Which do you think are the main quantum geometry (QG) lines recently appearing to rival Loop? Do any look especially promising to you? Any favorites? Here are some to consider. The authors shown aren't a complete list! Just enough for an arxiv search: [Colored] tensor models (e.g. Gurau, Ryan, ...) Cellular quantization (Bonzom, Smerlak) Shape dynamics (Koslowski, Gomes, ...) AsymSafe (Saueressig, Reuter, ...) For the purposes of this thread, to qualify as QG the central object of study should be the geometry of a 4D spacetime region or slice thereof, mathematically realized as a quantum state of geometry. The theory should be clearly background independent. I'm not sure that AsymSafe QG qualifies as fully background independent--that's something for us to discuss and decide what we think. As I recall there's some recent work indicating it might be. These theories are work in progress and are sufficiently similar to Loop QG that one can expect them to have testable predictions about early universe cosmology. Each should say something that can be confronted with observation. That's a question to discuss case by case, and resolve if we can. I'll get some links. See how it looks to you.
What struck me forcefully this morning was the news that James Ryan has accepted a tenure track faculty position at Morelia. That made the lights come on for colored tensor models. http://arxiv.org/abs/1109.4812 Colored Tensor Models - a review Razvan Gurau, James P. Ryan (Submitted on 22 Sep 2011) Colored tensor models have recently burst onto the scene as a promising conceptual and computational tool in the investigation of problems of random geometry in dimension three and higher. We present a snapshot of the cutting edge in this rapidly expanding research field... 80 pages Could this develop into a rival testable QG? I recall not too long ago (April 2011) both Gurau and Ryan (then postdoc working with Bianca Dittrich at AEI) gave talks at the ILQGS. Let's see what those were about. btw here are Ryan's papers: http://arxiv.org/find/gr-qc/1/au:+Ryan_J/0/1/0/all/0/1 Ryan's talk: http://relativity.phys.lsu.edu/ilqgs/ryan041211.pdf http://relativity.phys.lsu.edu/ilqgs/ryan041211.wav Gurau's talk: http://relativity.phys.lsu.edu/ilqgs/gurau042611.pdf http://relativity.phys.lsu.edu/ilqgs/gurau042611.wav A recent paper of Gurau et al suggests that one can do without the coloring (which is demoted to a useful but non-physical book-keeping device.) http://arxiv.org/abs/1202.3637 Random tensor models in the large N limit: Uncoloring the colored tensor models Valentin Bonzom, Razvan Gurau, Vincent Rivasseau (Submitted on 16 Feb 2012) Tensor models generalize random matrix models in yielding a theory of dynamical triangulations in arbitrary dimensions. Colored tensor models have been shown to admit a 1/N expansion and a continuum limit accessible analytically. In this paper we prove that these results extend to the most general tensor model for a single generic, i.e. non-symmetric, complex tensor. Colors appear in this setting as a canonical book-keeping device and not as a fundamental feature... 15 pages
Reuter's name for Asymptotic Safe QG is "Quantum Einstein Gravity". Here's a review article that just came out. http://arxiv.org/abs/1202.2274 Quantum Einstein Gravity Martin Reuter, Frank Saueressig (Submitted on 10 Feb 2012) We give a pedagogical introduction to the basic ideas and concepts of the Asymptotic Safety program in Quantum Einstein Gravity. Using the continuum approach based upon the effective average action, we summarize the state of the art of the field with a particular focus on the evidence supporting the existence of the non-trivial renormalization group fixed point at the heart of the construction. As an application, the multifractal structure of the emerging space-times is discussed in detail. In particular, we compare the continuum prediction for their spectral dimension with Monte Carlo data from the Causal Dynamical Triangulation approach. Comments: 87 pages, 13 figures, review article prepared for the New Journal of Physics focus issue on Quantum Einstein Gravity So apparently the New Journal of Physics is doing a focus issue. The background independence question is important. Without being fully free of prior geometry I don't expect a theory can tell much about what happened just before and after the start of expansion, or resolve the initial singularity (say, into a bounce). But there have been some interesting papers recently. http://arxiv.org/abs/1203.4207 The phase diagram of quantum gravity from diffeomorphism-invariant RG-flows Ivan Donkin, Jan M. Pawlowski (Submitted on 19 Mar 2012) We evaluate the phase diagram of quantum gravity within a fully diffeomorphism-invariant renormalisation group approach. The construction is based on the geometrical or Vilkovisky-DeWitt effective action. We also resolve the difference between the fluctuation metric and the background metric. This allows for fully background-independent flows in gravity. The results provide further evidence for the ultraviolet fixed point scenario in quantum gravity with quantitative changes for the fixed point physics. We also find a stable infrared fixed point related to classical Einstein gravity. Implications and possible extensions are discussed. 23 pages, 13 figures Cai and Easson: AsymSafe predicts long-lived microscopic BH which could explain dark matter. http://arxiv.org/abs/1007.1317 Bonanno: Inflation without exotic "inflaton field" simply from RG flow of G and Lambda, gets enough e-folds before naturally subsiding. http://arxiv.org/abs/1203.1962 Cai and Easson: same story, RG running takes care of inflation plus Higgs field naturally provides fluctuations seeding observed structure "Higgs Boson in RG running Inflationary Cosmology" http://arXiv.org/abs/arXiv:1202.1285
Cellular quantization is a possible competitor to the usual spinfoam quantization: http://arxiv.org/abs/1201.4996 Gauge symmetries in spinfoam gravity: the case for "cellular quantization" Valentin Bonzom, Matteo Smerlak (Submitted on 24 Jan 2012) The spinfoam approach to quantum gravity rests on a "quantization" of BF theory using 2-complexes and group representations. We explain why, in dimension three and higher, this "spinfoam quantization" must be amended to be made consistent with the gauge symmetries of discrete BF theory. We discuss a suitable generalization, called "cellular quantization", which (1) is finite, (2) produces a topological invariant, (3) matches with the properties of the continuum BF theory, (4) corresponds to its loop quantization. These results significantly clarify the foundations - and limitations - of the spinfoam formalism, and open the path to understanding, in a discrete setting, the symmetry-breaking which reduces BF theory to gravity. 6 pages My personal tendency is to take the alternative very seriously, but I cannot justify my impression and must simply present the paper for others more expert to judge. The mental picture where I see this might be applied is a deSitter spacetime bounded by initial and final (pre-bounce and post-bounce) S^{3} spin-network states of 3-d geometry. IOW not a boundaryless manifold, but one with two separate boundary components. (Bonzom and Smerlak at this point are concentrating on boundaryless manifolds.)
I think the really one competitor to loops is shape dynamics. I cannot see a way to unify its principles, even as an approximation, with LQG.
I think you are right that there is no way to merge Shape Dynamics QG with Loop. Or anyway I can't imagine how. In the other cases I can't say. Maybe, perhaps by radically changing the current Loop formulation. BTW I too found John Ralston's http://arxiv.org/abs/1203.5557 Quantum Theory without Planck's Constant quite interesting. It sounds strange at first but could be a sensible proposal. Glad you noticed it! Another approach, which, if not actually rival, would nevertheless involve a complete revision of Loop if it were adopted, is Wieland's use of COMPLEX Ashtekar variables. The whole Barbero-Immirzi idea was to get rid of complex Ashtekar variables and now 20 years later they reappear in Wieland's work. Hey, they don't look so hairy this time around http://arxiv.org/abs/1107.5002 Twistorial phase space for complex Ashtekar variables Wolfgang M. Wieland (Submitted on 25 Jul 2011, revised 24 Jan 2012) We generalise the SU(2) spinor framework of twisted geometries developed by Dupuis, Freidel, Livine, Speziale and Tambornino to the Lorentzian case, that is the group SL(2,C). We show that the phase space for complex valued Ashtekar variables on a spinnetwork graph can be decomposed in terms of twistorial variables. To every link there are two twistors---one to each boundary point---attached. The formalism provides a new derivation of the solution space of the simplicity constraints of loop quantum gravity. Key properties of the EPRL spinfoam model are perfectly recovered. 18 pages, to appear in Classical and Quantum Gravity
Let's not say it hasn't been tried! : http://arxiv.org/pdf/1302.7037.pdf and see also: http://arxiv.org/pdf/1305.1487.pdf (where he goes a bit further).
Shall we count this as a separate potential rival? Or should we view it as an alternate approach to quantizing Shape Dynamics? http://arxiv.org/abs/1304.5205 On the Geometric Quantization of Canonical Gravity (Submitted on 17 Apr 2013) One of the hardest problems to tackle in the dynamics of canonical approaches to quantum gravity is that of the Hamiltonian constraint. We investigate said problem in the context of formal geometric quantization. We study the implications of the non uniqueness in the choice of the vector field which satisfies the presymplectic equation for the Hamiltonian constraint, and study the implication of the same in the quantization of the theory. Our aim is to show that this non uniqueness in the choice of said vector field, which really stems from refoliation invariance leads to a very ambiguous notion of quantum evolution. We then investigate the case of a theory where the problem of the Hamiltonian constraint has been dealt with at the classical level, namely Shape Dynamics, and attempt to derive a time dependent Schrodinger equation for the quantum dynamics of this theory. 12 pages
Happy as am to see some of what I dabbled in here, I personally feel, with regard to the dynamics that Tim's approach with the effective Hamiltonian is much more pragmatic and far reaching. If you do really want to see my most recent view on how GQ could best be used to quantise SD, see my LOOPS parallel sessions talk. It was always my intention to update this article with that material, but alas, I haven't got the time yet (surely enough I will). For more on quantization of SD, see Henrique Gomes' recent paper on Weyl Anomalies which I think is very elegant.