Reconcile Planck-scale discreteness and the Lorentz-Fitzgerald contraction

TL;DR

This paper demonstrates that in loop quantum gravity, the minimal observable length remains invariant under Lorentz boosts, resolving apparent conflicts between quantum discreteness and Lorentz invariance by analyzing the transformation of the area operator.

Contribution

It shows that the minimal area in loop quantum gravity is invariant under Lorentz boosts, with the probability distribution changing rather than the eigenvalues.

Findings

The minimal area eigenvalues are invariant under boosts.

Boosts change the probability distribution of measuring minimal area.

A proposed generator of local boosts acts unitarily under certain conditions.

Abstract

A Planck-scale minimal observable length appears in many approaches to quantum gravity. It is sometimes argued that this minimal length might conflict with Lorentz invariance, because a boosted observer could see the minimal length further Lorentz contracted. We show that this is not the case within loop quantum gravity. In loop quantum gravity the minimal length (more precisely, minimal area) does not appear as a fixed property of geometry, but rather as the minimal (nonzero) eigenvalue of a quantum observable. The boosted observer can see the same observable spectrum, with the same minimal area. What changes continuously in the boost transformation is not the value of the minimal length: it is the probability distribution of seeing one or the other of the discrete eigenvalues of the area. We discuss several difficulties associated with boosts and area measurement in quantum gravity. We compute the transformation of the area operator under a local boost, propose an explicit expression for the generator of local boosts and give the conditions under which its action is unitary.