Classical and Quantum Physical Geometry

TL;DR

This paper explores the intersection of classical and quantum geometry, proposing a quantum principle of equivalence and general covariance, and suggesting that space-time points lack invariant meaning in quantum gravity.

Contribution

It introduces a quantum principle of equivalence, extends Einstein's hole argument to quantum gravity, and proposes quantum diffeomorphisms as fundamental symmetries.

Findings

Quantum wave free fall suggests a quantum equivalence principle

Sign change of Fermion fields implies supersymmetry in quantum geometry

Quantum diffeomorphisms lead to a background-independent quantum gravity framework

Abstract

The task of quantizing gravity is compared with Einstein's relativization of gravity. The philosophical and physical foundations of general relativity are briefly reviewed. The Ehlers-Pirani-Schild scheme of operationally determining the geometry of space-time, using freely falling classical particle trajectories, is done using operations in an infinitesimal neighborhood around each point. The study of the free fall of a quantum wave suggests a quantum principle of equivalence. The principle of general covariance is clarified. The sign change of a Fermion field when rotated by $2\pi$ radians is used to argue for a quantum mechanical modification of space-time, which leads naturally to supersymmetry. A novel effect in quantum gravity due to the author is used to extend Einstein's hole argument to quantum gravity. This suggests a quantum principle of general covariance, according to which the fundamental laws of physics should be covariant under `quantum diffeomorphisms'. This heuristic principle implies that space-time points have no invariant meaning in quantum gravity.