Unboundedness of triad-like operators in loop quantum gravity

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) the quantum Einstein equations are nonsingular which hints at the avoidance of the global initial singularity. This rests on (1) a key technique developed for LQG and (2) the fact that there are no inhomogeneous excitations. We display the result of a calculation for LQG which proves that the (analogon of the) inverse scale factor, while densely defined, is 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 the 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.

Graphical calculus of volume, inverse volume and Hamiltonian operators in loop quantum gravity

“…Note that, besides the regularization introduced by Thiemann [16], other proposal for the regularization of the Hamiltonian constraint operator is also available [36]. On the contrary to the conclusion in [27], our calculation shows that the inverse volume operator is bounded (zero-valued) at a trivalent non-planar vertex of the gauge-invariant spin-network states, which is a non-trivial eigenstate with zero-eigenvalue of the volume operator. This result opens a possible way to lift the result of singularity resolution of LQC to LQG.…”

“…If γ differs from γ , e.g., there is an edge e with spin j e belonging to γ but not γ , then the orthogonality relation, 27) implies that the corresponding integration in (3.26) becomes…”

“…In particular, it is shown in [26] that in spatially curved anisotropic models inverse volume effects may become important to bind expansion and shear scalars. However, it is shown in [27] that the inverse volume operator with certain ordering in full LQG is unbounded on the zero volume eigenstates (at a gauge-invariant trivalent vertex). This throws doubt on whether one can generalize the conclusions of LQC to LQG.…”

Semiclassical dynamics of horizons in spherically symmetric collapse

“…Introducing the new parameter Θ(t, r) := θ + θ − , we get 13) which in terms of the phase space variables, and working on the comoving gauge, by fixing N = 1, can be written as…”

“…Results from LQC leads to the conclusion that the cosmological singularities are resolved in quantum gravity [11]. However, in view of the full theory, there is still considerable ambiguity and none of those results are fully satisfactory [12,13].…”