Introducing a helpful taxonomic scheme, Chris Isham proposed to divide the many approaches to formulating a full, i.e. not semi-classical, quantum theory of gravity into four broad types of approaches: first, those quantizing GR; second, those “general-relativizing” quantum physics; third, construct a conventional quantum theory including gravity and regard GR as its low-energy limit; and fourth, consider both GR and conventional quantum theories of matter as low-energy limits of a radically novel fundamental theory.
The first family of strategies starts out from classical GR and seek to apply, in a mathematically rigorous and physically principled way, a “quantization” procedure, i.e. a recipe for cooking up a quantum theory from a classical theory such as GR. Of course, quantization proceeds, metaphysically speaking, backwards in that it starts out from the dubious classical theory – which is found to be deficient and hence in need of replacement – and tries to erect the sound building of a quantum theory of gravity on its ruin. But it should be understood, just like Wittgenstein’s ladder, as a methodologically promising means to an end. Quantization procedures have successfully been applied elsewhere in physics and produced, among others, important theories such as quantum electrodynamics.
The first family consists of two genera, the now mostly defunct covariant ansatz (Defunct because covariant quantizations of GR are not perturbatively renormalizable, a flaw usually considered fatal. This is not to say, however, that covariant techniques don’t play a role in contemporary quantum gravity.) and the vigorous canonical quantization approach. A canonical quantization requires that the theory to be quantized is expressed in a particular formalism, the so-called constrained Hamiltonian formalism. Loop quantum gravity (LQG) is the most prominent representative of this camp, but there are other approaches.
Secondly, there is to date no promising avenue to gaining a full quantum theory of gravity by “general-relativizing” quantum (field) theories, i.e. by employing techniques that permit the full incorporation of the lessons of GR into a quantum theory. The only existing representative of this approach consists of attempts to formulate a quantum field theory on a curved rather than the usual flat background spacetime. The general idea of this approach is to incorporate, in some local sense, GR’s principle of general covariance. It is important to note that, however, that the background spacetime, curved though it may be, is in no way dynamic. In other words, it cannot be interpreted, as it can in GR, to interact with the matter fields.
The third group also takes quantum physics as its vantage point, but instead of directly incorporating the lessons of GR, attempts to extend quantum physics with means as conventional as possible in order to include gravity. GR, it is hoped, will then drop out of the resulting theory in its low-energy limit. By far the most promising member of this family is string theory, which, however, goes well beyond conventional quantum field theory, both methodologically and in terms of ambition. Despite its extending the assumed boundaries of the family, string theory still takes conventional quantum field theory as its vantage point, both historically and systematically, and does not attempt to build a novel theory of quantum gravity dissociated from “old” physics. Again, there are other approaches in this family, such as topological quantum field theory, but none of them musters substantial support among physicists.
The fourth and final group of the Ishamian taxonomy is most aptly characterized by its iconoclastic attitude. For the heterodox approaches of this type, no known physics serves as starting point; rather, radically novel perspectives are considered in an attempt to formulate a quantum theory of gravity ab initio.
All these approaches have their attractions and hence their following. But all of them also have their deficiencies. To list them comprehensively would go well beyond the present endeavour. Apart from the two major challenges for LQG, a major problem common to all of them is their complete lack of a real connection to observations or experiments. Either the theory is too flexible so as to be able to accommodate almost any empirical data, such as string theory’s predictions of supersymmetric particles which have been constantly revised in light of particle detectors’ failures to find them at the predicted energies or as string theory’s embarras de richesses, the now notorious “landscape problem” of choosing among 10500 different models. Or the connection between the mostly understood data and the theories is highly tenuous and controversial, such as the issue of how and whether data narrowly confining possible violations of Lorentz symmetry relate to theories of quantum gravity predicting or assuming a discrete spacetime structure that is believed to violate, or at least modify, the Lorentz symmetry so well confirmed at larger scales. Or the predictions made by the theories are only testable in experimental regimes so far removed from present technological capacities, such as the predictions of LQG that spacetime is discrete at the Planck level at a quintillion (1018) times the energy scales probed by the Large Hadron Collider at CERN. Or simply no one remotely has a clue as to how the theory might connect to the empirical, such as is the case for the inchoate approaches of the fourth group like causal set theory.