Graph superspositions and improved regularisations

In loop quantum gravity, the elements of a certain basis in the Hilbert space
can be (roughly) interpreted as lattices. Generic quantum states can be constructed
as superpositions thereof and a priori have different properties which cannot
be realised at the level of individual lattices. 

Let us pick up the topic of extracting effective cosmological dynamics from loop quantum gravity again. Today’s post is about a recent paper by Emanuele Alesci and Francesco Cianfrani within their framework of “Quantum reduced loop gravity”. This version of loop quantum gravity is arrived at by a gauge fixing to the diagonal metric gauge at the quantum level (as opposed to the classical level as e.g. here), along with a truncation of the Hamiltonian constraint consistent with a spatially homogeneous setting (in particular, the non-trivial shift resulting from the gauge fixing is dropped). The simplified quantum dynamics resulting form this gauge fixing have allowed to compute the expectation value of the Hamiltonian constraint in suitable coherent states, leading to the effective Hamiltonian that one finds in loop quantum cosmology.

An open issue in this context has been to properly derive the so called “improved “ dynamics of loop quantum cosmology, which are consistent with observation and do not feature some unphysical properties of the original formulation. This is somewhat tricky if one uses standard connection variables along with a gauge group like SU(2) or U(1) for the following reason:

In loop quantum gravity, only holonomies of the connection are well defined operators. In the process of regularising the Hamiltonian, the connection, or rather, the field strength thereof, has to be approximated by holonomies. In practise, this roughly means that one performs the substitution $A \rightarrow \sin (A)$, which is clearly good only as long as A is small. If one now integrates the connection A along a “long” path, one finds that in the context of cosmology the resulting object scales as “matter energy density” times “distance”. In quantum gravity, we expect deviations from known physics as soon as we reach the Planck scale, i.e. when the matter energy density is at the order of the Planck density. Therefore, in order to be consistent with standard physics at large distances, one needs to modify the polymerisation $\int A \rightarrow \sin (\int A)$ to $\int A \rightarrow \sin (\bar \mu \int A) / \bar \mu$, where $\bar \mu$ scales as the inverse distance. This prescription was originally put forward using a coarse graining argument and rederived here and here using the above argument.

The problem now is that the inverse distance scaling necessitates to use non-integer representation labels in the quantum theory, which is not possible with SU(2) or U(1). A way out is to construct the theory from beginning using the gauge group $\mathbb R_{\text{Bohr}}$ after a classical gauge fixing, which was done here. However, this leaves open how the improved LQC dynamics can arise also from standard loop quantum gravity.

In their paper, Emanuele and Francesco now show that at least at the level of the expectation value of the Hamiltonian constraint, one can also get the improved dynamics without resorting to $\mathbb R_{\text{Bohr}}$. Due to the above argument, this cannot be achieved on a fixed graph. Instead, they propose to use a superposition of graphs of all possible sizes, weighted by a natural statistical factor. After a suitable saddle point approximation in the resulting state, they show that the expectation value of the Hamiltonian is given by that of loop quantum cosmology with the improved dynamics. Also, next-to-leading order corrections are computed, with the result that they don’t affect the LQC dynamics qualitatively $\omega < 1$ (see here for what $\omega$ is in cosmology), as well as for $\omega = 1$ imposing the universe at the bounce point is sufficiently large w.r.t. the Planck scale.

This is, to the best of my knowledge, the first example where graph superpositions were used in loop quantum gravity to derive a result which one cannot simply get by working on a fixed graph. The relevance of such graph superpositions for the theory, in particular for the classical limit and the issue of Lorentz invariance, has been discussed in an earlier post. The possibility to consider quantum superpositions of graphs is the key feature of loop quantum gravity that distinguishes it from approaches based on fixed lattices.

To sum up, the paper contains a very interesting new result based on a feature of loop quantum gravity (graph superpositions) that has been mostly ignored so far. The insights gained here can be potentially very important in studies of the classical limit of the theory.