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I have expanded the Idea-section at 3d quantum gravity and reorganized the remaining material slightly.
I feel unsure about the pointer to “group field theory” in the References. Can anyone list results that have come out of group field theory that are relevant here?
I find the following noteworthy, and I am not sure if this is widely appreciated:
the original discussion of the quantization of 3d gravity by Witten in 1988 happens work out to be precisely along the lines that “loop quantum gravity” once set out to get to work in higher dimensions: one realizes
that the configuration space is equivalently a space of connections;
that these can be characterized by their parallel transport along paths in base space;
that therefore observables of the theory are given by evaluating on choices of paths (an idea that goes by the unfortunate name “spin network”).
All this is in Witten’s 1988 article. Of course the point there is that in the case of 3d this can actually be made to work. The reason is that in this case it is sufficient to restrict to flat connections and for these everything drastically simplifies: their parallel transport depends not on the actual paths but just on their homotopy class, rel boundary. Accordingly the “spin networks” reduce to evaluations on generators of the fundamental group, etc.
Notice that in 4d the analog of this step that Witten easily performs in 3d was never carried out: instead, because it seemed to hard, the LQG literature always passes to a different system, where smooth connections are replaced by parallel transport that is required to be neigher smooth nor in fact continuous. These are called “generalized connections” in the LQG literature. Of course these have nothing much to do with Einstein-gravity: because there the configuration space does not contain such “generalized” fields.
For these reasons I feel a bit uneasy when the entry refers to LQG or spin foams as “other approaches” to discuss 3d quantum gravity. First of all, the existing good discussion by Witten did realize the LQG idea already in that dimension, and it did it correctly. So in which sense are there “other approaches”?
Which insights on 3d quantum gravity do “spin foam”s or does “group field theory”add? If anyone could list some results with concrete pointers to the literature, I’d be most grateful.
Urs, I agree that one shoudl question these questions. But these are physical models, like phi^4 theory. Mathematically one can study a model and find its predictions. It is social that spin foam models are classified into the field of 3d quantum gravity. String theory similarly arose from dual models for particle physics, and now is a model with completely different interpretation.
Now, I do not see about 3d vs 4d. If this is a toy model about 3d world it does not need to have anything in common with 4d world. In 2d in condensed matter there are statistics like braid statistics which are not allowed in 4d by spin-statistics theorem. So 4d is not an argument against studying something in 3d. Of course, there are people who extrapolate, and in that case this argument is valid. But we shuld take use in $n$Lab just of the interpretations which make sense, or even less, of mathematical models.
the existing good discussion by Witten did realize the LQG idea already in that dimension, and it did it correctly. So in which sense are there “other approaches”?
Sometimes you have hard time calculating something in coordinate space and you go to momentum space. So existence of a good approach does not mean that other approaches are not useful. There is no proof that those other models are not eventually equivalent, but that in pertirbation series some other aspects may be easier to extract or made explicit.
If anyone could list some results with concrete pointers to the literature, I’d be most grateful.
How do you expect to get an asnwer without getting into it ? And, according to the philosophy yo propagate here, the best way to do research is by doing $n$Lab notes along isn’t it ? So how can you ask for a later stage before the first stage is performed ?
Hey Zoran,
I don’t understand what the first half of your reply is referring to. Sorry.
Concerning your very last question: since you added a sentence to the effect that spin foam models and gorup field theory have been used to study 3d gravity, I would just like to add a list of references that demonstrates this claim. You must have had something in mind. We should therefore add something like: “For instance property X of 3d quantum gravity was understood using group field theory in paper ABC”.
I don’t know such a paper and I really would rather not spend time searching for one. But I also would not have announced that it exists. Since you did, it would be nice if you could point me to some article that you have in mind.
If you don’t have any article in mind (and nobody here knows one) then, I agree, we should eventually try to get into this. But then we should also remove the announcement sentence for the moment, because in that case it is a sentence that makes a statement which none of us feels like being able to defend.
You call Einstein-Hilbert action a 3d quantum gravity. Some other physicist calls the spin foam model in 3d. In both cases, the world in 3d does not exist. So the sentence is just recording that social fact, regardless weather there is or there is no proof that two models of non-existing world are equivalent.
What is incomprehensible in the first part of the text I wrote ?? It is plain English. I said that these are mathematical models and as such are OK to study. Regardless of your concern of relation to Einstein-Hilbert theory.
Hm. You mean we shouldn’t agree to say that the Einstein-Hilbert action in dimension other than 4 defines “gravity”, since “gravity” should only mean the observed force in our 4 dimensions?
I don’t think that’s a good move. And I think it is also in contradiction to the standard established use of the terms.
And I also don’t think that people doing spin foams want to be claiming this. They don’t claim that it doesn’t matter whether their model reproduces EH-dynamics at low energies, by arguing that we are free to redefine 3d gravity at will.
Physically, yes, the gravity is only one physical phenomenon which can be observed in our space-time. Even in 4d there are alternatives to Einstein approach. I am not saying that I am favorable for them. For example, Logunov cf. here can also get Mercur perhelium etc. effects out of another “covariant” theory he has a fixation on. Maybe the world is noncommutative at low scales and we need a completely different theory, defined on a noncommutative space-time.
I se eno problem with “standard established use” as one can always say “Newton gravity”, “Cartan-Einstein gravity” and other frameworks which are based on mathematical definitions. One lists models and this is a good mathematical classification. Once one lists the models, then one can add criticism of the community which misinterprets them.
I am not saying that the spin foam people want to be claiming that there is no relation to Einstein-Hilbert gravity. One can comment on misconceptions and myths there and criticize them. But we can record the valid thing and that is that there is a defined model which has some mathematical features of interest.
by arguing that we are free to redefine 3d gravity at will
3d quantum gravity is not a physical reality but a social designation of an intuition leading some group or researchers in theoretical physics. As long as some people make models following that intuitition and classify their models this way, we can record that they have that approach. We may ignore them if their models are crap, but I think that some of these models are intersting mathematically, at least.
But we can record the valid thing and that is that there is a defined model which has some mathematical features of interest.
That’s exactly what I am being suggesting sombody should do. Point out what are the mathematical features of interest.
I don’t know them. Seriously. But if you point out concrete statements, I’ll be interested in having a look.
3d quantum gravity is not a physical reality but a social designation
I very much disagree with this. This feeling of “anything goes” and “you can do what you like” has hurt the field so much that it has become hard to have a serious discussion about it.
Witten in 1988 has solved a well-defined problem: the quantization of the Einstein Hilbert action (with out without cc) on 3-manifolds of the form $\Sigma \times \mathbb{R}$. This is not a matter of choosing “approaches” or defining “frameworks”. Now I would like to know: what do various “approaches” have to add to this?
I am really interested in this. But it seems that you don’t know the answer, either, do you?
El-Nashie also has a proposal for how to quantize gravity. We don’t follow it, so we don’t mention it and don’t link to it. I think we should only mention work on the $n$Lab of which we have at least a rough idea for what it is actually about.
I see the discussion is around the mathematical interest of models. It is fascinating to me how Witten keeps on striking mathematical gold. Is there any indication that LQG, spin foams, group field theory, etc., have made good (deep, unexpected) contact with mathematics?
David, that’s what I am trying to formulate here.
As far as me, personally, am concerned, the answer is: no, as far as I have seen, they have not.
Therefore, naturally, I feel nervous about mentioning the “group field approach” for instance. But I haven’t seen everything. If somebody here knows the approach and can say why it is reasonable and what it achieves, this should be mentioned. But if nobody here knows anything about it, we should be careful. There is just too much activity in this subject that has not been connected to anything, either physics or math.
One has to be a bit careful with simply citing all the articles with “quantum gravity” in the abstract. That’s – to make a comparison for pure mathematicians – that’s as if on a page on the Riemann hypothesis we would blindly list articles that claim to have proven it.
I think.
One has to be a bit careful with simply citing all the articles with “quantum gravity” in the abstract.
I agree. Who is doing it ? If there is a paper in which you see a bad reasoning you should comment on it: we can also throw out some mistaken references at later stage if we decide they are not up to our expectations at first glance. $n$Lab’s ignorance will not help diminish craze about prevalence of bad and random models, they have enough many propaganda sites anyway. But clear counterarguments against some of the raised arguments will and are likely to get impact outside of $n$Lab. Notice also that
cited at spin foam, (from the abstract:) “aims at raising various issues which seem to challenge some of the methods and the results often taken as granted in these domains”, i.e. already tries to clean up some of the low standard methods there.
El-Nashie also has a proposal for how to quantize gravity. We don’t follow it, so we don’t mention it and don’t link to it.
El Naschie does not have any consistent and explicit definitions or nontrivial calculations. Group field theories are just a class of quantum field theories on group manifolds, with well defined action and a perturbation expansions of combinatorial nature, with lots of mathematically interesting cross checks leading to identities and connections between various mathematical characters in the game (e.g. 3nj-coefficients, quantum groups, perturbaton expansions, “simplicial gravity” etc.). Look at the paper of Freidel-Majid 2005 about the mathematically very interesting quantum group Fourier transform which appeared in this context and which gives a completely new light for example to a very important topic of Duflo map in quantization theory of Lie algebras. If we were working in 2d the group field theory would reduce to the mathematics related to the role of ribbon graphs in matrix models, the topic which is central to the Kontsevich’s approach to quantization.
I rephrased the entry sentence at spin foam:
Spin foam models are particular models aimed at finding a theory of quantum gravity. Their relation to Einstein-Hilbert gravity in classical limit seems however not to be convincingly argued for in the literature.
You see, for Chern-Simons theory there is an Axelrod-Singer combinatorial quantization, independently found by Kontsevich (as exposed in his 1990, 1991 talks at Harvard). Kontsevich later wrote an article
in which he sketched the intuition and basic methodology without going into details. The graph homology kind of combinatorics is rather central in that. The very fact that the group field theory is defined in somewhat similar terms makes it potentially interesting for mathematical study.
Funny enough, some of the proponents of loop quantum gravity compare their achievements with string theorist’s in their own critical assesments. One reference is cited at critics of string theory:
By the way, is there any sound proposal from string theory about the origin of dark matter and dark energy ? I was asked about this by a guy working in certain cosmological model in Cartan-Einstein gravity, who has some proposal in his approach.
Group field theories are just a class of quantum field theories on group manifolds, with well defined action and a perturbation expansions of combinatorial nature, with lots of mathematically interesting cross checks leading to identities and connections between various mathematical characters in the game (e.g. 3nj-coefficients, quantum groups, perturbaton expansions, “simplicial gravity” etc.).
But how are they related to 3d quantum gravity?
I went to the review by Oriti that you link to. Apparently in there it is the first full paragraph on p. 10 that is devoted to this question. That paragraph says the following:
The above Feynman amplitudes, modulo the exact form of the edge amplitudes, can be obtained and justified in various ways, e.g. starting from a discretisation of classical BF theory and a subsequent imposition of the constraints [12, 13], and in particular there is a good consensus on the validity of the Barrett-Crane construction for the vertex amplitude. As a hint that at least some of the properties of gravity we want are correctly encoded in the above Feynman amplitudes, we cite the fact that there exists at least a sector of the geometric configurations summed over where an explicit connection with classical discrete gravity can be exhibited. In fact [31], for non-degenerate configurations (i.e. those corresponding to non-degenerate simplicial geometries) the asymptotic limit of the Barrett-Crane amplitude VBC(J) is proportional to the cosine of the Regge action for simplicial gravity, i.e. the correct discretion of the Einstein-Hilbert action for continuum General Relativity. It is therefore very close to the form one would have expected for the amplitudes in a sum-over-histories formulation of quantum gravity on a given simplicial manifold.
I that the state of the art?
I rephrased the entry sentence at spin foam:
Spin foam models are particular models aimed at finding a theory of quantum gravity. Their relation to Einstein-Hilbert gravity in classical limit seems however not to be convincingly argued for in the literature.
Okay, thanks.
You know, I think it is clear that state sum models such as attempted in “spin foam” literature are potentially a great source of constructions of TQFTs. This is clear because we already know that this is the case in 3d! It will be harder to achieve this in 4d, but all the more interesting.
But what I have trouble understanding is why the investigations into this question always go by “quantum gravity”. There will be lots of 4d TQFTs (or positive boundary TQFTs or other variants) and only some of them will maybe be related to gravity. Given that the evidence that any of the existing proposals is related to actual gravity is very slim, my suggestion would be to first just talk about constructions of TQFTs. Establishing that these actually exist seems to be a hard enough problem in the first place.
You see, for Chern-Simons theory there is an Axelrod-Singer combinatorial quantization, […] The very fact that the group field theory is defined in somewhat similar terms makes it potentially interesting for mathematical study.
Okay, good. So it’s potentially interesting. I am happy then to have an entry on group field theory.
Funny enough, some of the proponents of loop quantum gravity compare their achievements with string theorist’s
Not sure what’s funny about that. Everybody does.
But how are they related to 3d quantum gravity?
You are unreasonable here. David asks in 8 about mathematical interest in models. In 9 you say that this is exactly your point. In 10 I answer you the hints on the evidence of mathematical interest and now you divert the discussion to another topic of physical relevance. Can we discuss issues one by one and classify things correctly ?
Not sure what’s funny about that. Everybody does.
The article says that they criticise string theory comparing it with the STANDARDS in his own field, i.e. loop quantum gravity. This is funny, as those standards seem to be even lower.
the state of the art?
It is about the standard in the field. Somewhat like in dualities of string theory: the isolated features are gradually found to match and nobody can make a comprehensive comparison. I am talking only about 3d work.
Given that the evidence that any of the existing proposals is related to actual gravity is very slim, my suggestion would be to first just talk about constructions of TQFTs.
An interested reader who heard about say Livine’s work will look for his work under the traditional classification of his work. So if you want your assesment of the work to be available than you should classify it in expected manner. Also these works cluster together and reading one is almost impossible without reading another. I placed Oriti’s review not because it has the *** word in the title or because I would like it, but because this review or other of his works are used much in the work of Livine etc. which I use in the study of quantum group Fourier transform etc. which are of my interest. You see, in the first approximation the pretriangulated A-infty and stable quasicategories are the same thing but you do not go to study both frameworks simultaneously to figure out something. I may not like the Oriti’s review, but I will not understand the references which have the substance I want to understand if I do not have it handy (including easy link) when in doubt with those.
Can we discuss issues one by one and classify things correctly ?
I am still talking about whether the entry 3d quantum gravity should link – as it currently does – to GFT and spin foams.
I think it should only link to it if it can provide some hint that and how these are related to 3d quantum gravity. If this is an “unreasonable” demand, as you now say, then we should remove those links.
I wrote:
If this is an “unreasonable” demand, as you now say, then we should remove those links.
But probably it is not an entirely unreasonable demand…
We can not work without having working links. You can always include warning signs and statements on the status of the theory, rather than ignoring. I changed the linking sentence to
Authors of spin foam models view them as an approach to quantum gravity.
As group field theory can be considered independently and is linked at the spin foam it does not need to be separately mentioned here, though it may confuse a reader who wants to find an opinion, statements and references to read and judge in $n$Lab.
coming back to #11, where you asked
is there any sound proposal from string theory about the origin of dark matter and dark energy ?
As a rule of thumb, there are lots and lots of semi-realistic and semi-consistent models that have been considered in string theory. (Both the “semi” are due to the fact that lots of approximations enter into these dicussions in order to make any progress at all.)
A natural candiate for dark matter has always been the LSP in any supersymmetric model, hence in string compactifications on CYs. One finds people giving their articles titles like
Natural Dark Matter from Type I String Theory (arXiv:hep-ph/0608135).
More generally, every KK-compactification of a critical string background by construction a priori has lots of scalar particles: the moduli. There is an abundance of them, so no lack of candidates for inflatons, Higgses and dark stuff. Before asking which, if any, of these is one of the scalar that is semi-observed (no scalar particle has been observed, but both the Higgs and the inflaton are (or have been since a few weeks back!!) treated as compelling by model building) before asking this, th big question over the last years has been how to see that there are vavua in which these “moduli are stabilized” in that they have potentials which give them high masses that are consistent with them not being observed in accelerators. It turns out that, indeed, they do pick up masses by coupling to the higher gauge fields in string backgrounds (“flux vacua”).
Originally there had been the vague hope (justified by nothing but hope) that there is an essentially unique solution that has stable moduli. That this is of course not the case, that the moduli space of string vacua a “large” space (has more than a few points!) is the evident observation that came to be associated with the awkward word “landscape”. (I just wish somebody had had the senses to use a proper technical term. But now it’s too late.)
dark matter and dark energy
I haven’t kept up on this, so is there any observational evidence yet that these are not what always seemed to me the most parsimonious possibilities: respectively right-handed neutrinos and the cosmological constant? (The last time that I asked this, I received the reply that there wasn’t, but that was several years ago. The current Wikipedia articles seem to accept these as possibilities.)
My understanding is that neutrino dark matter models don’t seem to be able to fit the data properly. For instance
Hasan Yüksel, John F. Beacom, Casey R. Watson, Strong Upper Limits on Sterile Neutrino Warm Dark Matter , 2008 (PRL)
But I am not following this in much detail. Maybe there are new insights.
Thank you very much for the idea about moduli giving interesting scalars. This looks very interesting. I knew only about LSP out of the possibilities you list but I never found it appealing, coming from strings or not.
haven’t read it yet, but today’s Canonical quantization of non-commutative holonomies in 2+1 loop quantum gravity almost looks as if written in reply to the discussion here.
By loop quantum gravity most physicist around me mean once you go to Ahstekar’s variables (what is 4d). The Ponzano-Regge spin foam model is a slightly different thing in 3d, but developed by the same community and similar conceptual background. Today’s is the version 2, the version 1 of this promising paper (it is new to me) is from May. Thanks for pointing out.
Today on the arXiv.
I can not judge if this fixes some of the open problems alluded to above, or just brings very nice mathematics to not so well founded subject.
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