Are the spectra of geometrical operators in Loop Quantum Gravity really discrete?

TL;DR

This paper questions the assumption that geometrical operators in Loop Quantum Gravity have a discrete spectrum by examining gauge invariance issues and showing that kinematical discreteness may not persist at the physical level.

Contribution

It demonstrates that gauge invariance considerations can alter the spectral properties of geometrical operators in simplified models, challenging the notion of fundamental discreteness in LQG.

Findings

Kinematical spectra may not be gauge invariant.

Gauge invariant completion affects the spectrum.

Discreteness at the kinematical level is not guaranteed physically.

Abstract

One of the celebrated results of Loop Quantum Gravity (LQG) is the discreteness of the spectrum of geometrical operators such as length, area and volume operators. This is an indication that Planck scale geometry in LQG is discontinuous rather than smooth. However, there is no rigorous proof thereof at present, because the afore mentioned operators are not gauge invariant, they do not commute with the quantum constraints. The relational formalism in the incarnation of Rovelli's partial and complete observables provides a possible mechanism for turning a non gauge invariant operator into a gauge invariant one. In this paper we investigate whether the spectrum of such a physical, that is gauge invariant, observable can be predicted from the spectrum of the corresponding gauge variant observables. We will not do this in full LQG but rather consider much simpler examples where field theoretical complications are absent. We find, even in those simpler cases, that kinematical discreteness of the spectrum does not necessarily survive at the gauge invariant level. Whether or not this happens depends crucially on how the gauge invariant completion is performed. This indicates that ``fundamental discreteness at Planck scale in LQG'' is far from established. To prove it, one must provide the detailed construction of gauge invariant versions of geometrical operators.