A quantum variant of Einstein's hole argument

TL;DR

This paper extends Einstein's hole argument into quantum mechanics, highlighting the necessity of a common background for defining quantum gravitational observables involving superpositions, demonstrated through a gravitational double-slit experiment.

Contribution

It introduces a quantum version of Einstein's hole argument, emphasizing the need for a shared background to define quantum gravitational observables in superpositional states.

Findings

Quantum observables require a common background for well-definition.

Transition probabilities can be computed using non-relativistic Schrödinger theory.

The approach applies to gravitational superpositions in simple experimental setups.

Abstract

We extend Einstein's hole argument into the quantum domain, and argue that quantum observables for quasiclassical superpositional states of gravitational fields require additional information to be well-defined, namely, relative positions of the gravitational fields involved in superpositional states. As a consequence, for the definition of these quantum observables we need a common background. The observable is a transition probability in a simple double-slit experiment with partial gravitational measurement of position. It may be easily computed in non-relativistic Schroedinger theory with Newtonian potential.