I was reading somewhere (I think wikipedia on quantum geometry?) that Loop Quantum Gravity is "noncommutative"...but I'm trying to figure this out on my own (naturally, there are no citations for this claim!). Now, my reasoning is that one would try to express various differential forms in Loop Quantum Gravity using "quantized" differential forms (see, e.g., Connes' Noncommutative Differential Geometry Page 10). If anyone knows any good reference, or has any idea whether this would be the correct method for verifying the claim that Loop Quantum Gravity is noncommutative, let me know The various forms that come to mind are: The Coframe 1-form: [tex]e^{I} = e_{\mu}^{I}dx^{\mu}[/tex] The Connection 1-form: [tex] \omega^{I}_{J} = \omega_{\mu}^{I}_{J}dx^{\mu}[/tex] and the Curvature 2-form [tex] \mathcal{R}^{I}_{J} = (1/2)*R_{\mu\nu}^{I}_{J}dx^{\mu}dx^{\nu} [/tex] (Boy I hope that tex code worked, if not I'll just edit the post) Would one simply find the "quantum" differentials [tex]dx^{\mu}=i[F,x^{\mu}] = i(Fx^{\mu} - x^{\mu}F)[/tex] and say "Shazam! I'm done! It's noncommutative, time for a beer!"? (I'm rather new to this noncommutative geometry idea, so I'm unsure if that would be enough to "demonstrate" in a hand-wavy manner that LQG is noncommutative.)
I've wondered whether the results of Conne's paper, including a prediction for the Higgs boson, applies to LQG.
Through my desperate googling for an answer, I did manage to come upon a rather interesting paper: Intersecting Connes Noncommutative Geometry with Quantum Gravity by Johannes Aastrup and Jesper M. Grimstrup. It talks more about the compatibility of noncommutative geometry in the standard model and loop quantum gravity, rather than the noncommutative geometric structure of loop quantum gravity.
Carlo Rovelli, a major LQG researcher, has collaborated with Alain Connes on noncommutative geometry but I'm not aware of Carlo Rovelli attempting to integrate noncommutative geometry with LQG. Alain Connes noncommutative geometry predicts the higgs boson mass to be 160 GEV, within LHC reach. In string theory there are proposal of little higgs that might be seen by Tevatraon in the 120-130 GEV range. It would be interesting if Alain Connes and Carlo Rovelli can embed noncommutative geometry in LQG (or vice versa) and the higgs boson remains 160 GEV and if this is what is seen at LHC.
AP, If you're still stuck looking for references on noncommutativity in LQG, Ashtekar's very recent review "Loop Quantum Gravity: Four Recent Advances and a Dozen Frequently Asked Questions" includes emergence of noncommutative QFT as one of the four advances. Unfortunately this seems to apply only to 2+1 gravity at the moment. There are a number of references in that paper that should provide the gory details.
Hi, I think (and someone can correct me if I am wrong) that LQG is noncommutatitive in the sense that certain geometrical operators fail to commute. For instance, the area operators associated with 2 surfaces that intersect fail to commute. For example, see section V in Ashtekar's review http://arxiv.org/abs/gr-qc/0404018 particularly the discussion after eq (5.18)
I didn't see anyone mention Laurent Freidel papers yet, he is one of a handful of top younger LQG researchers. the problem is which Freidel paper to point to. There are 39 here: http://arxiv.org/find/all/1/au:+Freidel/0/1/0/all/0/1 Some would have to do with NCG in one way or another, and others not. Here are the most recent 10. I have highlighted if "noncommutative" appears in the title. In others it might not be in the title, but could be in the abstract: 1. arXiv:hep-th/0701113 [ps, pdf, other] : Title: The Relativistic Particle: Dirac observables and Feynman propagator Authors: Laurent Freidel, Florian Girelli, Etera R. Livine Comments: 14 pages, Revtex4 2. arXiv:hep-th/0612170 [ps, pdf, other] : Title: From noncommutative kappa-Minkowski to Minkowski space-time Authors: Laurent Freidel, Jerzy Kowalski-Glikman, Sebastian Nowak Comments: 6 pages 3. arXiv:hep-th/0611042 [ps, pdf, other] : Title: Hidden Quantum Gravity in 4d Feynman diagrams: Emergence of spin foams Authors: Aristide Baratin, Laurent Freidel Comments: 28 pages (RevTeX4), 7 figures, references added Journal-ref: Class. Quantum Grav. 24 (2007) 2027-2060 4. arXiv:gr-qc/0607014 [ps, pdf, other] : Title: Particles as Wilson lines of gravitational field Authors: L. Freidel, J. Kowalski--Glikman, A. Starodubtsev Comments: 19 pages, some number of comments and clarifications added, to be published in Phys. Rev. D Journal-ref: Phys.Rev. D74 (2006) 084002 ... ... 9. arXiv:hep-th/0601004 [ps, pdf, other] : Title: Noncommutative Harmonic Analysis, Sampling Theory and the Duflo Map in 2+1 Quantum Gravity Authors: L. Freidel, S. Majid Comments: 54 pages latex 2 ;eps figs 10. arXiv:hep-th/0512113 [ps, pdf, other] : Title: 3d Quantum Gravity and Effective Non-Commutative Quantum Field Theory Authors: Laurent Freidel, Etera R. Livine Comments: 4 pages, to appear in Phys. Rev. Letters, Proceedings of the conference "Quantum Theory and Symmetries 4" 2005 (Varna, Bulgaria), v2: some clarifications on the Feynman propagator and slight change in title Journal-ref: Phys.Rev.Lett. 96 (2006) 221301
The arxiv search engine does not appear to be working properly because it will not come up with the most recent Freidel paper that has "noncommutative" in the abstract (when that is used as a keyword). However we can search manually. For example this January 2007 paper: http://arxiv.org/hep-th/0701113 The Relativistic Particle: Dirac observables and Feynman propagator Laurent Freidel, Florian Girelli, Etera R. Livine 14 pages (Submitted on 12 Jan 2007) "We analyze the algebra of Dirac observables of the relativistic particle in four space-time dimensions. We show that the position observables become non-commutative and the commutation relations lead to a structure very similar to the non-commutative geometry of Deformed Special Relativity (DSR). In this framework, it appears natural to consider the 4d relativistic particle as a five dimensional massless particle. We study its quantization in terms of wave functions on the 5d light cone. We introduce the corresponding five-dimensional action principle and analyze how it reproduces the physics of the 4d relativistic particle. The formalism is naturally subject to divergences and we show that DSR arises as a natural regularization: the 5d light cone is regularized as the de Sitter space. We interpret the fifth coordinate as the particle's proper time while the fifth moment can be understood as the mass. Finally, we show how to formulate the Feynman propagator and the Feynman amplitudes of quantum field theory in this context in terms of Dirac observables. This provides new insights for the construction of observables and scattering amplitudes in DSR." I have always found the arxiv search engine to work fine, so it is very surprising to me that just now it is failing. =================== EDIT, BETTER RESULTS NOW--the seach engine does not recognize "non-commutative" as a word, so one must say "commutative". I tried the arxiv.org search engine with Author=Freidel and Abstract=commutative and this time I got several Freidel papers with the keyword "noncommutative" in the asbstract: http://arxiv.org/find/all/1/AND+au:+Freidel+abs:+commutative/0/1/0/all/0/1 1. arXiv:hep-th/0701113 [ps, pdf, other] : Title: The Relativistic Particle: Dirac observables and Feynman propagator Authors: Laurent Freidel, Florian Girelli, Etera R. Livine Comments: 14 pages, Revtex4 2. arXiv:hep-th/0512113 [ps, pdf, other] : Title: 3d Quantum Gravity and Effective Non-Commutative Quantum Field Theory Authors: Laurent Freidel, Etera R. Livine Comments: 4 pages, to appear in Phys. Rev. Letters, Proceedings of the conference "Quantum Theory and Symmetries 4" 2005 (Varna, Bulgaria), v2: some clarifications on the Feynman propagator and slight change in title Journal-ref: Phys.Rev.Lett. 96 (2006) 221301 3. arXiv:hep-th/0502106 [ps, pdf, other] : Title: Ponzano-Regge model revisited III: Feynman diagrams and Effective field theory Authors: Laurent Freidel, Etera R. Livine (PI) Comments: 46 pages, the wrong file was first submitted Journal-ref: Class.Quant.Grav. 23 (2006) 2021-2062 ... ... But this misses two which we can get using Title=noncommutative http://arxiv.org/find/all/1/AND+au:+Freidel+ti:+noncommutative/0/1/0/all/0/1 1. arXiv:hep-th/0612170 [ps, pdf, other] : Title: From noncommutative kappa-Minkowski to Minkowski space-time Authors: Laurent Freidel, Jerzy Kowalski-Glikman, Sebastian Nowak Comments: 6 pages 2. arXiv:hep-th/0601004 [ps, pdf, other] : Title: Noncommutative Harmonic Analysis, Sampling Theory and the Duflo Map in 2+1 Quantum Gravity Authors: L. Freidel, S. Majid Comments: 54 pages latex 2 ;eps figs I guess the lesson is that sometimes the word or phrase is written "noncommutative" and sometimes "non commutative" or "non-commutative". So one can run into trouble doing keyword searches.
After reviewing some of the papers Marcus has linked (and exhaustive google scholar searching) I couldn't find what I was looking for. Thanks for the help though Marcus, some of the papers were absolutely fascinating regardless. No matter, there is a lingering question on the back of my mind: from my understanding, noncommutative geometry is essentially a sort of "quantum geometry"; supposing this were so, is there any way to take the classical limit of noncommutative geometry to get back a "classical" geometry?
Im not an expert on this stuff, but one has to be a little careful about what we mean when we say 'noncommutative geometry'. Generally speaking its reformulating geometry and topology concepts as algebraic statements concerning the C* algebra of functions on the manifold. One then drops the commutativity of this algebra. The context dependant part arises b/c different authors use the words 'noncommutative geometry' in different ways. For instance, I gather Connes et all work in configuration space, other authors use different constructions. Anyway as to your question. I don't think its possible *uniquely* to pass to a 'commutative/classical' limit, but there are procedures (i gather) to view things as suitable deformations thereof. The part that really confuses me is the topology. In principle we are doing tremendous violence to our geometry by imposing these conditions, it seems intuitively obvious that there will exist heavy topological obstructions from passing between spaces.