Quantum Gravity: are we there yet?

Abstract

The turn of the millennium was a time of optimism about an approach to noncommutative geometry inspired by rich mathematical objects called `quantum groups' and its applications to quantum spacetime. This would model quantum gravity effects as noncommutativity of spacetime coordinates and was arguably going to solve quantum gravity itself. It took a further 20 years from that point to develop a particularly suitable formalism of `quantum Riemannian geometry', but this was largely done and has begun to be used to construct baby quantum gravity models. In this article, we obtain new results for state of the art fuzzy sphere and n-gon models in this approach. We also review what are some elements of quantum gravity that we can already see and what are the critical conceptual and mathematical elements that are still missing to more fully achieve this goal.

Figures (6)

• Figure 1: Expectation value of Ricci scalar scalar showing rapid convergence of ${\langle}R{\rangle}L$ to $-3/4$ and first quantum gravity correction.
• Figure 2: Numerical behaviour of $Z$ for $L<<6$ and $Z/Z_0$ for fixed $z={L\over\epsilon}>1$ showing convergence to 1 as $\epsilon\to 0$.
• Figure 3: Expectation values in the deep quantum gravity $Z_0$ phase of the fuzzy sphere as a function of small and large $z$ respectively.
• Figure 4: For the QRG of a graph, arrows are differential forms and a quantum metric is an assignment of a non-zero real 'square length' to each edge. The quantum metric for the $n$-gon defines a function $a$ which at vertex $i$ is $a_i$, and similarly for the square, two functions $a,b$ with $a_{10}=a_{00}, a_{11}=a_{01}$ and $b_{01}=b_{00}, b_{11}=b_{10}$.
• Figure 5: $G$-dependence of vacuum expectation values and partition function for quantum gravity on a triangle.