Consistency check on volume and triad operator quantization in loop quantum gravity: I

Journal article

Publication Details

Author(s): Giesel K, Thiemann T
Journal: Classical and Quantum Gravity
Publication year: 2006
Volume: 23
Journal issue: 18
Pages range: 5667-5691
ISSN: 0264-9381


The volume operator plays a pivotal role for the quantum dynamics of loop quantum gravity (LQG). It is essential to construct triad operators that enter the Hamiltonian constraint and which become densely defined operators on the full Hilbert space, even though in the classical theory the triad becomes singular when classical GR breaks down. The expression for the volume and triad operators derives from the quantization of the fundamental electric flux operator of LQG by a complicated regularization procedure. In fact, there are two inequivalent volume operators available in the literature and, moreover, both operators are unique only up to a finite, multiplicative constant which should be viewed as a regularization ambiguity. Now on the one hand, classical volumes and triads can be expressed directly in terms of fluxes and this fact was used to construct the corresponding volume and triad operators. On the other hand, fluxes can be expressed in terms of triads and triads can be replaced by Poisson brackets between the holonomy and the volume operators. Therefore one can also view the holonomy operators and the volume operator as fundamental and consider the flux operator as a derived operator. In this paper we mathematically implement this second point of view and thus can examine whether the volume, triad and flux quantizations are consistent with each other. The results of this consistency analysis are rather surprising. Among other findings we show the following. ( 1) The regularization constant can be uniquely fixed. ( 2) One of the volume operators can be ruled out as inconsistent. ( 3) Factor ordering ambiguities in the definition of triad operators are immaterial for the classical limit of the derived flux operator. The results of this paper show that within full LQG triad operators are consistently quantized.

FAU Authors / FAU Editors

Giesel, Kristina Prof. Dr.
Professur für Theoretische Physik
Thiemann, Thomas Prof. Dr.
Chair for Theoretical Physics III (Quantum Gravity)

How to cite

Giesel, K., & Thiemann, T. (2006). Consistency check on volume and triad operator quantization in loop quantum gravity: I. Classical and Quantum Gravity, 23(18), 5667-5691.

Giesel, Kristina, and Thomas Thiemann. "Consistency check on volume and triad operator quantization in loop quantum gravity: I." Classical and Quantum Gravity 23.18 (2006): 5667-5691.


Last updated on 2018-06-07 at 01:23