One the candidate ideas for the elusive theory of quantum gravity, loop quantum gravity (LQG) is considered the primary rival of string theory.
Among the current attempts to construct a theory of quantum gravity, string theory and loop quantum gravity have been the most successful and made the most progress, though they certainly aren't the only ones (see, for instance, so the so-called random triangulation theories, the casual approach to quantum gravity etc.). Among those two, string theory outnumbers LQG by far - whether you count active researchers or published articles. Nevertheless, LQG remains an active branch of research, not falling behind string theory in many crucial tests a viable theory of quantum gravity should be able to pass.
The difference of between the two approaches is quite radical. String theory is on the on-set an attempt to create a unified field theory which should explain the whole rich spectrum of known particles and forces from fundumental principles. LQG only strives to give a description of the gravitation compatible with the principles of quantum mechanics. String theory adheres to quite radical (with respect to existing theories) ideas such as 10 (or 11) dimensional space-time and the introduction of strings and other extended objects as the fundumental degrees of freedom rather than fields or particles. LQG uses a comparatively conservative canonical quantization approach applying it to the classical theory of gravititation - Einstein's general relativity.
An important feature of LQG, from which may of its stregths stem is so-called background independence or diffeomorphism invariance. This is a property inherent in general relativity, and in fact, it might be claimed equivalent to one of the fundumental principles form which Einstein derived the theory - the equivalence principle. To-date, this feature have not been fully implemented in the theory, i.e., it has not been proven LQG is diffeomorphism invariant to full extent. The problems involved are related to the so-called problem of time in quantum gravity. Nevertheless, the principle has been a guideline all the way throughout the construction of LQG, in contrast to string theory, where the perturbative formulation of theory (which is also the only one fully understood) displays a total lack thereof.
LQG derives its name from the fundumental quantum degrees of freedom of space-time arising from its quantization procedure - loops or spin-networks. Those are also extended object which makes it somewhat similar to string theory. The difference is that those objects are not postulated but derived from the quantization of the gravitational field. Spin-networks were originally devised by the famous Cosmologist Roger Penrose for altogether different purposes.
The difference between the vantage point of LQG and the (largely unsuccessful) so-called covariant quantum gravity, which also attacks the quantum gravity problem via canonical quantization of general relativity combined with the diffeomorphism invariance principle is the different classical formulations of general relativity used. Covariant quantum gravity uses the original formulation initially derived by Einstein, where the fundumental variable of the theory is the space-time metric tensor. LQG uses the connection formulation of general relativity derived by Abhay Ashtekar, which is in itself a further development of formulations constructed by Plebanski and Palatini. In this form, general relativity becomes much more similar to gauge theory which was so successful in describing the other three forces of nature except gravity.
Classically, both the metric formulation and the connection formulation are equivalent, but after quantization they lead to radically different behavior. Most of the successes of LQG can be accounted to the fact it uses a connection formulation. It is also interesting note the origins of the formulation can be traced back to Einstein's work on unified field theory in his later period of life, generally considered unsuccessful.
Among the attractive features of LQG one can list:
- The ability to construct quantum mechanical operators corresponding the geometrical physical observables such as areas, volumes, etc. This is to be expected in a quantum theory of gravity where the geometry of space-time becomes a dynamical variable of theory.
- The recent derivation of the mysterious black hole entropy from the fundumental principles of LQG.
- The relation to the path integral based approach to quantum gravity know as "spin foam". The spin foams can be thought of as space-time histories of spin networks or as a generalization of Feynman diagrams.
- The recent work due to Martin Bojowald exploring quantum cosmology from the point of view of LQG, where he proves the initial singularity disappears from the theory. Instead, one obtains a collasping universe "inverting" and turning into an expanding universe. The reason there is no singularity in the middle is the quantum, discretized nature of space-time in the theory.
- The so-called Kodama state is known, which provides a LQG description of the de Sitter universe. In contrast, string theory
experiences much trouble with describing a de Sitter type universe, getting along much better with the anti-de Sitter type. Recent research shows the de Sitter universe might be modeled using string theory
but the resulting states are unstable and quite unelegant. Recent astronomical observations prove out universe is expanding with acceleration, i.e., it is of de Sitter type.
Among the drawbacks of LQG are:
- The appearance of the so-called Barbero-Immirzi parameter which is an ambiguity in the quantization procedure and which has to be fitted by hand to get compatibility with the semi-classical limit. Recent work shows a possibility to attack this problem using Bohr-Sommerfield quantization of black hole quasinormal modes.
- To date it is still unclear what is correct quantum mechanical operator for the Hamiltonian constraint, and there is no proof there is a sufficient amount of quantum states satisfying the constraint. This is paramount both to background invariance and deriving sensible dynamics out of theory. However, LQG stands much better in this respect than the above-mentioned covaraint quantum gravity in which the Hamiltonian constraint is the infamous Wheeler-de Witt equation, all attempts to make rigorous sense out of which have failed.