Are the spectra of geometrical operators in loop quantum gravity really discrete?

Journal article


Publication Details

Author(s): Dittrich B, Thiemann T
Journal: Journal of Mathematical Physics
Publisher: AMER INST PHYSICS
Publication year: 2009
Volume: 50
Journal issue: 1
ISSN: 0022-2488


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 the Planck scale geometry in LQG is discontinuous rather than smooth. However, there is no rigorous proof thereof at present. Because the aforementioned 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 the Planck scale in LQG" is far from established. To prove it, one must provide the detailed construction of gauge invariant versions of geometrical operators.



FAU Authors / FAU Editors

Thiemann, Thomas Prof. Dr.
Lehrstuhl für Theoretische Physik


How to cite

APA:
Dittrich, B., & Thiemann, T. (2009). Are the spectra of geometrical operators in loop quantum gravity really discrete? Journal of Mathematical Physics, 50(1). https://dx.doi.org/10.1063/1.3054277

MLA:
Dittrich, Bianca, and Thomas Thiemann. "Are the spectra of geometrical operators in loop quantum gravity really discrete?" Journal of Mathematical Physics 50.1 (2009).

BibTeX: 

Last updated on 2018-10-08 at 19:38