The December issue of Scientific American is out, and it has an article by Garrett Lisi and Jim Weatherall about geometry and unification entitled A Geometric Theory of Everything. Much of the article is about the geometry of Lie groups, fiber-bundles and connections that underpins the Standard Model as well as general relativity, and it promotes the idea of searching for a unified theory that would involve embedding the SU(3)xSU(2)xU(1) of the Standard Model and the Spin(3,1) Lorentz group in a larger Lie group.
The similarities between (pseudo)-Riemannian geometry in the “vierbein” formalism where there is a local Spin(3,1) symmetry, and the Standard Model with its local symmetries makes the idea of trying to somehow unify these into a single mathematical structure quite appealing. There’s a long history of such attempts and an extensive literature, sometimes under the name of “graviGUT”s. For a recent example, see here for some recent lectures by Roberto Percacci. The Scientific American article discusses two related unification schemes of this sort, one by Nesti and Percacci that uses SO(3,11), another by Garrett that uses E8. Garrett’s first article about this is here, the latest version here.
While I’m very sympathetic to the idea of trying to put these known local symmetry groups together, in a set-up close to our known formalism for quantizing theories with gauge symmetry, it still seems to me that major obstructions to this have always been and are still there, and I’m skeptical that the ideas about unification mentioned in the Scientific American article are close to success. I find it more likely that some major new ideas about the relationship between internal and space-time symmetry are still needed. But we’ll see, maybe the LHC will find new particles, new dimensions, or explain electroweak symmetry breaking, leading to a clear path forward.
For a really skeptical and hostile take on why these “graviGUT” ideas can’t work, see blog postings here and here by Jacques Distler, and an article here he wrote with Skip Garibaldi. For a recent workshop featuring Lisi, as well as many of the most active mathematicians working on representations of exceptional groups, see here. Some of the talks feature my favorite new mathematical construction, Dirac Cohomology.
One somewhat unusual aspect of Garrett’s work on all this, and of the Scientific American article, is that his discussion of Lie groups puts their maximal torus front and center, as well as the fascinating diagrams you get labeling the weights of various representations under the action of these maximal tori. He has a wonderful fun toy to play with that displays these things, which he calls the Elementary Particle Explorer. I hear that t-shirts will soon be available…
Update: T-shirts are available here.
Aaron wrote:
This is often a gauge field (gauge mediated susy breaking) or a graviton (gravity mediated supersymmetry breaking).
This is not what “gravity mediated supersymmetry breaking” means. “Gravity mediated supersymmetry breaking” means “supersymmetry breaking mediated by any set of Planck-suppressed operators.” In some sense, the minimal version of it is what’s known as “anomaly mediation,” but it encompasses a huge range of models (and some not-quite-models, like “mSUGRA” or “minimal supergravity” which is more of an ansatz than a model, and which unfortunately is most of what experimentalists have been setting limits on for decades).
The trouble with generic Planck-suppressed operators is the flavor problem. As John said, there are about a hundred parameters in the MSSM with soft SUSY breaking, but phenomenology imposes strong restrictions so that really only about 20 are completely independent. If you tried to wander very far outside of this low-dimensional subspace in the 100-dimensional parameter space, you would be in gross conflict with observations. To give an example, if selectrons and smuons are both light, they have to be almost the same mass. So gravity mediation requires extra structure to explain these phenomenological facts, and this structure must be present at or near the Planck scale and survive running down to low energies.
As for John’s question:
Does someone know how to get a supersymmetric extension of the Standard Model where supersymmetry is broken spontaneously?
It’s important to note that particles beyond those of the MSSM are needed for supersymmetry to be broken spontaneously, which is why models always involve a hidden sector. This was realized quite early on; the paper by Dimopoulos & Georgi that introduced the MSSM with soft SUSY breaking explained that without a hidden sector (i.e. if SUSY is broken spontaneously in the MSSM alone), the theory would always have a scalar lighter than the up or down quark.
John: Thanks for this — I hope we can clarify a lot, and maybe even make some new progress.
Excellent point. Here’s what happens. We start with an E8 principal bundle with connection (not a superconnection). The symmetry breaks when this connection gets a vacuum expectation value (VEV), [tex]A \simeq E_0[/tex] (I’m not sure if TeX is working here in the comments, as it once was), which leaves the curvature 0. (One way this could happen spontaneously, starting with an E8 invariant action, is described in the paper with Lee and Simone, but the particular mechanism isn’t so important.) This spontaneous symmetry breaking picks out some directions in E8 as special, allowing all other generators in E8 to be identified (and named) with respect to these, based on their Lie brackets. Since it’s key, let me describe this in more detail. If we describe the E8(-24) Lie algebra as
e8 = spin(12,4) + 128^+_S
then the VEV of the connection is [tex]E_0 = 1/4 e_0 \phi_0[/tex], in which [tex]\phi_0[/tex] is the VEV of a Higgs multiplet that transforms as a 12 vector under a spin(11,1) subalgebra of the spin(12,4), and [tex]e_0[/tex] is the 1-form frame field of deSitter spacetime, transforming as a 4 vector under a spin(1,3) subalgebra of spin(12,4), such that the nonzero VEV is in the complement of spin(1,3) and spin(11,1) in the spin(12,4) of e8. It had to be deSitter spacetime if the curvature of the connection is to be 0, with cosmological constant related to the Higgs VEV. Personally, I think this symmetry breaking mechanism — combining cosmogenisis with a Higgs model — is… awesome. I’d enjoy getting your feedback on it.
The “bosonic and fermionic” parts of the connection can only mix before spontaneous symmetry breaking — which is to say, before our universe technically exists. However, if an appropriate action has been chosen that is independent of the “fermion” parts of the E8 connection, then there is a prescription for replacing the “fermion” parts of the connection (1-forms valued in the parts e8 that we’re calling the fermion part, based on [tex]E_0[/tex]) with Grassmann fields, which are identified as fermions (or pre-fermions, if you like). Now, based on our action, and on [tex]E_0[/tex], we could separate out the [tex]128^+_S[/tex] as the fermion part, or, as I consider preferable, we could break e8 up as
e8 = spin(4,4) + spin(8) + 8×8^+_S + 8×8^-_S + 8x8_V
and consider those last three blocks of 64 as pre-fermion Grassmann fields. This works because spin(4,4) + spin(8) is reductive in e8.
Ah, as I’m reading this, I see I have an email from you…
John Baez,
A recent, up to date, small (20 pages) concise review with a comparison of the various mechanisms (pros and cons) and potential string theory realizations is the following
arXiv:1006.0949 by Alwis
This is not what “gravity mediated supersymmetry breaking” means.
Geez. I go away for a few years and I already start forgetting things. Enh. Phenomenology was never my thing anyways.
For Garrett, you still haven’t explained whether the infinitesimal generators of your symmetry are all commuting or if some or Grassman.
And do you still claim to be able to reproduce any part of the standard model action?
I do apologize for confusing your paper with an earlier one of Lee’s which was fermion free, however. However, your paper seems more along the lines of Percacci’s earlier paper where fermions are considered separately and not in the same multiplets as bosons.
I will probably have to bow out of this discussion now, however.
“This works because spin(4,4) + spin(8) is reductive in e8”
How the Standard Model gauge group sits inside of spin(4,4) + spin(8) ?
“Personally, I think this symmetry breaking mechanism — combining cosmogenisis with a Higgs model — is… awesome.”
Does that mean that you predict the cosmological constant is electro-weak scale in size?
Except that you have still completely failed to answer Distler’s criticism. Allow me to quote the relevant excerpt from your paper:
This is completely misleading. For one thing, the phrase is ambiguous. There are models of particle physics which impose exact parity symmetry, which requires introducing so-called . However, in this case, the extra particles are charged under a different gauge group to the visible particles, and are easy to ‘hide’. What you have is nothing like this.
In the standard model, no fermion mass terms are allowed, because they violate the gauge symmetry. The reason is that the left-handed fermions are in a complex representation of the gauge group. After electroweak symmetry breaking, all fermions are vector-like with respect to the remaining gauge symmetry, and mass terms can be written down. In practice, these come from Yukawa couplings to the Higgs field. However, in your model, the fermion content is doubled, such that the left-handed fermions now fall into a real representation of the standard model gauge group (in fact, R + R-bar, where R is the standard model rep). Therefore there is nothing to forbid mass terms for all the fermions, and in fact these should be generated radiatively in the absence of supersymmetry. So generically, all fermions in your theory should have masses roughly of the cut-off scale (probably the Planck scale here).
This is a serious problem, and why Distler rightly calls your model a ‘zero-generation’ model. You can’t just wave your hands about it — you have to at least provide a solution in principle, or there is no reason to think your model is anything more than a pretty exercise in group theory.
Let me finish by explaining why this is so different to supersymmetry. Before SUSY breaking, there are no mass terms in the MSSM. For the fermions, the reason is the same as for the standard model, and the bosons are related to the fermions by SUSY. After SUSY breaking, nothing stops us writing down mass terms for the bosons, but those for the fermions are still forbidden by chirality. That’s why it is natural for the (unseen) scalar partners to be significantly more massive than the standard model fermions. The Higgs is different, because there is an up-type Higgs and a down-type Higgs, and together they form a real representation, so one can write down a supersymmetric mass term.
Garrett suggested to sombody:
I suppose you have followed Alain Connes’ construction (here is a survey and links) of the standard model by a Kaluza-Klein compactification in spectral geometry. It unifies all standard model gauge fields, gravity as well as the Higgs as components of a single spin connection. Connes finds a remarkably simple characterizaiton of the vector bundle over the compactification space such that its sections poduce precisely the standard model particle spectrum, three chiral generations and all.
Alain Connes had computed the Higgs mass in this model under the big-desert hypothesis to a value that was in a rather remarkable chain of events experimentall ruled out shortly afterwards by the Tevatron. But the big desert is a big assumption and people got over the shock and are making better assumptions now. We’ll see.
Apart from being a nice geometrical unification of gravity and the other forces (credits ought to go all the way back to Kaluza and Klein, but in spectral geometry their orginal idea works out better) Connes’ model has some other striking features:
the total dimension of the compactified spacetime in the model as seen by K-theory is and has to be, as they showed, to produce exactly the standard model spectrum plus gravity: D= 4+6.
Now “as seen by K-theory” was shown by Stolz and Teichner and students to mean in a precise sense: as seen by quantum superparticles (here is some link — you can ask me for a better link). In fact what they consider is almost exactly the spectral triples that Connes considers, with some slight variation and from a slightly different angle. For the relation see the nLab entry on spectral triple (ask me to expand that entry…).
As also indicated at that entry: there is a decent theory of how to obtain a spectral triple as the point particle limit of a superconformal 2-dimensional CFT. Yan Soibelmal will have an article on that in our book Precisely because Connes’ model turns out to have real K-theory dimension 4+6 does it have a chance to be the point particle limit of a critical 2D SCFT. That would even give it the UV-completion — as they say — that would make its quantization consistent (which, remember, contains gravity).
I think there is some impressive progress here. It is not coming out of the physics departments, though, but out of the math departmens. For some reason.
(I’m continuing through the comments chronologically, picking up where I left off, trying not to miss anything directed to me that’s important. Peter, thanks for allowing the discussion.)
John:
Getting the CKM and MNS matrices is the goal, and it is true, E8 Theory is not there yet — and I have been completely candid about this at every opportunity. But I do think there is hope, and my work may soon have something to say about these. How can this possibly work? Well, it all has to start with symmetry breaking, as described in my previous comment. After that, the masses of all other particles are determined by how they interact with the Higgs.
There is potentially a way to fit three generations of fermions into E8 and avoid the mirror fermion problem. The basic idea is that, using an inner-automorphism of E8 related to triality, we can independently gauge transform the 64 mirror fermions, and 64 pre-fermions in the 8_Vx8_V of a so(4,4)+so(8) subalgebra of e8, into generations of usual fermions that will all interact differently with the Higgs. I don’t yet know if CKM and MNS will come out of this, but I’m working on it. In the meantime, yes, it’s fair to say the model is incomplete, and the burden is on me (or some other researcher) to figure out how this can work, but it’s premature to say it can’t work.
Mark:
Did that.
Working on it. But, I also encourage skepticism.
The testable predictions are that if any new particles are found that don’t fit E8, such as superparticles (which many expect to see), then this theory is wrong.
In progress.
What people label me, whether it’s crackpot, next-einstein, or surfer-physics dude, isn’t really up to me. And I’m not selling anything. I am, however, helping friends sell things I think are cool, and see no problem with that. Also, I did get paid to write the SciAm article, and will use the money to buy a new surfboard.
What powerful financial and media forces?
If you look at my list of (1)-(6) issues in my comment above, you’ll see that there has been good progress on several of them.
Brian: Peter has addressed your points. My only addition is that I have published work on the theory, without difficulty, with coauthors. But as the Bogdanovs showed, this means little.
Peter:
Precisely. Thank you.
ned:
My life hasn’t been all roses. But I am working on making it easier for other scientists to come spend some time in Maui — that’s what the Pacific Science Institute is about.
Aaron:
The superconnection is the formal sum of a 1-form and an anti-commuting Grassmann field, both valued in different parts of some algebra. That algebra can be a Lie algebra, such as E8, or it can be a Lie superalgebra — the necessary restriction is that the 1-form be valued in a reductive subalgebra.
Yes, see the paper with Lee and Simone.
True. We wanted to make this less unfamiliar and upsetting to people by keeping the bosons and fermions separate.
Sorry to lose your input.
Wolfgang:
Excellent observation. It doesn’t. The Standard Model and gravity fits in spin(12,4), and spin(4,4)+spin(8) fits in that. There are two SM generators, W^+ and W^-, that occupy the complement. When considering triality, one needs to use spin(4,4) and/or spin(8). I’m not yet sure whether those W’s will remain in the complement as bosons, displacing two fermion degrees of freedom, or whether things will work some other way. A key idea that came from Banff is that the spin(4,4)+spin(8) subalgebra of e8 which relates to triality might be a different subalgebra than the one containing the Standard Model, including having different Cartan subalgebras. This will, of course, give a mixing mess, but the question will be whether that mess is the CKM and MNS mess.
Yes! That seems terrible, but the hope is that the cosmological constant runs from this value at the unification scale down to the tiny value we see at low energies.
“There are two SM generators, W^+ and W^-, that occupy the complement. When considering triality, one needs to use spin(4,4) and/or spin(8). I’m not yet sure whether those W’s will remain in the complement as bosons, displacing two fermion degrees of freedom, or whether things will work some other way.”
Didn’t you explain that all the generators of e8 are either in spin(4,4)+ spin(8) or in one of the 8×8 blocks exchanged by your triality?
So, if some of 8×8 block is bosons, where are rest of fermions? And, if Ws live in 8×8 block, does that mean they form triplet under triality? Do different generations have different gauge bosons coupling to them?
Sorry for so many questions, but this is very confusing!
Rhys:
No, Distler and Garibaldi’s claim is that one cannot even get one generation of fermions in E8. This paper answers that directly by showing explicitly how a generation of fermions does fit in E8. (And it explains it via a direct identification of generators, which is quite nifty.)
No, it is direct. It is Distler’s language that is misleading. When he says “there are no chiral generations,” what he actually means is that there is a generation and what he calls an anti-generation and I call mirror fermions. For him and you to use this twist of mathematical language to say “there are no generations” is a lie.
The top Google hits and I disagree.
If I had said “mirror matter,” then yes, it would have been ambiguous. But I did not.
That’s such a fun word, “should.” With that one word, you are presuming what nature does — when the truth is that we just don’t know. It is fine if you want to say “it should not be,” it is a lie to say “it can not be,” at least until we know what nature actually does. And in that same phrase, you are incorrectly presuming, as do Distler and Garibaldi, that my theory must have mirror fermions in it, when I have said several times that I expect these to be gauge transformed to usual fermions.
This is a valid point. I cannot say I have a complete theory of everything, and I do not, until these problems are solved. But I can and do wave my hands about how they might be solved in principle. And even if it ends up having been a pretty exercise in group theory, I won’t have considered it a waste of time, because I think there is a lot here — especially the dodge of the Coleman-Mandula theorem via symmetry breaking — that is true about nature, even if we don’t yet have the full picture.
This may not be the proper place to bring this up. But does anyone have a view on Penrose’s ‘conformal cyclic cosmology’? I am reading his book now and but have only seen a couple of arxiv articles mentioning it…..
Urs: I agree that Alain Connes’ model is fascinating and deserves more attention from physicists. But it has not been fruitful in making successfully tested new HEP predictions — nor has any model in the past 30 years. It was not my intent to be discouraging, or particularly disparaging of other ideas — I was mostly counter-snarking Aaron.
Steve,
I’ve heard Penrose talk about this, but didn’t really understand the point. I look forward to reading his book (it isn’t out in the US yet), and might write about it then. But, I’m no cosmologist, best to find a blog run by someone who is to discuss the subject.
It has. Within weeks even. It was experimentally verified that the big desert assumption is inconsistent in this model with experiment.
It’s an impressive model. And I didn’t quite say that “it deserves more attention from physicists”. Let them spend their time with what pleases them. Instead I mentioned this in reply to your insinuation that there is nothing promising in fundamentall model building out there besides your idea. It occurred to me that you might actually think that’s true. And maybe because the most impressive progress in fundamental physics these days does not quite percolate through the physics community.
Wolfgang (and John):
Here we are on the edge of what I’m working on. So I don’t yet know what the complete picture is, and my remarks here will be speculative. We do know that under the decomposition
e8(-24) = spin(12,4) + 128^+_S
that one generation of fermions can be the 64^+_S rep (part of the above 128) of a spin(11,3) subalgebra of the above spin(12,4). And in fact, the known gravitational and Standard Model bosons can fit in a spin(5,3)+spin(6) subalgebra of spin(11,3). But, spin(5,3) and spin(7,1) don’t have triality automorphisms. However, spin(4,4) and spin(8) do, so we can decompose e8 as
e8(-24) = spin(4,4) + spin(8) + 8x8_V + 8x8_+ + 8x8_-
and consider innner automorphisms of e8, corresponding to so(4,4) and so(8) triality, that interchange those three blocks of 64. If we put gravitational spin(1,3) in the spin(4,4), and strong su(3) in the spin(8), and the photon and the Z in both, then we’re stuck with at least the W^+ and the W^- in that 8x8_V. And you’re right that if we’re identifying those three blocks of 64 by triality that we’re probably going to be missing at least three sets of two fermion degrees of freedom to accommodate those W’s. Maybe nature has chosen to exclude the right-handed components of neutrinos in this way? Or, an even weirder speculation, maybe that particular spin(4,4)+spin(8) is not completely in the spin(12,4)? There would have to be a lot of mixing angles to describe the geometry of how these spin groups are mutually related, but we want something like that to come out anyway that corresponds to CKM and MNS. The bottom line is that I don’t know how this works yet, but it’s really fun and interesting! (And please do correct me if I’ve made any mistakes here.)
Urs: I didn’t mean to insinuate that there aren’t other promising models. But I don’t consider a prediction proven false to be a “fruitful” prediction in the usual sense, though these impressive events and the positive aspects of Connes’ model are not lost on me. I do consider this E8 Theory to be even more fascinating and promising, but I’m biased. Although, it indisputably looks really good on the new T-shirts. 🙂
Garrett, about
e8(-24) = spin(4,4) + spin(8) + 8x8_V + 8x8_+ + 8x8_-
where you put
gravity spin(1,3) in the spin(4,4)
and
color su(3) in the spin(8)
but
are “stuck with at least the W+ and W- in that 8x8_V”
could you find within “that 8x8_V” a spacetime base
manifold that is an 8-dim Kaluza-Klein M4 x CP2
where
M4 is 4-dim Minkowski spacetime
and
CP2 is internal symmetry space.
Then since CP2 = SU(3)/U(2)
you would have the electroweak U(2) (weak bosons and photon)
naturally included in your structure,
and
the added benefit of getting not only fermions and bosons,
but spacetime itself as part of your E8.
The 8-dim Kaluza-Klein idea is not mine,
but is due to N. A. Batakis
who wrote Class. Quantum Grav. 3 (1986) L99-L105 in which he showed
that “… In a standard Kaluza-Klein framework,
M4 x CP2 allows the classical unified description of an SU(3) gauge field with gravity
… an additional SU(2) x U(l) gauge field structure is uncovered …
As a result,
M4 x CP2 could conceivably accommodate the classical limit of a fully unified theory
for the fundamental interactions and matter fields …”.
Roughly, he uses the structure CP2 = SU(3)/U(2) with the local U(2) giving electroweak and the global SU(3) working for color since its global action is on CP2 which is, due to Kaluza-Klein structure,
local with respect to M4 Minkowski spacetime.
As to why Batakis is not well known and his model fell into obscurity,
Batakis never handled fermions properly in his model.
Since he had nothing to work with but M4xCP2 Kaluza-Klein,
he was reduced to introducing fermions sort of ad hoc by hand,
and he could not show that they worked nicely with his gauge bosons.
However,
since all your structures (spacetime, gauge bosons, fermions) would come from E8 they can be shown to work together nicely.
Tony
“If we put gravitational spin(1,3) in the spin(4,4), and strong su(3) in the spin(8), and the photon and the Z in both, then we’re stuck with at least the W^+ and the W^- in that 8x8_V.”
If I understand what you are saying, there are three W^+s and three W^-s (the ones in the 8x8_V and their triality partners in the other 8×8 blocks). How to reconcile that with there being only one SU(2), whose gauge field corresponding to the diagonal generator occurs only once?
How does the SU(2) Yang Mills action look?
I understood that you wanted the physics of having three generations of fermions, but I don’t think you want three generations of W’s.
Tony:
Your idea sounds pretty close to constructing a Cartan geometry starting from E8. However, if we mod E8 by spin(4,4)+spin(8), we get not only the 8x8_V but also the 128 spinor as the base, which is waaaay too big. It seems much cleaner to consider an E8 principal bundle over a 4D base. If it turns out that structure isn’t rich enough, then I’m open to re-considering an 8D base and KK with CP2. I agree it looks pretty good, but I want to see what I can do with just a 4D base, E8, and triality first.
Wolfgang:
You are understanding things perfectly.
But when looking for a mixing mechanism, I think it’s probably good to have these issues in mind but not focus on them too hard. The W’s are important, but they’re not the only problem. We also have to either get rid of or give large masses to all the X bosons somehow — the various gauge fields other than those of the SM. And, of course, we want to mix the fermions (including mirrors) and get the CKM and MNS for them — all of this in one go, with limited options.
I think a good thing to try is going to be using the so(4,4)+so(8) decomposition to calculate a set of E8 inner automorphisms related to triality, then try applying those inner automorphisms back in the so(12,4) + 128 decomposition to see how it can mix elements. I want to see what is learned from trying that before focusing on specifics.
> Also, I did get paid to write the SciAm article, and will use the money
> to buy a new surfboard.
this is the coolest statement on this blog yet 🙂
and kudos that you resisted invoking the anthropic principle … yet
Peter,
Penrose’s book has been available on Amazon since late October.
“The W’s are important, but they’re not the only problem. We also have to either get rid of or give large masses to all the X bosons somehow”
OK. But other gauge symmetries could be broken at Planck scale. Standard Model gauge symmetries are supposed to be unbroken down to low energies. That seems much more restrictive.
One more question:
In previous comments, you seemed to say that even if triality idea doesn’t work out, E8 theory is still OK.
My understanding from trying to read distler-garibaldi paper is that they show two things,
1) 128 (out of 248) fields are fermions
2) fermion spectrum is non-chiral
so (assuming no triality), fermions are at best 1 generation and 1 mirror-generation of Standard Model. From this they conclude E8 theory is not viable.
Do you say E8 theory with 1 generation and 1 mirror-generation (even if no triality) is still viable theory of Nature?
If so, could you explain how?
The lack of response to Connes’ theory is indeed interesting. I think the problem is that nonbody has been able to explain in a language that particle theorists can understand whether this is indeed a new idea (and if so, what the new idea is) or whether this is just a complicated way to formulate an old idea (GUT’s or maybe Gravi-GUT’s). Where is Witten when you need him?
Sorry. I have one more question. About cosmological constant being electro-weak scale in size, you said:
“Yes! That seems terrible, but the hope is that the cosmological constant runs from this value at the unification scale down to the tiny value we see at low energies.”
Does that mean the relation between electro-weak scale and cosmological constant is accidental feature of your classical lagrangian? Is there a more general lagrangian where they are independent parameters?
I ask because in renormalizable theory all counterterms should appear as possible terms in classical lagrangian.
Didn’t Connes predict a 170 GeV Higgs? Which was the first region to be ruled out by the Tevatron?
Just to be clear,
my intent was not to suggest
“… mod E8 by spin(4,4)+spin(8) …”
in which case “… we get not only the 8x8_V but also the 128 spinor as the base …”
but to suggest
mod E8 by both spin(4,4)+spin(8) and also the 128 spinor
(that may be a 2-stage process)
so that
we get only the 8x8_V as the base
and then
to let the 8x8_V represent an 8-dim M4 x CP2 Kaluza-Klein.
Tony
Yes. Generally my impression is that the number of theoretical physicists actively aware of or at least interested in the issues of what it means to find a conceptual or even axiomatic framework for fundamental physics is currently much lower than it used to be. It seems to me that in the early 90s or so the situation has been very different. In fact from that time date a few articles by string theorists who had read Connes, had understood what he is after and had tried to connect it to string theory.
Because the curious thing is: what Connes suggests is precisely the 1-dimensional version of the very idea of perturbative string theory (which is the 2d version of an even more general idea):
regard the algebraic data characterizing a d-dimensional super QFT as a stand-in for the geometric data characterizing the target space of which this QFT would be the sigma-model, if it were one. What in Connes’s setup is a spectral triple is a vertex operator algebra for the string.
(References that discuss how to make this statement precise are at spectral triple and 2-spectral triple).
And why do you necessarily need him?
Lately Witten seems to be busy providing more evidence for the holographic principle of higher category theory (scroll down to see what i mean).
Wolfgang:
What I was saying was slightly different. I do think the triality idea is going to have to work out for E8 Theory to be a good theory. However, since I only have some rough ideas on how triality might work out, I have been forced by critics to defend the theory without it. Without triality, the best I can say is that the theory is incomplete and unattractive, but not necessarily wrong. The best way to look at it, in my opinion, is that we currently know exactly how gravity and the Standard Model gauge fields along with one generation of fermions can embed in E8, which is incredibly cool. And there are some indications of how to get the other two generations, with a much tighter fit, but that is not yet clear.
This is a straw man setup. I disagree with Distler and Garibaldi at step (1) — they insisted on using this [tex]Z_2[/tex] grading of E8 even though I said the theory would rely on other options. However, even this silly straw man is not easy to knock down, because mirror fermions have not been completely ruled out, even if they make it ugly.
Essentially, I think E8 Theory is about half way done. We’ve got gravity, the Standard Model, a generation of fermions, and a nice symmetry breaking mechanism. And the triality-related gauge transformations I’m working with are very encouraging. It is kind of stupid to assess a half-done theory as if was supposed to be a complete theory of nature. It’s like looking at a half-built house and saying “oh, that’s no good — it’s leaky.” Rather, one needs to assess E8 Theory as a research program moving towards a complete ToE. And, from that point of view, it’s doing pretty well.
Wolfgang:
The relation between Higgs VEV and cosmological constant is even more fundamental than the Lagrangian. If the bosonic connection is
[tex]H = \frac{1}{2} \omega + \frac{1}{4} e \phi + A[/tex]
then its curvature is
[tex]F = \frac{1}{2}(R – \frac{1}{8}\phi^2 e e) + \frac{1}{4} ( T \phi – e D \phi) + F_A[/tex]
and the relationship between Higgs VEV and cosmological constant, [tex]\Lambda = \frac{3}{4} \phi_0^2[/tex], comes from [tex]F_0 = 0[/tex], which I consider more fundamental than the Lagrangian. One might be able to cook up a way to change that relationship, but I wouldn’t recommend it.
(Hmm, that “8211;” above is a “-“. I don’t know why it did that.)
Tony: I think one is only allowed to mod out by subgroups.
“This is a straw man setup. I disagree with Distler and Garibaldi at step (1) — they insisted on using this Z_2 grading of E8 even though I said the theory would rely on other options.”
Maybe I expressed myself badly.
Fermions transform as Lorentz spinors. Your triality idea is to change how fields in the 248 transform under Lorentz group. Without triality (which remains to be worked out), fields transform according to the “naive” transformation rule. Do you agree that naive transformation rule gives 128 fermions or is even that part wrong?
Put differently: if the triality idea doesn’t work out, do you have another way to avoid distler-garibaldi conclusion?
“The relation between Higgs VEV and cosmological constant is even more fundamental than the Lagrangian. … One might be able to cook up a way to change that relationship, but I wouldn’t recommend it.”
If you don’t change it, how do you avoid cosmological constant of order the electro-weak symmetry breaking scale?
In earlier comment, you said “the hope is that the cosmological constant runs from this value at the unification scale down to the tiny value we see at low energies.” But if Higgs VEV and cosmological constant are tied together as you say how can one be big (250 GeV) and the other tiny?
Garrett, you say “one is only allowed to mod out by subgroups”.
Maybe you and I are not using “mod out” in the same sense,
and maybe (since it is a term with which I am not very familiar)
I have been misusing it, so here is what I am trying to say in
terms of graded Lie algebras:
Consider Thomas Larsson’s 7-grading of E8 which is of the form
E8 = g_-3 + g_-2 + g_-1 + g_0 + g_1 + g_2 + g_3
with graded dimensions
E8 = 8 + 28 + 56 + (sl(8) + 1) + 56 + 28 + 8
The odd graded part of E8 has 8+56 + 56+8 = 64+64 = 128 dimensions
and corresponds to your 128 spinor.
The even graded part of E8 has 28 + 64 + 28 = 120 dimensions
and corresponds to your D8 Lie algebra so(4,12)
My first stage is to “mod out” the odd graded 128 spinor,
which leads to the next stage about the D8.
The D8 Lie algebra has a 3-grading which is of the form
D8 = g_-3 + g_-2 + g_-1 + g_0 + g_1 + g_2 + g_3
with graded dimensions
D8 = 28 + (sl(8) + 1) + 28
The odd graded part of D8 has 28 + 28 = 56 dimensions
and corresponds to your D4 + D4 Lie algebras so(4,4) and so(8)
The even graded part of D8 has 64 dimensions
and corresponds to your 8x8_V.
My second stage is to “mod out” the odd graded D4 + D4,
which leaves your 64-dimensional 8x8_V to represent
an 8-dim spacetime that can (by breaking octonionic symmetry
down to quaternionic) give you a M4 x CP2 Kaluza-Klein
with the Batakis structure giving you the U(2) from CP2= SU(3)/U(2).
Then you can construct a nice Lagrangian as follows:
Base Manifold from the 8-dim Kaluza-Klein in the 8x8_V
Gauge Boson terms from D4 + D4 “modded out” in stage 2
Fermion terms from 128 half-spinor “modded out” in stage 1.
Then,
if you look at the geometry of the octonionic/quaternionic
symmetry breaking down to 4+4 dim Kaluza-Klein you see
that Meinhard Mayer’s mechanism (Hadronic Journal 4 (1981) 108-152)
(he is physics professor emeritus at U. C. Irvine)
gives the Higgs scalar.
Tony
Garrett,
What about this: whenever you talk about using triality on a part of E(8), it seems you are not talking about E(8) anymore, but a semiderect product of SO(8)XE(8), SO(8) being the group that “insert” the triality. Now, what do you think of this?
Hello Peter & Garrett,
Well, it seems that Peter is quite skeptical of Garrett’s theory, and that Garrett is too, if less so. The question, then, is why does it keep getting so much attention and funding?
Peter writes, ”
Tristan,
My understanding is that Garrett is well aware that his proposal has problems. In the Scientific American article he writes:
“All new ideas must endure a trial by fire, and this one is no exception. Many physicists are skeptical—and rightly so. The theory remains incomplete.”
I have no problem with skepticism, I’m skeptical about many of Garrett’s ideas too. If Jacques wants to make a clean technical argument showing the nature of the problems with Garrett’s proposal, that’s great, and could be potentially worthwhile. But I don’t see any reason for the hostile, sneering tone of Jacques’s blog posting explaining these points. This is not the way to professionally make a credible technical argument. ”
Are there not a lot of other theories out there which we can be skeptical about? So why is Garrett’s “theory” getting all the attention from television, magazines, and the press? Who is pushing/promoting this and why?
Insights? Ideas? Thanks!
Wolfgang:
That is correct.
No, without triality, we’re stuck with mirror fermions. But the Distler-Garibaldi conclusion that “the theory can’t work” would still be untrue, because mirror fermions could exist. But I don’t think they do — I think triality will work.
I haven’t done the calculation, but perhaps they run independently, with the effective cosmological constant getting contributions from gravity, and the Higgs mass from Standard Model and other interactions.
Tony:
If one were to try and build a universe by deforming the E8 Lie group, the nicest way to do it would probably be to use Cartan geometry, by which the base spacetime is modeled on the (too large) symmetric space obtained by moding E8 out by a subgroup.
Of course, you’re also welcome to just start with an 8D base and a principal bundle, which is less restrictive, and play with different gradings and KK schemes as you are here.
Daniel:
When I am talking about E8 triality I am talking about the triality outer automorphisms of the so(4,4) and so(8) subalgebras, and the corresponding inner automorphisms of E8.
Gregor:
Since you cannot accept that the media has been attracted to a story about an unusual physicist who has come up with an interesting new theory, the attention must be because I am so incredibly handsome.
Happy Thanksgiving!
“No, without triality, we’re stuck with mirror fermions. But the Distler-Garibaldi conclusion that “the theory can’t work” would still be untrue, because mirror fermions could exist.”
Sorry. That I don’t understand.
Without triality, you are stuck with one generation and one mirror-generation. I don’t see how you can say that “works” as a theory of nature. Could you explain?
” But I don’t think they do — I think triality will work.”
Maybe it will. But it faces serious obstacles (see above discussion about W bosons).
“I haven’t done the calculation, but perhaps they run independently,”
Every independently-running coupling constant corresponds to an independent term you can add to classical lagrangian. You just explained that in your theory electro-weak scale and cosmological constant are not independently adjustable coupling constants. So how can they run independently?
>“No, without triality, we’re stuck with mirror fermions.
> But the Distler-Garibaldi conclusion that
> “the theory can’t work” would still be
> untrue, because mirror fermions could exist.”
> Sorry. That I don’t understand.
i guess he just means that you can build a model that looks like the SM at low energies without chiral fermions. and he is right – you can. you just need fine tuning, which is “ugly” but not forbidden.
and of course you can add 2 carbon copy generations and by hand add CKM mixing. it’s not pretty – but who says that top-color assisted extending walking technicolor is? 🙂
I’ve had to delete repeated anonymous comments by someone who couldn’t be bothered to either look things up for himself or read Garrett’s previous response to the same question:
http://www.math.columbia.edu/~woit/wordpress/?p=3292&cpage=1#comment-69686
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