Quantum gravity, the as yet unconsummated marriage between quantum physics and Einstein's general relativity, is widely (though perhaps not universally) regarded as the single most pressing problem facing theoretical physics at the turn of the millennium. The two main contenders, ``Brane theory/ String theory'' and ``Quantum geometry/ new variables'', have their genesis in different communities. They address different questions, using different strategies, and have different strengths (and weaknesses).
Brane theory/ string theory grew out of the high-energy particle physics community, and views quantum physics as paramount . The consensus feeling in the brane community is that to achieve the quantization of gravity they would be willing to take quite drastic steps, to mutilate the geometrical foundations of general relativity and if necessary to force general relativity to fit into the brane framework. In contrast, the general relativity community views the geometrical nature of Einstein's gravity as sacrosanct, and would by and large be quite willing to do a little drastic surgery to the foundations of quantum physics if they felt it unavoidable . ``Lorentzian lattice quantum gravity'' does a little of both: it adopts some aspects of each of these approaches, and violates other cherished notions of these two main candidate models.
On the one hand, ``Lorentzian lattice quantum gravity'' has grown out of the lattice community, itself a subset of the particle physics community. In lattice physics spacetime is approximated by a discrete lattice of points spaced a finite distance apart. This ``latticization'' process is a way of guaranteeing that quantum field theory can be defined in a finite and non-perturbative fashion. (Indeed currently the lattice is the only known non-perturbative regulator for flat-space quantum field theory. This technique is absolutely essential when carrying out computer simulations of quantum field theories, and in particular, computer simulations of quarks, gluons, and the like in QCD.) In addition to these particle physics notions, ``Lorentzian lattice quantum gravity'' has strongly adopted the geometric flavour of general relativity; it speaks of surfaces and spaces, of geometries and shapes.
On the other hand, ``Lorentzian lattice quantum gravity'' has irritated both brane theorists and general relativists (and more than a few lattice physicists as well): It does not have, and does not seem to require, the complicated superstructure of supersymmetry and all the other technical machinery of brane theory/ string theory. (A critically important feature of brane theory/ string theory which justifies the amount of time spent on the model is that in an appropriate limit it seems to approximate key aspects of general relativity; and do so without the violent mathematical infinities encountered in most other approaches. Of course, there is always the risk that there might be other less complicated theories out there that might do an equally good job in this regard.) Additionally, ``Lorentzian lattice quantum gravity'' irritates some members of the relativity community by not including all possible 4-dimensional geometries: The key ingredient that makes this Lorentzian approach different (and successful, at last in a lower-dimensional setting) is that it to some extent enforces a separation between the notions of space and time, so that space-time is really taken as a product of ``space'' with ``time''. It then sums over the resulting restricted set of (3+1)-dimensional geometries; not over all 4-dimensional geometries (that being the traditional approach of the so-called Euclidean lattice quantum gravity).
Technically, Lorentzian lattice quantum gravity restricts the sum over 4-dimensional geometries to cover only that subset of 4-dimensional geometries compatible with the existence of 3+1 space+time dimensions. (The condition used is a discretized version of stable causality; in the sense of the existence of a global time function.) The result of this topological/ geometrical restriction is that the model produces reasonably large, reasonably smooth patches of spacetime that look like they are good precursors for our observable universe. (Euclidean lattice quantum gravity, and variants thereof such as Matrix theory, have an unfortunate tendency to curdle into long thin polymer-like strands that look nothing like the more or less flat spacetime in our immediate vicinity; Quantum geometry based on new variables likewise encounters technical difficulties in generating an approximately smooth manifold in the low-energy large-distance limit.)
The good news is that once reasonably large, reasonably flat, patches of spacetime exist, the arguments leading to Sakharov's notion of ``induced gravity'' almost guarantee the generation of a cosmological constant and an Einstein-Hilbert term in the effective action through one-loop quantum effects ; and this would almost automatically guarantee an inverse-square law at very low energies (large distances). The bad news is that so far the large flat regions have only been demonstrated to exist in 1+1 and 2+1 dimensions -- the (3+1)-dimensional case continues to pose considerable technical difficulties.
All in all, the development of ``Lorentzian lattice quantum gravity'' is extremely exciting: It is non-perturbative, definitely high-energy (ultraviolet) finite, and has good prospects for an acceptable low-energy (infra-red) limit. It has taken ideas from both the quantum and the relativity camps, though it has not completely satisfied either camp. Keep an eye out for further developments.
Key papers on Lorentzian lattice quantum gravity:
J. Ambjorn, A. Dasgupta, J. Jurkiewicz and R. Loll, ``A Lorentzian cure for Euclidean troubles,'' Nucl. Phys. Proc. Suppl. 106 (2002) 977-979 arXiv:hep-th/0201104
J. Ambjorn, J. Jurkiewicz and R. Loll, ``3d Lorentzian, dynamically triangulated quantum gravity,'' Nucl. Phys. Proc. Suppl. 106 (2002) 980-982 arXiv:hep-lat/0201013.
J. Ambjorn, J. Jurkiewicz, R. Loll and G. Vernizzi, ``Lorentzian 3d gravity with wormholes via matrix models,'' JHEP 0109 (2001) 022 arXiv:hep-th/0106082
J. Ambjorn, J. Jurkiewicz and R. Loll, ``Dynamically triangulating Lorentzian quantum gravity,'' Nucl. Phys. B610 (2001) 347-382 arXiv:hep-th/0105267.
A. Dasgupta and R. Loll, ``A proper-time cure for the conformal sickness in quantum gravity,'' Nucl. Phys. B 606 (2001) 357-379 arXiv:hep-th/0103186.
J. Ambjorn, J. Jurkiewicz and R. Loll, ``Non-perturbative 3d Lorentzian quantum gravity,'' Phys. Rev. D 64 (2001) 044011 arXiv:hep-th/0011276.
R. Loll, ``Discrete Lorentzian quantum gravity,'' Nucl. Phys. Proc. Suppl. 94 (2001) 96-107 arXiv:hep-th/0011194.
J. Ambjorn, J. Jurkiewicz and R. Loll, ``Computer simulations of 3d Lorentzian quantum gravity,'' Nucl. Phys. Proc. Suppl. 94 (2001) 689-692 arXiv:hep-lat/0011055.
J. Ambjorn, J. Jurkiewicz and R. Loll, ``A non-perturbative Lorentzian path integral for gravity,'' Phys. Rev. Lett. 85 (2000) 924-927 arXiv:hep-th/0002050.
A survey of brane theory and quantum geometry:
G. Horowitz, ``Quantum Gravity at the Turn of the Millennium'',
MG9 -- Ninth Marcel Grossmann meeting, Rome, Jul 2000,
Sakharov's induced gravity:
A.D. Sakharov, ``Vacuum quantum fluctuations in curved space and the theory of gravitation'', Sov. Phys. Dokl. 12 (1968) 1040-1041; Dokl. Akad. Nauk Ser. Fiz. 177 (1967) 70-71.