Geometrical Incompatibility of Quantum Measurements

TL;DR

This paper explores how quantum measurements of geometrical quantities in noncommutative geometry can lead to incompatible descriptions of space-time, highlighting fundamental issues in quantizing gravity.

Contribution

It introduces operator analogues for geometric quantities and demonstrates how different measurement procedures yield non-equivalent space-time geometries.

Findings

Quantum measurements can produce incompatible geometrical descriptions.

Operator analogues for curvature and Einstein tensor are proposed.

A concrete example illustrates measurement-induced paradoxes.

Abstract

The problem of quantization of general relativity is considered in the framework of noncommutative differential geometry. Operator analogues for interval, scalar curvature, values of the Einstein tensor are proposed. Quantum measurements of these observables lead to a paradox: different procedures of measurements can supply non equivalent geometrical pictures of space-time. A concrete example of such situation is provided.