3.8  Loop Quantum Gravity

In 1986, Abhay Ashtekar (b. 1949) reformulated Einstein’s General Relativity in a language closer to that of the rest of fundamental physics. Shortly after that, Ted Jacobson (b. 1954) and Lee Smolin (b. 1955) realized that the formal equation of Quantum Gravity, called the Wheeler-DeWitt equation, admitted solutions labeled by loops when rewritten in the new Ashtekar variables. Smolin and Carlo Rovelli (b. ) defined a non-perturbative and background-independent quantum theory of gravity in terms of these loop solutions. They also showed that the quantum operators of the theory associated to area and volume have a discrete spectrum, which means that geometry is quantized. This result defines an explicit basis of states of quantum geometry, which turned out to be labeled by Penrose’s spin networks, which are graphs labeled by spins. Spin foam is a topological structure made out of two-dimensional faces that represents one of the configurations that must be summed to obtain a Feynman’s path integral description of Quantum Gravity.

The canonical version of the dynamics was put on firm ground by Thomas Thiemann (b. 1967), who defined an anomaly-free Hamiltonian operator, showing the existence of a mathematically consistent background-independent theory. The covariant or spin foam version of the dynamics developed during several decades from the joint work of various research groups, leading to the definition of a family of transition amplitudes, which in the classical limit can be shown to be related to a family of truncations of General Relativity. The finiteness of these amplitudes was proven in 2011. It requires the existence of a positive cosmological constant, and this is consistent with observed acceleration in the expansion of the Universe.

In Strings Theory one generally starts with quantized excitations on top of a classically fixed background. This theory is thus described as background dependent. Particles like photons as well as changes in the space-time geometry, or gravitons, are both described as excitations on the string world-sheet. The background dependence of Strings Theory can have important physical consequences, such as determining the number of quark generations. In contrast, Loop Quantum Gravity, like General Relativity, is manifestly background independent, while both aim to overcome the non-re-normalizable divergences of quantum field theories.

Loop Quantum Gravity never introduces a background and excitations living on this background, so it does not use gravitons as building blocks. Instead one expects that one may recover a kind of semi-classical limit or weak field limit where something like gravitons will show up again. In contrast, gravitons play a key role in Strings Theory where they are among the first massless level of excitations of a super-string.

Loop Quantum Gravity differs from Strings Theory in that it is formulated in 3 and 4 dimensions and without super-symmetry or Kaluza-Klein extra dimensions, while the latter requires both, although there is no experimental evidence to support that. Rovelli, however, regards the fact that Loop Quantum Gravity is formulated in 4 dimensions and without super-symmetry as a strength of the theory, since it represents the most parsimonious explanation, consistent with current experimental results, over its rival Strings Theory. The latter, however, demonstrably reproduces the established theories of General Relativity and Quantum Field Theory in the appropriate limits, which Loop Quantum Gravity has struggled to do. Additionally, Loop Quantum Gravity does not have any direct references to matter particles, the fermions, whereas Strings Theory also addresses unification of all known forces and particles as manifestations of a single entity, by postulating extra dimensions and so-far unobserved additional particles and symmetries. Contrary to this, Loop Quantum Gravity is based only on Quantum Theory and General Relativity and its scope is limited to understanding the quantum aspects of the gravitational interaction. On the other hand, the consequences of Loop Quantum Gravity are radical, because they fundamentally change the nature of space and time and provide a tentative but detailed physical and mathematical picture of quantum space-time.