The Plebanski action extended to a unification of gravity and Yang-Mills theory
(Via Physics Forums). This is the new paper by Lee Smolin: http://arxiv.org/abs/0712.0977.
We study a unification of gravity with Yang-Mills fields based on a simple extension of the Plebanski action to a Lie group G which contains the local lorentz group. The Coleman-Mandula theorem is avoided because the theory necessarily has a non-zero cosmological constant and the dynamics has no global spacetime symmetry. This may be applied to Lisi’s proposal of an E8 unified theory, giving a fully E8 invariant action. The extended form of the Plebanski action suggests a new class of spin foam models.
Notice the citation to Lisi’s recent paper.
I read Lisi’s paper and I’ll read this one. Sounds very interesting indeed, and a quite “fast link” to Lisi’s idea, that is, a connection of that idea with the LQG framework.
If you would like to discuss that paper in technical terms, I’ll be glad to host your comments here. Recall that this is a highly moderated, very high signal to noise ratio blog. Be educated.
Update: A personal view can be found here.
December 11, 2007 at 8:37 am
For me, the most important statement is the claim that this is a novel class of quantum theories to which the Coleman-Mandula theorem does not apply, but this claim is not defended at any length. Instead there is a purely classical discussion of Lisi’s symmetry-breaking method, and then a brief comment that you could make a spin-foam model out of this.
Several times over, in loop quantum gravity, big promises turned out to be wrong. Spin networks did not offer an independent prediction of black hole entropy. Thomas Thiemann’s method of quantizing a string, which was supposed to work in any dimension (and not just d=10), is physically wrong even for a simple harmonic oscillator. The Barrett-Crane model does not work.
The explanation given here sounds rather peculiar. Allegedly there exists a nonsupersymmetric quantum theory with a symmetry linking fields of different spins. How does this get past Coleman-Mandula? Because the theory has no S-matrix, except in a limit where the symmetry is safely broken! Even to me in my ignorance, that sounds dubious beyond belief. OK, so you can’t make an S-matrix for quantum fields in de Sitter space; but you can still have correlation functions. Won’t the full symmetry show up there, then? And won’t that run up against some generalization of Coleman-Mandula? If it doesn’t, then that’s a major result, but there’s no evidence here that this has been examined at all. Instead, there is just the formal manipulation of an action until one gets the theory with reduced symmetry, and then one says, ‘and obviously that can be quantized’. Whether the quantum theory defined by the UN-modified BF action exists has not been demonstrated at all.
After the most recent discussions at Physics Forums, my impression is that the theory with reduced symmetry in Lisi’s paper probably does exist, but implicitly it is descended from a full E8 theory which may not exist at all (except classically), in which case the manipulations describing that descent are purely formal in character. If I’m right, the way it all works is as follows: You can indeed construct a classical E8 theory, with 248 fields whose couplings come straight from the group algebra. Then you add some more terms, along the lines described in Smolin’s paper, and now you have a classical theory with a much smaller symmetry – a Pati-Salam model coupled to gravity. Then you quantize that; and now you have a parameter-free theory of everything, because the couplings are inherited from that earlier stage. But that earlier stage cannot be quantized – at least, no-one has given any evidence yet that it can be quantized.
December 11, 2007 at 4:39 pm
[...] calmed down (although we can still hear some fresh echoes of “Lisi’s wave”), a new paper by Smolin, citing Lisi’s work, appeared in the [...]
December 11, 2007 at 4:55 pm
Mitchell Porter,
Thanks for your comment. You wrote:
Several times over, in loop quantum gravity, big promises turned out to be wrong. Spin networks did not offer an independent prediction of black hole entropy. Thomas Thiemann’s method of quantizing a string, which was supposed to work in any dimension (and not just d=10), is physically wrong even for a simple harmonic oscillator. The Barrett-Crane model does not work.
This is quite a recurrent discussion. I’m not sure there is a consensus on those issues. I would like to ask those who follow discussions here to include references to any definite claims. I do not ask this as a nitpicking exercise. I’m trying to learn from it and I am sure there are many readers that can benefit from it.
I have no comments for the rest of your post, for the moment.
Best,
Christine
December 11, 2007 at 9:50 pm
Hi – Taking those in reverse:
1) The graviton propagator in the Barrett-Crane model is wrong. There is now a follow-up paper which says that by making a “rather artificial” modification you can get it right.
2) LQG quantization of the harmonic oscillator: original paper, follow-up, discussion #1, discussion #2 (a link to your old blog there!).
3) Immirzi parameter / black hole entropy: I don’t know a good review paper, but see Luboš Motl’s comments, last three paragraphs.
December 12, 2007 at 6:59 am
Hi Mitchell,
That helps a lot. I’ll try to organize those links/references into a kind of FAQ in order not to loose sight from those recurrent questions in LQG. I’ll also try to recover some essential links from my old blog in order to make that organization. Actually, blogs are not the best way to organize information, but I’ll try to make those essential points more visible.
Best,
Christine
December 13, 2007 at 3:07 am
Some further thoughts, resembling your personal view–
Lisi’s paper contains a group-theoretic construction and a field-theoretic construction. The group theory has received the bulk of the attention. Smolin’s paper is an opportunity to focus on the field theory.
Especially in the wake of Distler’s critique, there are already alternative but Lisi-inspired proposals for E8-based unification floating around, some of which appear in Smolin’s paper. My philosophy is that for purposes of self-education, it is worth focusing on Lisi’s original proposal for a while longer. The personal milestone one should aim for is to understand Lisi’s original constructions completely, and also to be able to understand completely by oneself, from first principles, any errors they may contain. From this perspective, further ideas about “fermions from disordered locality”, etc, are a distraction from an opportunity to achieve a new level of rigor in one’s own understanding. Lisi’s original proposal is self-contained and offers numerous lessons. Even for an aspiring string theorist, the glimpse into the guts of E8 might be useful, to understand something like Bourjaily’s ideas.
So I think the key field-theoretic issue here is the status of these “extended Plebanski” models(?) with respect to the Coleman-Mandula theorem (with fermions-from-BRST in second place). I made some statements, similar to my first comment here, at Cosmic Variance, and Lee Smolin chimed in to repeat his contention that there is no problem, since the actual symmetry is not G, it’s just (spacetime symmetry)+(some subgroup of G). But a few comments down he also says that he “give[s] an action which is fully gauge invariant, not ad-hoc and is directly amenable to quantization with known techniques”. That would refer to his section 3.
I suspect the basic problem is that any consistent quantization of the unbroken BFE8 theory – as would be defined by Lisi’s action without the two gauge-fixing terms – would be purely bosonic. But the broken theory has fermionic fields, so where do they come from? This is why Lisi wants to get his fermions from BRST ghosts, and why Smolin speculates about getting them from “disordered locality” – because there’s a problem! (if one wants to get the phenomenological theory from the unbroken theory somehow). But there are other things that still confuse me. I thought the whole motivation to put fermions and bosons into the one E8 representation was because E-series exceptional groups have classical representations mixing spinors and tensors. But doesn’t that mean that the spinors ought to become fermionic when quantized, because of spin-statistics? In which case even the idea that there’s a purely bosonic “quantum E8 theory” would be wrong. Or maybe Lisi’s BRST-quantized E8 theory, with its fermionic ghosts, is just a disguised form of supergravity!
But in any case, even if we put aside the trouble with fermions, these extended Plebanski models are still putting spin-1 and spin-2 bosons into the same representation, and that is still at odds with Coleman-Mandula. I finally looked up the original CM paper, and it looks quite comprehensible. My study plan would be to focus on this tip from hep-th/0611133 –
“The Coleman-Mandula theorem has a simple intuitive explanation. Considering a general two-body scattering process in field theory, we know that Lorentz invariance fixes the incoming and outgoing trajectories to lie all on the same plane and translation invariance fixes the energies of the outgoing particles. So, the only unknown in this process is the scattering angle. Allowing for an internal symmetry that mixes with the Poincare symmetry would further constrain the kinematics of the process and would result in a quantization of the scattering angle. Analyticity of the S-matrix would then imply that the transition element is independent of the angle, i.e. that the theory is trivial.” (section 2.1.1)
–and then try to understand the nine successive lemmas of the CM paper in terms of that simplified case.
But while reading the CM paper is probably good practice for anyone who wants to think like a field theorist, I’m not sure how helpful it will be in understanding Smolin and Lisi, because they say that the CM theorem doesn’t apply because their models have no S-matrix, or only have it in a certain limit. Lubos Motl, among others, has pointed out that even if you have an S-matrix only in an approximation, Coleman and Mandula’s reasoning should still be approximately valid, and so problems should remain. But Smolin and Lisi can counter with the argument that by the time you make the approximation, the symmetry group is safely reduced, as above. But then I don’t understand the sense in which the full symmetry ever applies. Again, I want to see how this is supposed to work. The idea seems to be, full symmetry in a topological phase, broken symmetry in a spacetime phase with a metric and an S-matrix. My concept of what a “topological phase” looks like is limited to Witten 1988 – expectation values of holonomies along loops. I don’t understand how to combine that and ordinary field theory into the one theory; but that was twenty years ago, maybe it’s been done by now.
Further confusing the issue, sometimes they (Smolin and Lisi) emphasize that an “extended Plebanski model in Lisi gauge” (my terminology) is of necessity in de Sitter space rather than Minkowski space, as if that is the reason why the CM no-go doesn’t apply. But dS field theory isn’t just topological, right? You don’t have an S-matrix, but you do have correlation functions – in which case Motl’s argument that the CM theorem should apply approximately remains. Or would they say that the full symmetry is necessarily broken in de Sitter space too?
December 13, 2007 at 7:28 am
Mitchell,
You raise very interesting questions, thank you for sharing them here. And I completely agree with your points in the first paragraphs.
I’ll go into the links you provide, thanks.
One thing important for the readers: please do probe for links with your mouse over the text. The present style of this blog does not show clearly the links. Mitchell included several links in his comment that do not look visibile without probing with the mouse. If someone knows how to change this style please let me know!
Christine
December 13, 2007 at 2:07 pm
In the last comment by Mitchell, there is a bug in one of his links. The link to Witten 1988 is this:
http://projecteuclid.org/handle/euclid.cmp/1104161738
Christine
December 13, 2007 at 2:27 pm
Dear Mitchel,
I don’t have time at the moment to address all your points, will do so in the next few days when there is time. But this stuck out: “But Smolin and Lisi can counter with the argument that by the time you make the approximation, the symmetry group is safely reduced, as above. But then I don’t understand the sense in which the full symmetry ever applies. Again, I want to see how this is supposed to work.” In the version in my paper the full symmetry G is a LOCAL GAUGE invariance. This is never broken-gauge symmetries never are-. In the solution I present there is a global symmetry which is smaller, in that case the deSitter group + H, where H=the largest compact subgroup of
G/dS.
The CM theorem applies only to global symmetries, and only to the limit where the cosmological constant vanishes so there could be an S matrix. In this limit the global symmetry of the solution I present is (euclian) lorentz + H. There is no conflict with CM. As you note I ssaid this already in the paper and on CV.
The key point is to distinguish global symmetries from local gauge invariances. The former are transformations between distinct physical states. The latter are different coordinizations of the same physical state or solution. If you do not make this distinction clearly there will be many issues in gauge theories and GR that will be confusing.
As for your polemical intro, you mistate the situation with Thiemann’s proposal for a different quantum theory of the string. It is-as he says clearly in his paper-not a different approach to the standard string-it is a distinct quantum theory arising from a different quantization procedure. This is neither surprising nor unusual. Different quantizations of the same classical theory often give different quantum theories. The fact that this is true also of a system with one degree of freedom is also unsurprising and irrelevent.
And BC is not wrong, but there are proposals for other actions that may have better properties. This is just the standard progress. No one ever claimed there was a unique dynamics for spin foam models, from the beginning different choices of the spin foam amplitude have been studied, what has happened is the discovery of some new interesting alternatives.
Thanks,
Lee
December 13, 2007 at 10:07 pm
Or maybe Lisi’s BRST-quantized E8 theory, with its fermionic ghosts, is just a disguised form of supergravity!
Indeed, and since Lisi makes use of the real split form of E8, namely E8(8), my first guess would be 4D maximal N=8 supergravity, where E8(8) is realized as a quasiconformal group in the 57D charge entropy space of BPS black hole solutions. See Gunaydin’s hep-th/0409263 for more details on this particular supergravity.
December 15, 2007 at 12:20 pm
Hello everyone… In a discussion at this level, I really ought to be lurking in the audience; too much of what I “know” is second-hand. Nonetheless–
As I wrote above, what I would like to understand is the field-theoretic construction in Lisi’s paper. In discussion, he writes that a similar theory could be constructed for the smaller group F4; and assuming the decomposition f4=so(9)+9, I think working through Lee Smolin’s argument for the case G = SO(9) might help with the construction of that simpler case. But let me see if I even understand how that argument runs:
We start with a topological Yang-Mills theory with gauge group G. We find a solution to the classical equations of motion which is interpretable as gravity coupled to topological Yang-Mills with gauge group H, H a subgroup of G. Then somehow we use that to build a quantum theory of gravity coupled to Yang-Mills, possibly in the form of a spin-foam model. (And this all happens in Euclideanized de Sitter space.) For G = E8, this theory will be the bosonic part of Lisi’s, plus higher-order corrections.
That sounds like a plan. But I do see in the literature a lot of headscratching about how to formulate quantum theory in de Sitter space and how to formulate spin foams in general, so this may not be as straightforward as it sounds.
December 15, 2007 at 1:24 pm
Hi Christine:
I’m not using wordpress, but the style is probably set in your css style sheet, seems to be in this folder (wp-content/themes/pub/redoable-lite/style.css) look for something with ‘link’ and ‘comments’ and change the style (i.e. underline, bold, color) etc. how you like. Since it works in your post, maybe just use the same style for the comments. You find info on css style sheets in abundance on the web.
While I am at it, I’d like to repeat my question that I’ve previously asked in several places (you don’t have to answer it). All this might work for gauge fields, so even though I can’t quite follow the details of Lee’s paper I find it possible. But whatever you do with your E8, if you put the fermions in there, write down an action with unbroken symmetry, and then break it, I can’t see how the Fermionic part for the Lagrangian can possibly come out with the correct order of derivatives.
The reasoning is just if you start treating fermions like gauge fields, then how come they can eventually couple differently? Having fields that transform under the right Lorentz representation isn’t sufficient. The gauge field part is DADA, the fermionic part is *not* D\psiD\psi. There is one D too much. Lisi fixes this in his paper by picking a Lagrange multiplier that in the Fermionic case is just \psi without derivatives.
( That’s why I’ve been nitpicking on the coupling constants and dimensions of the fields. Since the Lagrangian for Fermionic fields is psi D psi, the dimension of Fermionic fields is differnt to gauge fields. I have no problem adding things that transform differently, but I have a problem adding things with different dimensions.)
Best,
B.
December 15, 2007 at 2:43 pm
Hi Bee,
Thanks for your help. But as went to make the changes, it appears that I have to purchase the service in order to edit my CSS style sheet. For the moment, I’m using the free service.
And thanks for posting your question. It’s sounds reasonable. Of course the action must come out with the right dimension…
Best,
Christine
December 15, 2007 at 9:03 pm
Hi Christine:
Now that you mention it, I think I actually tried wordpress before I set up the blog at blogger, because wordpress insisted I’d need to ‘upgrade’ my account or stick with the default template. Either, for whatever reason your blog seems to be the only one that shows this funny icon with the cat I must have chosen sometime somehow. All the best,
B.
December 16, 2007 at 8:08 am
Hi Bee,
Eh eh… Yeah, that’s a bad point in chosing WordPress. But I am satisfied with the rest. It’s superior in many senses to blogger. Concerning the cat, I don’t know whether I can change it. It’s funny that it is only my blog that is doing this to you. Sorry. (But the cat is pretty).
Best
Christine
December 23, 2007 at 3:36 am
There is a new paper by Roberto Percacci (co-inventor of the graviweak unification used by Lisi) on the gravitational frame-Higgs mechanism (used by Lisi and by Smolin): http://arxiv.org/abs/0712.3545
February 25, 2008 at 12:27 pm
[...] posts on this (on inverse chronological order): here, here and [...]