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

TL;DR

This work probes the consistency of volume, triad, and flux quantisations in full Loop Quantum Gravity by constructing an alternative flux operator from Poisson brackets with the volume and comparing it to the usual flux operator across two volume regularisations and two canonical configurations. It shows that the Ashtekar–Lewandowski volume yields exact consistency after fixing the regularisation constant to $C_{reg}=\tfrac{1}{48}$, while Rovelli–Smolin is inconsistent; importantly, the spin representation label $\ell$ drops out in the semiclassical limit and the signum operator $\widehat{\cal S}$ is crucial for nontrivial results. This consistency check tightens the quantum foundation of LQG, clarifying the correct triad quantisation and removing regularisation ambiguities, with detailed proofs provided in a companion paper. The results favor the AL-volume framework for implementing quantum dynamics in LQG and illuminate the role of orientation and sign factors in the quantum geometry of volume.

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.

Figures (5)

• Figure 1: Smearing of the surface $S$ into the third dimension. We obtain an array of surfaces $S_t$ labelled by the parameter $t$ with $t\in\{-\epsilon,+\epsilon\}$. The original surface $S$ is associated with $t=0$.
• Figure 2: Edges of type up, down, in and out with respect to the surface $S$.
• Figure 3: Partition ${\cal P}_t$ of the surface $S_t$ into small squares with an parameter edge length $\epsilon'$.
• Figure 4: An non-vaishing contribution to the matrix element of ${}^{ (\ell)}\!\!\:\widehat{\widetilde{E}}_k$ on a given arbitrary SNF $T_{s}$, i.e. $\;< T_{s'}\,|\,{}^{ (\ell)}\!\!\:\widehat{\widetilde{E}}_k(\Box)\,|\, T_s\!\!>$ can only be achieved if $T_s$ contains edges of type up and/or down, respectively with respect to the surface $S_t$. Moreover, the edges $e_{ 3}({ \Box}),e_{ 4}({ \Box})$ have to be attachted to $T_s$ in this specific way.
• Figure 5: The SNF $|\beta^{j_{ 12}}\,{ n_{ 12}} \!\!>$ is transformed into an new SNF $|\alpha^{ J}_{i}\,{ M}\!\!>$ by the action of ${}^{ (\ell)}\!\!\:\widehat{\widetilde{E}}_{k,{ tot}}(S_t)$.