Consistency Check on Volume and Triad Operator Quantisation in Loop Quantum Gravity I

TL;DR

This paper investigates the consistency of volume, triad, and flux operator quantizations in Loop Quantum Gravity, revealing that some operators are inconsistent and fixing ambiguities, thus ensuring the internal consistency of the theory.

Contribution

It provides a systematic consistency check of different operator quantizations in LQG, fixing regularisation ambiguities and ruling out inconsistent operators.

Findings

The regularisation constant can be uniquely fixed.

One volume operator is inconsistent and can be discarded.

Factor ordering ambiguities do not affect the classical limit.

Abstract

The volume operator plays a pivotal role for the quantum dynamics of Loop Quantum Gravity (LQG). It is essential in order 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 quantisation of the fundamental electric flux operator of LQG by a complicated regularisation 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 regularisation 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 therefore one can also view the volume operator as fundamental and consider the flux operator as a derived operator. In this paper we examine whether the volume, triad and flux quantisations are consistent with each other. The results of this consistency analysis are rather surprising. Among other findings we show: 1. The regularisation 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. In this paper we present ideas and results of the consistency check. In a companion paper we supply detailed proofs.