The Raychaudhuri equation for a quantized timelike geodesic congruence

TL;DR

This paper critically examines a recent proposal for a quantum Raychaudhuri equation, identifies its limitations, and suggests a limited domain where a quantum version can be formulated to explore singularities in quantum gravity.

Contribution

It analyzes the problems in existing quantum Raychaudhuri equations and proposes a salvageable form applicable in certain scenarios, ensuring probability conservation in arbitrary dimensions.

Findings

Identifies inherent issues in previous quantum Raychaudhuri formulations.

Shows the possibility of a limited domain for a consistent quantum version.

Ensures probability conservation through a self-adjoint Hamiltonian extension.

Abstract

A recent attempt to arrive at a quantum version of Raychaudhuri's equation is looked at critically. It is shown that the method, and even the idea, has some inherent problems. The issues are pointed out here. We have also shown that it is possible to salvage the method in some limited domain of applicability. Although no generality can be claimed, a quantum version of the equation should be useful in the context of ascertaining the existence of a singularity in the quantum regime. The equation presented in the present work holds for arbitrary $n + 1$ dimensions. An important feature of the Hamiltonian in the operator form is that it admits a self-adjoint extension quite generally. Thus, the conservation of probability is ensured.