Second-Order Moment Quantum Fluctuations and Quantum Equivalence Principle

TL;DR

This paper explores how second-order quantum fluctuations relate to the equivalence principle, distinguishing dynamic and geometric parts, and proposes a geometric-only quantum gravity model consistent with general covariance.

Contribution

It introduces a novel decomposition of second-order quantum fluctuations into dynamic and geometric parts, linking the geometric part to an effective gravity theory and extending the equivalence principle to quantum regimes.

Findings

Dynamic part is mass-dependent and coordinate-dependent, can be canceled by coordinate transformation.

Geometric part is mass-independent, universal, and leads to an effective Einstein gravity.

Geometric fluctuations induce coordinate transformation anomalies, suggesting a quantum origin of gravity.

Abstract

The second-order moment quantum fluctuations or uncertainties are mass-dependent, and the incompatibility between the quantum uncertainty principle and the equivalence principle is at the second-order moment (variation) level, but not the first-order moment (mean) level. To reconcile the two fundamental principles, we find that the second-order moment quantum fluctuations are actually distinguished into two parts: a dynamic part and a geometric part. The dynamic part is indeed mass-dependent and governed by a non-zero Hamiltonian in a non-general-covariant inertial frame, and the geometric part is mass-independent and comes from coarse-graining and/or geometric effects. The dynamic part is coordinate dependent, it can be canceled away by a coordinate transformation, and hence it plays no role in general covariant theories whose Hamiltonian automatically vanishes. However, the geometric part is valid for general coordinate, and it can not be eliminated by a coordinate transformation. On the contrary, the geometric part of second-order moment fluctuation of quantum spacetime leads to coordinate transformation anomaly, which induces an effective Einstein's gravity theory. The geometric part is mass-independent and universal, so it is only this part measures the universal second-order moment quantum fluctuation of the spacetime, while the dynamic part plays no role in the general covariant description. The observation generalizes the classical equivalence principle to the quantum level. And according to the principle, a general covariant theory with only geometric part quantum fluctuation, i.e. a non-linear sigma model, is proposed as a theory of a material quantum reference frame system. The effects of the universal second-order moment quantum fluctuations in the material quantum reference system and its implications to an effective gravity theory are also discussed.