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

TL;DR

The paper investigates whether the discrete spectra of Loop Quantum Gravity geometric operators at the kinematical level endure once promoted to gauge-invariant, Dirac observables via Rovelli's partial and complete observables. Using simple toy models with one or two constraints, it shows that discreteness can disappear or reappear depending on the gauge-invariant completion and the choice of clock, highlighting the crucial role of the global structure of the reduced phase space. The results demonstrate that kinematical discreteness does not automatically imply physical discreteness, underscoring the need for explicit gauge-invariant constructions of geometric operators in LQG. This has important implications for claims about Planck-scale discreteness in quantum gravity and motivates further study of gauge-invariant geometric observables in more realistic settings.

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.