On (Cosmological) Singularity Avoidance in Loop Quantum Gravity

TL;DR

This paper critically assesses whether Loop Quantum Cosmology’s (LQC) singularity-resolution features extend to full Loop Quantum Gravity (LQG). It shows that the LQC-like boundedness of the inverse scale factor does not generalize to full LQG, where the inverse scale factor analogue is unbounded on zero-volume states, highlighting the crucial role of inhomogeneous degrees of freedom. Nevertheless, the authors demonstrate that the expectation value of this operator within kinematical coherent states peaked on homogeneous and isotropic data remains bounded at the Big Bang, suggesting a different mechanism for singularity avoidance that does not rely on operator boundedness alone. They argue that definitive conclusions require the physical Hilbert space and Dirac observables, and they propose an approximate scheme, including the Master Constraint Programme and relational observables, to determine the presence or absence of the initial singularity in full LQG. Overall, the work underscores the need to distinguish LQC as a valuable but distinct toy model from full LQG and outlines a concrete path toward testing singularity resolution in the complete theory.

Abstract

Loop Quantum Cosmology (LQC), mainly due to Bojowald, is not the cosmological sector of Loop Quantum Gravity (LQG). Rather, LQC consists of a truncation of the phase space of classical General Relativity to spatially homogeneous situations which is then quantized by the methods of LQG. Thus, LQC is a quantum mechanical toy model (finite number of degrees of freedom) for LQG(a genuine QFT with an infinite number of degrees of freedom) which provides important consistency checks. However, it is a non trivial question whether the predictions of LQC are robust after switching on the inhomogeneous fluctuations present in full LQG. Two of the most spectacular findings of LQC are that 1. the inverse scale factor is bounded from above on zero volume eigenstates which hints at the avoidance of the local curvature singularity and 2. that the Quantum Einstein Equations are non -- singular which hints at the avoidance of the global initial singularity. We display the result of a calculation for LQG which proves that the (analogon of the) inverse scale factor, while densely defined, is {\it not} bounded from above on zero volume eigenstates. Thus, in full LQG, if curvature singularity avoidance is realized, then not in this simple way. In fact, it turns out that the boundedness of the inverse scale factor is neither necessary nor sufficient for curvature singularity avoidance and that non -- singular evolution equations are neither necessary nor sufficient for initial singularity avoidance because none of these criteria are formulated in terms of observable quantities.After outlining what would be required, we present the results of a calculation for LQG which could be a first indication that our criteria at least for curvature singularity avoidance are satisfied in LQG.

Figures (16)

• Figure 1: Simplest 3 valent gauge invariant graph $\gamma$: $V(\gamma)=\{v_1,v_2\},E(\gamma)=\{e_1,e_2,e_3\}$
• Figure 2: Plot for $j_1=j_2=\frac{j_3}{2}$ where $j_3 \in \mathbbm{N}$ with $1\le j_3 \le 40$. The graph oscillates between 0 (if $j_3$ even) and an increasing value (if $j_3$ odd)
• Figure 3: Plot for $j_1=j_2=\frac{j_3+1}{2}$ where $j_3 \in \mathbbm{N}$ with $1\le j_3 \le 40$. The graph oscillates between 0 (if $j_3$ odd) and an increasing value (if $j_3$ even)
• Figure 4: Plot for $j_1=\frac{3}{2}$, $j_2=j_3+\frac{1}{2}$ where $j_3 \in \mathbbm{N}$ with $1\le j_3 \le 40$. The graph first decreases for $1\le j_3 <3$ and is 0 for $j_3=3$ . It increases for $j_3>3$
• Figure 5: Plot for $j_3=5$