GR demonstrates the need for a quantum theory of gravity rather directly, as it relates to which is a quantum operator. In the usual approach to quantize gravity one needs to select a (vacuum) state for the metric which is a solution to the classical equations of motion and expand around it:
Nevertheless, the three most prominent approaches to non-perturbative quantum gravity, which are viewed as being very probable solutions, are:
In 1995 Smolin and Rovelli took Ashtekar's non-perturbative approach a step further by introducing the ideas of so-called spin networks, developed by Penrose in the 1960s (see [Maj99]); [RoSm95] and [Sm97]. It is interesting to note that spin networks also appear in gauge theories, topological QFT's, conformal field theories and knot theory; compare with appendix B. But there is also a direct link to section 3 of appendix A.7 given by the fact that the so-called loop representation of quantum gravity (or loop gravity) yields a discrete picture of space-time; [BMS95]. The loop representation is a formulation of QFT suitable when the degrees of freedom of the theory are given by a gauge field or a connection, as is the case with the new variables theory. Spin networks can be employed to characterize quantum gravity in the loop representation.
There are many approaches to non-perturbative quantum geometry. Penrose's twistor theory developed in 1986 is an example; [PR86], [WW90] and [HT94] . Ashtekar bases his line of attack on canonical quantization and the absence of any background geometry. Interestingly, ideas originating in twistor theory and spin networks play a key role in this version of quantum geometry. String/M-theory also modifies the (general) relativistic notion that space-time is a smooth manifold. The geometrical structure used to describe the physics of string/M-theory is also referred to as quantum geometry, to note its distinctiveness from classical (differential) geometry. This term is somewhat arbitrary, as it depends upon which probe is used to study the underlying geometry. Hence a string or a D-brane will reveal a different geometry, in effect concluding the rich structure of the quantum geometry associated with string/M-theory. In addition, it is possible to study a kind of classical geometry arising within the context of non-quantum string theory, sometimes referred to as `classical stringy geometry'. So quantum geometry in this context is understood as the new geometrical discipline whose basic ingredients are the observables of string/M-theory, which in an appropriate limiting case reduces to standard geometry (algebraic geometry and topology). Hence the dilemma of pre-string theory, that although one had a gauge theory and a QFT for spin-1 force carriers there existed only a `gauge theory' (meaning GR) for spin-2 gravitons and no appropriate quantum field formulation, is maybe overcome. It appears that the corresponding `QFT' for gravity is quantum geometry. Consult [Gre97].
Reference [H-et.al.98] contains contributions from Ashtekar (quantum gravity and quantum geometry), Connes (non-commutative geometry), Smolin (spin networks) and in addition covers the topics of twistors among many other aspects of the geometry of space-time. Rovelli summarizes all the approaches to quantum gravity in [Ro98], among the topics are string theory, loop gravity, QFT in curved space-time, twistors, non-commutative geometry and topological QFT. Twistors are covered in [HSW99] within the context of integrable systems. Also, see [Ho95] for a note on supersymmetry and twistors. In [FGR97] an additional connection between quantum gravity and non-commutative geometry is given. It is also argued that a combination of GR and and quantum theory leads to the prediction that space-time cannot be a classical manifold and that the basic degrees of freedom should be associated with extended objects (e.g. strings). Quantum gravity and its philosophical implications are discussed in [BI99].