. We argue that proper time quantization provides a superior approach to the problem of time in canonical quantization.