Currently the two definitive papers are 1004.1780 and 1010.1939, with several others applying spinfoams to cosmology e.g. 1003.3483. LQG is about where one could have predicted back in Fall 2008, with the merger of the canonical, covariant, and cosmological versions. I think in fact one or more people here at PF did observe that trend and predict that. It unifies the theory and brings it closer to testability, because early-universe cosmology is a potential venue for testing. The present form of LQG is at the intersection of lines of work by Ooguri, Atiyah, Feynman, Regge, Penrose. The October paper mentions that it follows from 3 separate approaches: 1. Canonical quantization of the conventional phase space of General Relativity 2. Polyhedral quantum geometry 3. Covariant lattice quantization For details, see 1010.1939 Thus there are signs that the present form of LQG is a satisfactory theory of quantum geometry/gravity without matter. Matter still has to be introduced. So the question concerns the logical next step. Assuming that what we see will turn out to be satisfactory, how can matter be laid on to the spacetime foundation it provides? At first sight, in the one-page formulation given in the October paper, you see a list of FEYNMAN RULES GOVERNING TWO-COMPLEXES. There is a half-page section on page 1 of 1010.1939 called "Feynman Rules" which at the end says "This completes the definition of the model." The 4 Feynman rules determine how to calculate transition amplitudes, for the two-complexes. That defines LQG. So at first sight, and this may be correct as well, the theory is a theory of two-complexes, so if matter is to be added to the picture it must carried by the two complexes. That's one possibility. I'd like to hear any ideas about how this could be done, or about other schemes for including matter.
LQG is not satisfactory. The physical innner product is probably divergent (in addition to the IR divergence). My own guess is that it needs GFT renormalization. And I would prefer if gravity and matter should both emerge together from a GFT.
As an outsider and total ignoramus about LQG, the statement that LQG is OK except for the need for coupling to matter seems a little implausible. Maybe I'm just an idiot, but to me, this would seem to imply that LQG was currently able to reproduce any result that you could get in the classical limit from the vacuum field equations of GR, and to examine the quantum effects when the classical limit doesn't apply. Is this really true? Do the LQG folks have a Schwarzschild metric with quantum corrections? Do they have plane gravitational waves with appropriate modes of polarization?
http://arxiv.org/abs/1010.1939 "The second source of divergences is given by the limit (26)." http://arxiv.org/abs/1010.5437 "We have observed that under certain general conditions, if this limit exist ..." In a different but related context (GFT, not spinfoam LQG), http://arxiv.org/abs/gr-qc/0607032 "That such a sum can be defined constructively thanks to the simplicial and QFT setting is already quite an achievement, and to ask for it to be finite would be really too much!" I don't know, of course, if I did ... But I am hoping for further developments following http://arxiv.org/abs/hep-th/0512113 http://arxiv.org/abs/0903.3475 http://arxiv.org/abs/1004.0672 Some background as to whether gravity and matter should be unified is given in the last reference: "Several approaches to coupling matter within spin foams were embarked upon [2–7]. The most tractable and indeed most successful of these procedures embedded the Feynman diagrams of the field theory into the spin foam. Remarkably, summing over the gravitational degrees of freedom, the effective matter amplitude was seen to arise as the Feynman diagram of a non-commutative field theory [8]. To add to this position, it was shown that an explicit 2nd quantised theory of this gravity matter theory could be provided by group field theory, while later the non-commutative field theory was seen to arise as a phase around a classical solution of a related group field theory [9]. Of course, one may approach the subject with the view that one should discretise the field directly on the spin foam, since in the continuum theory, we expect that the field has a non-trivial energy-momentum tensor, and should affect the state sum globally. This method has yielded to a succinct initial quantisation for Yang-Mills and fermionic theories [4–6], but due to the non-topological nature of the resulting amplitudes, further calculations proved unwieldy. Now, it was not our intention that this work would or should settle this debate, but we find that this theory is more in line with the arguments of the former way."
I am not making that statement. The author of the papers, which provide a kind of current status report, does not. You should look at the two papers, which have carefully qualified statements with a lot of references. I would say there are signs that the current form might be satisfactory----a kind of final version of LQG. That does not yet mean that it is RIGHT. (One still has to derive predictions, and test.) But if we are seeing something like a finished version of the theory, then a natural question to ask is how to add matter. So this is a kind of speculative experiment. If the present form were satisfactory, how would matter be added? I don't immediately see how,and I would like to get people's ideas of how it could go.
None of this has been achieved. The latest is summarized in http://arxiv.org/abs/1004.4550 . This is not sufficient, because eg. DT started from a similar point, but didn't produce anything sensible until it became CDT. Also, in CDT, although one starts with the Regge action, a continuum limit is supposed to be taken ultimately. Whereas in LQG, the Regge action is the classical limit, but that would seem to imply classical spacetime is discrete? Is another limit missing? Would that limit commute with the classical limit? Or will matching the free parameter in LQG make the discretization sufficiently fine? The free parameter is discussed http://arxiv.org/abs/1010.1939 "Let's call LPl the unit of length in which all the equations above hold. LPl is a fundamental parameter of the theory, setting the scale at which the theory is defined, namely the scale of the quantum granularity of space"
I'd be interested to hear about that too. The reference Atyy gave, with the quote, was http://arxiv.org/abs/1004.0672 The particle interpretation of N = 1 supersymmetric spin foams V. Baccetti, E. R. Livine, J. P. Ryan (Submitted on 5 Apr 2010) "We show that N = 1 supersymmetric BF theory in 3d leads to a supersymmetric spin foam amplitude via a lattice discretisation. Furthermore, by analysing the supersymmetric quantum amplitudes, we show that they can be re-interpreted as 3d gravity coupled to embedded fermionic Feynman diagrams." Here's what was quoted in Atyy's post: It isn't clear to me, and we are having company so I won't have time soon to try to figure it out. Would be grateful for any hints as to how this might work.
Atyy, I looked at the paper you indicated that you were quoting. I found this on page 15, in the conclusions: ==quote 1004.0672== Finally, the most interesting application to our formalism would be to study the insertion of actual physical non-topological fermionic ﬁelds. Starting in 3d, in the present work, we have tracked from the initial continuum action down to the ﬁnal discretised spinfoam amplitude how the explicit fermionic Feynman diagrams get inserted in the spinfoam amplitude. These fermionic observables come with precise weights (see e.g. eqn. (39)-(40)). These weights are ﬁne-tuned so as to ensure that the full model ‘gravity+fermions’ is topological. That shows that these spinfoam amplitudes provide the correct quantisation for our supersymmetric theory. As soon as we modify these weights, we would get non-topological amplitudes and it would be interesting to see how we could modify them in order to insert more physical fermionic ﬁelds. Then, we hope to apply the same procedure to the four-dimensional case by ﬁrst deriving the spinfoam quantisation of supersymmetric BF theory and studying how the fermions are coupled to the spinfoam background, and then seeing how this structure is maintained or deformed when we introduce the (simplicity) constraints on the B-ﬁeld in order to go from the topological BF theory down back to proper gravity. Another interesting outlook is to push our analysis to N = 2 supersymmetric BF theory, already in three space-time dimensions, following the footsteps of [7]. Indeed, such a theory already include a spin-1 gauge ﬁeld, and we could study in more detail how the full supersymmetric amplitudes decomposes into Feynman diagrams for the fermions and spin-1 ﬁeld inserted in the gravitational spinfoam structure. Then we would see how it is possible to deform this structure in such a way that the spin-1 ﬁeld represents standard gauge ﬁelds. This road would provide an alternative way to coupling (Yang-Mills) gauge ﬁelds to spinfoam models, which we could then compare to the other approaches developed in this direction [6]... ==endquote== I am thankful for any indication of how the researchers imagine adding matter to the picture. The LQG literature goes back and forth between Group Field Theory (GFT) and spinfoam and BF theory. It is becoming all one. So whichever way you can get matter in, seems fine. With this paper, they are working in 3D and so far can just hope to extend the method to 4D.
Afaik neither Schwarzschild nor de Sitter has been reproduced so far. The graviton propagator has been constructed over the last couple of years by Rovelli et al. and was shown to have thre correct limit. This is a kind of consistency check b/c one does not know whether the graviton propagator as constructed from standard GR at tree level is of any physical relevenace beyond (!) tree level (as standard GR fails to be consistend beyond tree level).
Tom responded as relates to plane gravitational waves (at least I think what he mentioned about the LQG graviton applies in that direction.) In LQG there are BH models which reproduce classical results with quantum corrections. There are many papers and you can judge for yourself how complete the program is in that department by looking at recent ones. I doubt that this is all that relevant to the main topic question of how to add matter. But here are some papers to glance at, if you are curious: http://arxiv.org/abs/1007.2768 http://arxiv.org/abs/1006.0634 http://arxiv.org/abs/0905.3168 The first one here, for example: Generic isolated horizons in loop quantum gravity Christopher Beetle, Jonathan Engle (Submitted on 16 Jul 2010) "Isolated horizons model equilibrium states of classical black holes. A detailed quantization, starting from a classical phase space restricted to spherically symmetric horizons, exists in the literature and has since been extended to axisymmetry. This paper extends the quantum theory to horizons of arbitrary shape. Surprisingly, the Hilbert space obtained by quantizing the full phase space of all generic horizons with a fixed area is identical to that originally found in spherical symmetry. The entropy of a large horizon remains one quarter its area, with the Barbero-Immirzi parameter retaining its value from symmetric analyses. These results suggest a reinterpretation of the intrinsic quantum geometry of the horizon surface." From my perspective as outside observer, I reckon that matterless LQG is now reaching a satisfactory stable formulation, so that it is time to ask how they are going to include matter. What approaches will be tried? What makes sense given the kind of "asymptotic" formulation (w/o matter) that we are now seeing emerge? What I THINK is that the best clues, or hints come from looking at last year's Oberwolfach workshop "Noncommutative Geometry and LQG" http://owpdb.mfo.de/show_workshop?id=783 What this shows me is a network of people, which includes Alain Connes and Vincent Rivasseau even thought they did not directly participate in the workshop. So it is a window on a network of potentially fertile ideas. You see there elements of GFT (group field theory) Noncommutative field theory (e.g. Richard Nest, Thomas Schücker) NCG (Marcolli and others). One of the workshop participants was Thomas Krajewski, a member of Rovelli's QG team at Marseille. I'll list his papers to see what things he has worked on.
BTW I think it would be naive to start asking if the theory is right or wrong, or to start making bets. What we see now are more like signs of maturity. What a I called an "asymptotic" version. The formulation is mathematically extremely nice. It is at a convergence of several lines of QG research, that I mentioned in the opening post. It is at an intersection, making contact with other things I mentioned (NGC, Ooguri, BF, Feynman diagrams, GFT, Regge). Notably also we see a strenghtened coherence, it's clear now that canonical=covariant=cosmology. A loose association has fused and taken shape. This is still happening, which is why I called the formulation asymptotic. But AFAICS it is time to assume that the main outlines (of matterless LQG) will remain as they are and to look ahead. It is how matter is added that could change things now. ================ So, looking for clues as to how that could go, I looked at Krajewski's list of papers. He has 24 on Spires going back to before his 1998 PhD thesis. Look at the network of topics and collaborators: http://www.slac.stanford.edu/spires/find/hep/www?rawcmd=a+Krajewski,+Thomas&FORMAT=WWW&SEQUENCE= My sense is that somewhere in that "hotbed" of mathematical topics, that you see in the list, there are the seeds of how to put matter into LQG. And it is not the person (Krajewski in this case) but the web of mathematically fertile ideas that you see. The person or persons could be anybody---someone we have heard of or not heard of. I am trying to comprehend what is comprised in this mathematical "hotbed". BTW Alain Connes and Vaughn Jones were on Krajewski's 1998 Thesis committee, at Marseille. The more I hear about CPT Luminy at Marseille the more I like it. It seems to enjoy a good intellectual climate.
I think this particular cluster is spinfoams-GFT-non commutative field theory and comes from a paper http://arxiv.org/abs/hep-th/0512113 and a manifesto http://arxiv.org/abs/hep-th/0505016 .
Two more graduates of the ENS Lyon! Must be something in the water at the École Normale :-D. But then tell me if you see: what mathematical form would the matter take? The basic object here is the set of square-integrable complex-valued functions on a cartesian product of (just any number) K copies of a compact group G. L_{2}(G^{K}) (It looks good already: L_{2} spaces and compact groups are some of the really nice things in mathematics.) In matterless LQG the group G is SU(2). And K is the number of links in a graph. Afterwards the "graph goes to infinity" but the theory is initially built on finite graphs. Could it be that one adds matter to the picture simply by enlarging the group G? This could have been what was happening in the paper you quoted only a few posts back. Baccetti Livine Ryan. As I recall the group UOSP(1|2) appeared in that paper. I don't know that group. I suppose there are other possibilities. the L_{2} space could be a set of functions not from G^{K} to the complex numbers x+iy but to some other number system, or to matrices. That does not immediately make sense to me, so I am inclined to prefer thinking about the first possibility---an enlarged group G manifold---for the time being. ===================== Here is a 1980 paper about the group UOSP(1|2) by Berezin and Tolstoy http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.cmp/1103908695 Free open access provided by Project Euclid. To recall the paper that uses UOSP(1|2):
Whatever happened to the idea that matter is already included in LQG (and similar approaches) via the Bilson-Thompson topological preon construction (see, e.g. here)? I seem to recall a claim by Smolin that such models may be 'already unified', and thus, one wouldn't have to add matter as much as to just find it.
Skyrmions have been used to model baryons. It has been predicted that they could be created in a multicomponent Bose–Einstein condensate. Skyrmion is a particular case of a topological soliton. Is this theory a mainstream now or something beyond the mainstream ?
Skyrmions are valid in the context of chiral effective theories using pions (and other mesons) as degrees of freedom. Afaik there is no reason why Skyrmions should be treated as something more fundamental. But that is certainly not relevant in the LQG context.