Tuesday, November 06, 2007

A Theoretically Simple Exception of Everything

Garrett Lisi, who was featured in our inspiration series back in August, has a new paper on the arxiv about his recent work

    An Exceptionally Simple Theory of Everything
    arXiv: 0711.0770

I met Garrett at the Loops '07 in Morelia, and invited him to PI. He gave a talk here in October, which confirmed my theory that the interest in a seminar is inversely proportional to the number of words in the abstract. In his case the abstract read: "All fields of the standard model and gravity are unified as an E8 principal bundle connection," and during my time at PI it was the best attended Quantum Gravity seminar I've been at.

Anyway, since I've spend some time trying to understand what he's been doing (famously referred to as 'kicking his baby in the head') here is a brief summary of my thoughts on the matter.

Preliminaries

In the 50's physicists were faced with a confusing, and still growing multitude of particles. By introducing new quantum numbers, it was clear that this particle zoo exhibited some kind of pattern. Murray Gell-Mann realized the particles could be classified using the mathematics of Lie-groups. More specifically, he found that the baryons with spin 3/2 known at this time correspond to the weight diagram of the ten-dimensional representation of the group SU(3) [1].



He matched the nine known spin 3/2 baryons (4 Δs, 3 Σ*s, 2 Ξ*s) with the weights of this representation, but there was one particle missing in the pyramid. He therefore predicted a new particle, named Ω-, which was later discovered and had the correct quantum numbers to complete the diagram [2]. Because of the ten baryons in the multiplet, this is also known as the 'baryon decuplet'.


A similar prediction could later be made for the baryon octet, where the center of the diagram should be doubly occupied. The existence of the missing Σ0 was later experimentally confirmed.

After this, the use of symmetry groups to describe nature has repeatedly proven to be an enormously powerful and successful tool. Besides being useful, it is also aesthetically appealing since the symmetry of these diagrams is often perceived as beautiful [3].

GUTs and TOEs

Today we are again facing a confusing multitude of particles, though on a more elementary level. The number of what we now believe are elementary particles hasn't grown for a while, but who knows what the LHC will discover? Given the previous successes with symmetry principles, it is only natural to try to explain the presently known particles in the standard model - their families, generations, and quantum numbers - as arising from some larger symmetry group in a Grand Unified Theory (GUT). One can do so in many ways; typically these models predict new particles, and so far unobserved features like proton decay and lepton number violation. This larger symmetry has to be broken at some high mass scale, leaving us with our present day observations.

Today's Standard Model of particle physics (SM) is based on a local SU(3)xSU(2)xU(1) gauge symmetry (with some additional complications like chirality and symmetry breaking). Unifying the electroweak and strong interaction would be great to begin with, but even then there is still gravity, the mysterious outsider. A theory which would also achieve the incorporation of gravity is often modestly called a 'Theory of Everything' (TOE). Such a theory would hopefully answer what presently is the top question in theoretical physics: how do we quantize gravity? It is also believed that a TOE would help us address other problems, like the observed value of the cosmological constant, why the gravitational interaction is so weak, or how to deal with singularities that classical general relativity (GR) predicts.

Commonly, gravity is thought of as an effect of geometry - the curvature of the space-time we live in. The problem with gravity is then that its symmetry transformations are tied to this space-time. A gauge transformations is 'local' with respect to the space-time coordinates (they are a function of x), but the transformations in space-time are not 'local' with respect to the position in the fibre, i.e. the Lie-Group. That is to say, usually a gauge transformation can be performed without inducing a Lorentz transformation. But besides this, the behavior of particles under rotations and boosts - depending on whether dealing with a vector, spinor or tensor - looks pretty much like a gauge transformation.

Therefore, people have tried to base gravity on an equal footing with the other interactions by either describing both as geometry, both as a gauge theory, or both as something completely different. Kaluza-Klein theory e.g. is an approach to unify GR with gauge theories. This works very nicely for the vector fields, but the difficulty is to get the fermions in. So far I thought there are two ways out of this situation. Either add dimensions where the coordinates have weird properties and make your theory supersymmetric to get a fermion for every boson. Or start by building up everything of fermionic fields.

Exceptional Simplicity

On the algebraic level the problem is that fermions are defined through the fundamental representation of the gauge group, whereas the gauge fields transform under the adjoint representation. Now I learned from Garrett that the five exceptional Lie-groups have the remarkable property that the adjoint action of a subgroup is the fundamental subgroup action on other parts of the group. This then offers the possibility to arrange both, the fermions as well as the gauge fields, in the Lie algebra and root diagram of a single group. Thus, Garret has a third way to address the fermionic problem, using the exceptionality of E8.

His paper consists of two parts. The first is an examination of the root diagram of E8. He shows in detail how this diagram can be decomposed such that it reproduces the quantum numbers of the SM, plus quantum numbers that can be used to label the behaviour under Lorentz transformations. He finds a few additional particles that are new, which are colored scalar fields. This is cute, and I really like this part. He unifies the SM with gravity while causing only a minimum amount of extra clutter. Plus, his plots are pretty. Note how much effort he put in the color coding!

Garrett calls his particle classification the "periodic table of the standard model". The video below shows projections of various rotations of the E8 root system in eight dimensions (see here for a Quicktime movie with better resolution ~10.5 MB)




[Each root of the E8 Lie algebra corresponds to an elementary particle field, including the gravitational (green circles), electroweak (yellow circles), and strong gauge fields (blue circles), the frame-Higgs (squares), and three generations of leptons (yellow and gray triangles) and quarks (rbg triangles) related by triality (lines). Spinning this root system in eight dimensions shows the F4 and G2 subalgebras.]

However, just from the root diagram alone it is not clear whether the additional quantum numbers actually have something to do with gravity, or whether they are just some other additional properties. To answer this question, one needs to tie the symmetry to the base manifold and identify part of the structure with the behaviour under Lorentz transformations. A manifold can have a lot of bundles over it, but the tangential bundle is a special one that comes with the manifold, and one needs to identify the appropriate part of the E8 symmetry with the local Lorentz symmetry in the tangential space. The additional complication is that Garrett has identified an SO(3,1) subgroup, but without breaking the symmetry one doesn't have a direct product of this subgroup with additional symmetries - meaning that gauge transformations mix with Lorentz-transformations.

Garrett provides the missing ingredient in the second part of the paper where he writes down an action that does exactly this. After he addressed the algebraical problem of the fermions being different in the first part, he now attacks the dynamical problem with the fermions: they are different because their action is - unlike that of the gauge fields - not quadratic in the derivatives. As much as I like the first part, I find this construction neither simple nor particularly beautiful. That is to say, I admittedly don't understand why it works. Nevertheless, with the chosen action he is able to reproduce the adequate equations of motion.

This is without doubt cool: He has a theory that contains gravity as well as the other interactions of the SM. Given that he has to choose the action by hand to reproduce the SM (see also update below), one can debate how natural this actually is. However, for me the question remains which problem he can address at this stage. He neither can say anything about the quantization of gravity, renormalizability, nor about the hierarchy problem. When it comes to the cosmological constant, it seems for his theory to work he needs it to be the size of about the Higgs vev, i.e. roughly 12 orders of magnitude too large. (And this is not the common problem with the too large quantum corrections, but actually the constant appearing in the Lagrangian.)

To make predictions with this model, one first needs to find a mechanism for symmetry breaking which is likely to become very involved. I think these two points, the cosmological constant and the symmetry breaking, are the biggest obstacles on the way to making actual predictions [4].

Bottomline

Now I find it hard to make up my mind on Garrett's model because the attractive and the unattractive features seem to balance each other. To me, the most attractive feature is the way he uses the exceptional Lie-groups to get the fermions together with the bosons. The most unattractive feature are the extra assumptions he needs to write down an action that gives the correct equations of motion. So, my opinion on Garrett's work has been flip-flopping since I learned of it.

So far, I admittedly can't hear what Lee referred to in his book as 'the ring of truth'. But maybe that's just because my BlackBerry is beeping all the time. And then there's all the algebra clogging my ears. I think Garrett's paper has the potential to become a very important contribution, and his approach is worth further examination.

Aside: I've complained repeatedly, and fruitlessly, about the absence of coupling constants throughout the paper, and want to use the opportunity to complain one more time.

For more info: Check Garrett's Wiki or his homepage.

Update Nov. 10th: See also Peter Woit's post
Update Nov. 27th: See also the post by Jacques Distler, who objects on the reproduction of the SM.


[1] Note that this SU(3) classification is for quark flavor (the three lightest ones: up, down and strange), and not for color.
[2] For more historical details see Stefan's post
The Omega-Minus gets a spin (part 1) which is still patiently waiting for a part 2.
[3] See also my earlier post on
The Beauty of it All.
[4] If one were to find another action, the cc problem might vanish.


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284 comments:

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Aaron Bergman said...

As best I can tell, his Lagrangian explicitly breaks the E_{8+8} symmetry (such as it is), so it's tough to understand what role the symmetry actually plays in the theory.

Bee said...

Hi Trogl,

Thanks for your call on objective reason.

Hi Marcus,

G.L. described what he is working on as a "top down INSPIRED bottom up approach".

that is, or so I think, a bottom up built approach, which is inspired by a certain mathematical ideal.


He might have borrowed this expression from my talk in Morelia. The problem with that reasoning in this case is that the 'mathematical ideal' on its own, might qualify as a GUT but not as a TOE. See, the step that would make Garrett's approach more than a GUT is the identification with the Lorentz generators acting on the manifold, and this is the very step that he does by hand.

How can I possibly say that clearer: Given today's status, Garrett's model does *not* naturally lead to a unification of the SM interactions with gravity (he has to chose the action by hand that contains both), it does *not* allow us to understand quantum gravity (since there's nothing said about quantization), it does *not* explain the parameters in the SM (since there isn't yet a mechanism for symmetry breaking), it does *not* explain the cosmological constant or its value (as said above, to claim there has to be one, it would be necessary to show there's no way to do it without one), it does *not* explain the hierarchy problem (and I see no way to do so), it does *not* explain why we live in a spacetime with 3 spatial and 1 timelike dimensions, it does *not* in my very humble opinion yet qualify being called a Theory of Everything.

Besides these, there are the concerns that I have about whether one can possibly get the fermionic part of the action to come out right without assuming it by hand, and about the too large value of the CC that occurs.

As said above, I do think it is a very interesting approach that can open new points of view on various problems. It might be a promising path to follow, and I look really forward to further investigations. But there is a long, long way to go, I don't think one should overhype things that early.

Best,

B.

Justin Case said...

Garrett,

Don’t know whether you are/were aware of it, but there is a strange connection between E8 and the monster group which is currently (still) not well understood. I think the below two papers do reflect the most recent and comprehensive insights about it:

http://arxiv.org/abs/math/0403010

http://arxiv.org/abs/math/0503239

It might be of relevance.

J.C.

Justin Case said...

PS: and Witten's paper (Three-Dimensional Gravity Revisited") in which he states that "the monster theory may be the first in a discrete series of CFT's that are dual to three-dimensional gravity" might of course also be interesting... :-)

J.C.

Moshe said...

Thanks Garrett, the bosons and fermions you discuss are coordinates on some linear space, and since there are Grassmans involved it cannot be simply a Lie algebra, perhaps a graded Lie algebra like the ones discussed in the context of supersymmetry? In any event, the connection you discuss cannot be simply E8 connection which takes value in a (bosonic) Lie algebra.

Additionally, there is the question of how things transform under diffeomorphisms (which are distinct from frame rotations, acting on the tangent space). It makes little sense to add up things that transform differently under reparametrizations of the base manifold. If I understand things correctly the bosons you have are all 1-forms in that sense, but how about the fermions?

erm said...

that lumo guy seems a bit of an arse, i'd like to unify my fists on his face , theorize that biatch

QUASAR9 said...

Hi Garrett, nice to see the E8
Mathematical Pattern has made it into New Scientist

Alex said...

Lumo:"You haven't responded to a single of these basic questions because it is clearly not possible. The reason why you are adding bosons and fermions is simply that you don't understand the difference between them, much like the difference between spin-0, spin-1/2, spin-1 fields and the difference between the adjoint representation and others."

I wholehartedly concur with this assesment.

Both Moshe and Aaron (well respected physisists) have been more kind when poining out the major flaw in Garrett's construct but he keeps insisting that the quarks and leptons can be treated as ghosts - a complete bullshit because ghosts violate the spin-statistics theorem!

Bee, you know particle physics, how can't you see this!!!

Bee said...

Hi Alex,

Bee, you know particle physics, how can't you see this!!!

I have commented on what actually is the content of Garrett's paper, and explained my opinion on it. I don't think this is the place to engage in speculation about how he thinks the weaknesses I have pointed out could be addressed.

Best,

B.

Dave said...

"Beautiful"

An off-topic serious question:

One sees this this a lot in the scientific literature and I would like to ask why.

Evolution built this admiration for symmetric features into us for it's purposes. Why do scientists persist in assuming that a criteria for animals to choose mates is appropriate for choosing theories.

wintermute said...

Having found this discussion fascinating even as a layman, it seems to me that if an idea can generate so much meaningless hate and all too obvious jealousy, there must be something to it.

Best of luck for the future Mr. Lisi and may I kindly suggest to the detractors to focus on developing theories of their own, instead of writing 3000 word blog posts ridiculing the author of this one. After all, a troll with a PhD is still nothing but a troll :)

John G said...

Since Smolin and David Finkelstein have both expressed optimism for this model, there are at least two very well repected physicists who don't see a fatal flaw in the paper (they of course could see things that need worked on).

Tony Smith, besides having his own E8 model, was a grad student of Finkelstein's. On Tony's website is this comment about ghosts:

"As van Holten indicates in hep-th/0201124, there is a 1-1 correspondence between Y-M gauge bosons and ghosts. In the case of the 28-dimensional Spin(8) gauge group of the D4-D5-E6-E7-E8 VoDou Physics model,, the corresponding 28 ghosts can be regarded as antisymmetric pairs of 8 pre-ghosts, with one of the pre-ghosts in the pair being a particle and the other being an antiparticle. The 8 pre-ghosts are like the 8 gauge potentials that form 28 gauge field bosons by antisymmetric wedge product. From that viewpoint, you could say that the role of ghosts is played by first-generation fermion particle-antiparticle pairs, and that when you do path-integral sum-over-histories quantization you don't need to add ghosts in by hand, because virtual spinor fermion particle-antiparticle pairs will do what is needed.

In my opinion, the triality relation for Spin(8), which identifies half-spinors with vectors is a natural relationship that makes it unnecessary to throw in ad-hoc ghosts when you quantize, thus making Cl(8) with triality uniquely useful for physics, and that is an indication that the 8-dim Clifford algebra structure is not only basic from the point of view of 8-periodicity, but is also basic for modelling a quantum theory of gauge group physics with ghosts.

From a more geometric point of view, if the BRST transformation acts like the nilpotent cohomology operator on the cohomology of Spin(8), then, consider that Spin(8) cohommology looks like

S3 u S7 u S11 u S7

and

S7 is a fibre bundle made up of S3 and S4 = QP1 (quaternionic projective space)

S11 is a fibre bundle made up of S3 and QP2 (quaternionic projective plane).

The 28 ghosts for the 28 Spin(8) gauge bosons can be seen as:

S3
S3 u QP1
S3 u QP2
S3 u QP1
Note that S3 u S3 u QP2 = S3 u S11 is the structure of G2, the automorphism group of the octonions, and that the two S3 u QP1 = S7 7-spheres are each unit spheres in octonion space.

Since S3 looks like the quaternion unit sphere, and is generated by an associative triple of octonions, and QP1 and QP2 are obviously quaternionic, it seems obvious to ask what happens to the octonionic Spin(8) or Clifford(8) theory when a particular quaternionic subspace is frozen out, and that is exactly how in the D4-D5-E6-E7-E8 VoDou Physics model a M4xCP2 Kaluza-Klein space emerges at low temperatures (of our present world) from the full high-temperature octonion 8-dim spacetime."

Garrett said...

Sorry I haven't been able to comment in a while -- the media is... well, being the media. And my inbox exploded. I expect to have more time in a few days when the attention simmers down. However, if there are SHORT (one or two sentence) questions posted, that are CONSTRUCTIVE (i.e. will help clarify matters) I will come back and do my best to answer them. Long posts, and anything posted by people who have been mean in this thread, will be ignored. I'm hoping to help explain the math here, which will be familiar to some but not others. The theory might turn out to be wrong about nature, but the math does make sense.

Momchil said...

Probably it is too early to ask, but by not having supersymmetry, we loose
the most popular candidate for dark matter. Still I prefer the axions. Is there a chance to get an axion from the new particles in the theory? Maybe these new colored Higgs fields can help? If not, is there any chance to get the Peccei-Quinn symmetry, which is "almost" a Standard Model feature ?

SirBruce said...

Garrett,

It seems a big stumbling block for your theory is the deSitter space, which a lot of people are uncomfortable with. Do you have any thoughts as to how perhaps deSitter space could be avoided, or how it might be rendered more analytically friendly so physicists will find it palatable?

Secondly, how long will it be before the math can be worked out on the predicitions of some of the new theoretical particles and fields? Will you be doing this or will you be relying on others to do this?

garrett said...

Momchil:
I've actually tried to avoid considering dark matter, because there are so many options. So I don't have good answers for this. There are a handful of massive, weakly and strongly interacting scalar fields that pop up in this theory, but I don't know enough about dark matter to discuss the possibilities.

SirBruce:
The theory requires a cosmological constant -- the MacDowell-Mansouri approach to gravity is not defined without one, and it's an integral part of this theory. This may change as the theory develops, but that's the way it is now. So, with a positive cosmological constant, the vacuum GR solution is de Sitter. I'm going to be working on getting the particle masses out, yes -- that would be nice. I will work on this for as long as it takes, in this theory or in what comes next. I can't speak for others, but there is a lot of interest and a lot of very interesting, related ideas floating around.

Anonymous said...

When I first learned about ghosts in my grad student days I got excited and thought "wow maybe these really exist and you could get "fermions" automagically out of a pure gauge theory". I then learned that the ghosts are unphysical because they have negative probability and serve only to cancel out other negative probability terms which arise from the gauge fixing.

Do I understand correctly that the ghosts in this E8 model do not have this nasty unphysical property? If so why not?

Cheers,
anonymous35

garrett said...

anonymous35:
That the fermions are ghosts is one explanation for this mathematical construction, but I didn't include that hypothesis in this paper because others are possible. However, going with ghosts... the usual ghosts in QFT are Lie algebra valued Grassmann number fields that compensate for gauge degrees of freedom in the Lie algebra valued 1-form connection fields. They are not spinor fields, algebraically, but Lie algebra valued fields, and they can't have external lines in Feynman diagrams -- which means they aren't measured as physical particles, but only figure in to the calculations. Nevertheless, you can formally add the ghosts and connection fields in a kind of superconnection, compute its curvature, and write down an action for these fields that gives good predictions. Now, if we're working with one of the exceptional Lie groups as our gauge group, and we build a certain action, some of the ghost parts ARE, algebraically, spinor field multiplets with respect to some subgroups of the exceptional group. This means, mathematically, they are fermions -- spinor multiplets of Grassmann numbers. Using the same mathematical construction as in BRST, these can be formally added with the rest of the connection in a superconnection, as I have done in the paper. Looking at the Feynman rules for ghosts, there is one rule that says "ghosts can't be external lines." If we remove that rule -- which is now possible since these fields are algebraically spinors and satisfy spin-statistics -- then these ghosts are physical and their Feynman rules are the ones for fermions.

I hope that answers this question for you, and for others. If you don't like the physics of this construction, that's arguable. But the math is correct, as it's lifted straight from standard BRST techniques, with no alterations.

chris said...

hi garrett,

just a very stupid question. you say that spinor ghost fields (wrt a subgroup of E8) pop up in the lagrangean of your theory.

does that imply that the fundamental lagrangean in your theory is necessarily one with nonlocallocal gauge fixing already integrated? if no, how do ghosts show up at all? if yes, what if i choose another gauge or don't fix it at all?

best,

chris

alex said...

Garrett:"This means, mathematically, they are fermions -- spinor multiplets of Grassmann numbers. Using the same mathematical construction as in BRST, these can be formally added with the rest of the connection in a superconnection, as I have done in the paper."
That's exactly where you went wrong! Lorentz spinors CANNOT be "formally" added just like ghosts.

amused said...

Back to the Coleman-Mandula issue: First of all, regardless of the global structure of the spacetime manifold, Poincare symmetry must hold to an arbitrarily good approximation in sufficiently small spacetime regions (away from singularities). So if we consider the restriction of the principal E8 bundle over a small region of spacetime, say the spatial extent of Cern's LHC during time periods on the scale relevant for particle collisions, then Poincare symmetry must surely hold there to a very good approximation. Therefore, unless there is some kind of discontinuity (which I can't see any origin for), Coleman-Mandula effectively applies for any local QFT which purports to describe the physics of particle scatterings in the lab experiments (assuming the other conditions of the theorem are satisfied - I didn't bother to look them up). Since Lisi's theory contradicts this, and continues to contradict it for restrictions of the E8 bundle to arbitrarily small localized regions of the the base manifold (spacetime), the conclusion is that either Coleman-Mandula is wrong or Lisi is wrong.

Assuming the latter case, one possible explanation might be this: It seems to me that the gravitational spin connection (denoted `omega' in the paper) is being treated as if it is the gauge field associated with an *internal* so(3,1) symmetry (embedded in e8), rather than a gauge field associated with the *external* so(3,1) symmetry (i.e. the symmetry transformations the tangent spaces of the spacetime manifold). Since the C-M theorem is a statement about the impossibility of combining internal gauge symmetries in a non-trivial way with the external Poincare symmetry (which contains the aforementioned external so(3,1)), I guess this would explain it.

In fact, for E8 or any gauge group which contains SO(3,1) as a subgroup, you can decompose a connection/gauge field to get a piece, call it omega, that is a so(3,1) gauge field and therefore looks like a gravitational spin connection. But C-M implies that you can't actually identify this so(3,1) with the external so(3,1) symmetry of the spacetime tangent bundle, and therefore can't identify omega with the real gravitational spin connection.
(And if I got something wrong there hopefully someone will correct it.)

Anonymous said...

Garret:

Thanks for that clear explanation regarding the role of ghosts.

Taking this seriously, would this explain "why" fermions exist at all, assuming the pure gauge theory exists?

That seems quite profound...

Eric said...

Garrett,
Just curious if you have submitted your paper to a journal for publication (JHEP, NPB, or PRD perhaps?), yet. It seems to me that this is still the standard for how the scientific community judges the merits of a paper. Let's see if you can get this past the peer-review process.

Aaron Bergman said...

I think, for "amused", he should submit it to PRL :).

garrett said...

chris:

Not a stupid question at all. The gauge fixing is determined by the choice of "BRST potential," which determines the action for these "fermion ghosts." I've chosen this potential by hand such that the action that comes out is the Dirac action. A different choice, corresponding to a different gauge, would produce a different action. The important thing here, in my opinion, is that the mathematics works. People may be unhappy with calling these "ghosts," and I agree that a new name and interpretation might be in order, but the mathematics is the same.

alex:
From your name, I presume you're familiar with Jeopardy? Well, you haven't given your question in the form of a question. So, instead of an answer, I'll pose you a question: "Can Lie algebra valued 1-forms be formally added to Lie algebra valued Grassmann numbers?"

amused:
Your description is correct. However, you're combining an approximation with a mathematical proof. To the degree that this approximation is true, Coleman-Mandula holds and this E8 Theory is in agreement with it.

Let me take a step back though, and present this C-M argument from a different perspective. The C-M proof is a fairly complicated construction within a very detailed framework. It is true, but the conditions and setup are intricately constructed. We could spend a lot of time exhausting ourselves by pouring over the details of this proof, but that's not necessary. This E8 Theory is extremely straightforward, and it's easy to see how the gravitational degrees of freedom are incorporated and the dynamics determined, in agreement with general relativity and the standard model. This implies it's fitting through some loophole in the C-M proof. This theory is much more straightforward than the theorem and framework you're trying to use to argue it can't exist. And, there's history here: supergravity was invented and considered as a possibly valid theory before people figured out the loophole in C-M that allows it to exist. Please consider that history could be repeating itself in this case.

anonymous:
Yes.

Aaron Bergman said...

Garrett, you can formally add whatever you want -- that's what "formal" means. The question is whether what you get is meaningful.

Again, the BRST symmetry is a Grassman symmetry; yours is not.

Anonymous said...

amused:"In fact, for E8 or any gauge group which contains SO(3,1) as a subgroup, you can decompose a connection/gauge field to get a piece, call it omega, that is a so(3,1) gauge field and therefore looks like a gravitational spin connection. But C-M implies that you can't actually identify this so(3,1) with the external so(3,1) symmetry of the spacetime tangent bundle, and therefore can't identify omega with the real gravitational spin connection."

You have just hit the nail in the head! Indeed, C-M does apply in this case and implies presicely that as you have stated: "you can't actually identify this so(3,1) with the external so(3,1) symmetry of the spacetime tangent bundle, and therefore can't identify omega with the real gravitational spin connection."

My respect to amused!

garrett said...

eric:
I haven't submitted it yet, but I'm considering it. However, after the Reddit/Digg/Slashdot triple crown yesterday, I'm not sure this paper needs to get a broader readership. ;)

Eric said...

Garrett,
The point of submitting to a journal is not to get a broad readership, it's to validate work via peer-review. Bottom line, if you can't get it published then your fifteen minutes of fame will be up.

Moshe said...

Garrett, still trying to understand the formal addition, sorry for being slow. Two things bother me:

1. What is the resulting structure on the tangent space? it is not simply the E8 Lie algebra, is it some graded Lie algebra? if I got it right you are taking a linear combination of some basis of E8, with some coefficients bosons and some fermions. It is not clear to me you get a well defined structure: for example is it independent of choice of basis? is it closed under (anti) commutation? all these will be required to make sense out of gauge theory using that structure.

2. Similarly, I am still worried about diffeomorphisms of the base manifold. When formally adding objects that transform differently, seems to me the resulting object will not be diff. invariant, it will depend on what coordinate systm you do the addition. Note this is a new issue in gravity which does not arise in BRST quantization of gauge theories in flat space.

Jonathan said...

Eric,
At a guess, I would imagine Garrett is familiar with the point of the peer-review process, and was being facetious.

Given the knee-jerk attacks he's received from some who seem more interested in publishing their own personality shortcomings than in actually discussing the work (or even promoting the progress of science), I don't exactly blame him for attempting to inject a little levity.

Eric said...

Jonathan,
It looks to me like Lisi is not so familiar with the peer-review process. As far as I can tell, he has exactly one paper published in a journal and this is the not too prestigous Journal of Physics A, thirteen years ago. It's absolutely amazing to me how anyone could take this guy seriously.

alex said...

Garrett:"Can Lie algebra valued 1-forms be formally added to Lie algebra valued Grassmann numbers?"

Aaron:"Garrett, you can formally add whatever you want -- that's what "formal" means. The question is whether what you get is meaningful.

Again, the BRST symmetry is a Grassman symmetry; yours is not."

Thanks Aaron, you beat me by a few minutes.

Garrett, you can "formally" add a ghost (an anticommuting boson) to the gauge field. Even though the ghost fields are unphysical, in this case you get something meaningful because the ghosts appear only as internal states. But you cannot get anything meaningful when you treat quarks and leptons as ghosts.

Also, in your construct, the E8 is a Lie algebra, i.e. it does not have anticommutators, hence it's not a Grassmann symmetry like the BRST, as Aaron keeps telling you.

Jonathan said...

Eric,
Regardless of his publishing history, I would find it extremely surprising that anyone with a Ph.D in theoretical physics from an accredited institution would be unfamiliar with peer-review or the purpose of such. He can, of course, elaborate if he wishes to do so himself.

I would also like to see the work published in a peer-reviewed manner; however, it's also possible that what we're seeing here is very early (prior-to-publication) work that in pre-Internet eras wouldn't have been widely distributed. Garrett himself notes that he's still working out the details.

Eric said...

Jonathan,
Just having a PhD from an accredited institution does not mean that a person will have familiarity with the peer-review process. This only comes from writing papers and publishing in journals. Writing a dissertation and passing a final exam is not the same. If this paper of Lisi's is not ready to be published, then he shouldn't have submitted it to the arXiv.

Jonathan said...

Eric,
As I said, I'm sure Garrett can elaborate on his plans, and talk at length about peer review if that's what he wants to do.

I certainly wouldn't write the work off simply because it hasn't been published yet; I'm rather enjoying the discussion here. That's one reason why I stepped in on this topic: it seems to me the author is interested in focusing on the theory right now, which (on an online discussion group) is hardly inappropriate.

I share your desire to see this pass peer-review muster at some point, and I certainly hope it goes that way.

garrett said...

Moshe:

Let me write the extended connection out in local coordinates so we can clear this up:

A = dx^i H_i^B T_B + Psi_C T^C

This is over a four dimensional base manifold. The dx^i are coordinate basis 1-forms, the H_i^B are the gauge field coefficients, T_B are E8 Lie algebra basis elements that live in the so(7,1)+so(8) subalgebra, T_C are E8 Lie algebra basis elements which live in the rest of the Lie algebra and are acted on as spinor multiplets by the so(7,1)+so(8), and the Psi_C are Grassmann number coefficients. This A is a coordinate independent object.

Eric:

I think the peer review process is broken.

Alex:

"Garrett, you can 'formally' add a ghost (an anticommuting boson) to the gauge field."

Yes. Thank you. Now please consider the fact that the mathematics of this construction allows for the description of fermions. Just put aside your prejudice against ghosts for a second and consider how the math works out.

To All:

If I cared more about fame than physics, I'd be answering the 20 requests for interviews in my inbox instead of addressing these physics questions. I'm just trying to figure out how the universe works as best I can, and I posted this article because I think I've found something cool.

Aaron Bergman said...

It's probably worth saying that all these issues about symmetries, C-M, etc. aside, it would be interesting if one could see that all the various fields of the standard model could arise as a decomposition of the adjoint of E_8. Towards that, I've tried at various points to read through the first half of the paper and have failed, and I don't have the energy or motivation to work through the group theory myself. As best I can tell from what is in the text, it is not true, however that the standard model has arisen here. Instead, one obtains the three eight dimensional reps of spin(8). There is some handwaving about how these are related by triality, but triality is an automorphism of the group. It doesn't change the fact that these are still different representations.

Aaron Bergman said...

Now please consider the fact that the mathematics of this construction allows for the description of fermions. Just put aside your prejudice against ghosts for a second and consider how the math works out.

No, it doesn't. That's what people have been trying to tell you.

(one gets the feeling one is being ignored sometimes....)

Anonymous said...

Garret, in one way this *is* a peer reviewed process, so don't knock it quite yet. Personally I feel if you were to work on the theory a bit more and then publish it to one of the journals it could get accepted. I think publishing it now would be too hastey. Just a comment from a layman who only barely understands these concepts. Which in reality does lend to how simple it really is.

Eric said...

Garrett,
The peer-review process is not broken. This is just the typical cop-out excuse from crackpots who can't get their half-baked theories accepted.

Jonathan said...

Anonymous,
I agree with your comment, and I DO understand Mr. Lisi's problems with the peer-review process. Nonetheless, peer-review IS a valuable thing, and I hope (once the theory is fully developed) that he will consider publishing in a journal.

The biggest problem I have here is that it seems too many folks are simply looking to dismiss the work out-of-hand, not on scientific grounds, but rather for personal or procedural reasons. Sometimes dismissing something as crackpot is certainly warranted. It doesn't necessarily follow that everything that hasn't been peer-reviewed yet (or is simply different than one's own view) should be so dismissed.

In point of fact, the nature of much of the criticism I've seen leveled here has served, in my mind, to lend credibility to Mr. Lisi's work.

garrett said...

Aaron:

Your comment about the three representations are correct -- they are only equivalent under triality, but are otherwise distinct. I agree this is unsatisfactory, and needs more work, and I discuss this in the paper.

The BRST framework is a little strange, but the math does work out. (And I know that "being ignored" feeling.) Did you look in the references I gave for BRST?

Anonymous said...

Same anonymous as above (can't figure out how to set my name to something unique), eric, I think if he were to publish in the peer review process before he got at least a few things more clearly articulated, he'd get the same reaction he is getting in this blog. Frankly someone as intelligent as Garret to me cannot possibly be a "crackpot." Wrong, maybe, crackpot? Geez. Wait as long as you like Garret. :)

Anyway, doesn't String Theory try to "packaging Bosons and Fermions together"? I seem to recall something about that in David Gross' String Theory lecture recently at Berkley.

almida said...

Eric,
I assume you would have claim Perelman's attempt at proving the Poincaré conjecture is pure gibberish simply because he is an incompetent PH.D afraid of being peer-reviewed. Do you realize how inconceivable your judgment criteria are?

chris said...

hi garrett,

just to understand: do you finally expect that the dynamics will somehow fix this choice of gauge that you introduced at hoc for the moment?

best,

chris

Lumo said...

Dear "Almida",

I assure you that Perelman's precious results have been peer-reviewed several times, for example in Journal of Asian Mathematics.

Anonymous said...

I would be very glad if somebody could tell me how SO(3,1) is connected with gravity. As far as I know SO (3,1) is a rotation group in our 3D space so how it fits with gravity is for me unclear.

Justin Case said...

Talking about the peer-review process, it might also be good to see how the first string theory paper by Susskind was received:

LEONARD SUSSKIND: And I fiddled with it, I monkeyed with it. I sat in my attic, I think for two months on and off. But the first thing I could see in it, it was describing some kind of particles which had internal structure which could vibrate, which could do things, which wasn't just a point particle. And I began to realize that what was being described here was a string, an elastic string, like a rubber band, or like a rubber band cut in half. And this rubber band could not only stretch and contract, but wiggle. And marvel of marvels, it exactly agreed with this formula.

I was pretty sure at that time that I was the only one in the world who knew this.

BRIAN GREENE: Susskind wrote up his discovery introducing the revolutionary idea of strings. But before his paper could be published it had to be reviewed by a panel of experts.

LEONARD SUSSKIND: I was completely convinced that when it came back it was going to say, "Susskind is the next Einstein," or maybe even, "the next Newton." And it came back saying, "this paper's not very good, probably shouldn't be published."

I was truly knocked off my chair. I was depressed, I was unhappy. I was saddened by it. It made me a nervous wreck, and the result was I went home and got drunk.

BRIAN GREENE: As Susskind drowned his sorrows over the rejection of his far out idea, it appeared string theory was dead.

Source: http://www.pbs.org/wgbh/nova/transcripts/3013_elegant.html

garrett said...

chris:

Yes, the wishful thinking is that there's an action that makes sense from the top down and matches up with this one I've built to match the standard model from the bottom up.

Anonymous comment #248:
First, congratulations -- you win... absolutely nothing! Second, try Wikipedia for the Lorentz group.

Aaron Bergman said...

Lubos, that, ummm, might not be the best example for peer review. Nonetheless, Perelman's work has been checked many times, not the least of which by the above, Morgan and Tian, and Kleiner and Lott.

Garrett, yes I did. As I've pointed out to you a number of times, the BRST symmetry is a Grassman symmetry. I also pointed out where it says that in the reference you use for BRST. Now, it would be interesting if you did have a super-Lie algebra. Have you considered trying to work that out?

Eric said...

Garrett,
Just curious about an equation in your paper, in section 2.2.2 where you've written S0(4) = SU(2)_L + SU(2)_R. Shouldn't the correct expression be SO(4) = SU(2)_L x SU(2)_R, or do I just misunderstand you? Similarly, where you write SO(6) = SU(4), this is not strictly true. The two groups are isomorphic but not equal.

Bee said...

Anonymous: To post under a name, choose the option 'Other' under the box where you type the comment, it will open a field where you can enter a name.

amused said...

Aaron: lol yes, I think so. (re. PRL) Everyone should be sending their best work there.

Garrett,

"To the degree that this approximation is true, Coleman-Mandula holds and this E8 Theory is in agreement with it."

The SO(3,1) is embedded in E8 in a nontrivial way, so I don't see how the so(3,1) part of the gauge field can ever separate from the other parts - they will get mixed under E8 gauge transformations. Assuming the C-M theorem is correct (surely!), the implication is that one of the following must hold: (i) your model evades one or more of the assumptions of the C-M theorem, and continues to evade it when applied to physics in arbitrarly small spacetime regions such as lab experiments (it's a local theory, right?), or (ii) the so(3,1) part of your e8 gauge field can't be identified with the graviational spin connection, in which case the theory doesn't contain gravity.

The question then is: which, if any, of the C-M assumptions does your theory evade when applied, say, to lab experiments. It would be shocking if we can't assume for all practical purposes that Poincare symmetry holds in lab experiments. It seems to hold up pretty well so far! I don't know what the other C-M assumptions are (never studied C-M, can't be bothered) but no doubt they are reasonable and generic. If you can point to any other of the assumptions that your theory might evade in a lab setting, please do so.

One aspect of your theory that I guess could put it outside the C-M assumptions is the nature of your e8 gauge field, involving the sum of a usual 1-form and a grassmann field term. But the meaningfulness of this sum is, well, a bit controvesial at this point. So if you take that route to try to evade C-M then you will be basing you evasion on what appears to be an absence of meaninfulness in your theory. Perhaps you can discover some further mathematical structure that allows you to give well-defined meaning to the sum in your connection, but, for reasons other commentators have mentioned, that's looking fairly problematic at this point. Anyway, good luck!

Anonymous: thanks!

garrett said...

Aaron:

No, I haven't tried working this through with a super-Lie-algebra yet, since I'm allergic to superparticles, but please give it a try and see if you can get anything that works that way and makes sense to you.

The way I've formulated it is what I think makes the most sense, based on the BRST literature I've waded through.

Also, if it just irrecoverably turns your stomach to be adding these Grassmann fields and 1-forms in one connection, then go ahead and consider them as separate fields, inhabiting different parts of e8. The physics is the same.

almida said...

The lumo said:

"I assure you that Perelman's precious results have been peer-reviewed several times..."

The historical fact is this: Perelman put his work in the archive and never published it. He was crucified by smart guys like lomo, who were absolutely convinced the guy is an incompetent crank. (However, even those arrogant mathematicians did not regard their anonymous rival, who popped out from nowhere, as if he "belongs to different species" - using lumo's own words).

And the rest is history.

Eric said...

Garrett,
Also in your paper, when you write equations such as

SU(4) = U(1) + SU(3) + 3 + 3-bar

what exactly does this mean? Maybe it's just your notation, but this expression doesn't make any sense to me.

Bee said...

amused, regarding your question

The SO(3,1) is embedded in E8 in a nontrivial way, so I don't see how the so(3,1) part of the gauge field can ever separate from the other parts - they will get mixed under E8 gauge transformations.

Read the above exchange:

Bee:
How can you have the SM sector not at all mixing with the part you identify with the Lorentz group - as is usually the case. Or do these mix, and you have to fix that with the symmetry breaking?

Garrett:
They all mix, and we fix that with the symmetry breaking.


Hi Garrett,

If the SM gauge transformations mix with the Lorentz transformation, but the SM gauge transformations affect the way fermions transform, does this imply that the part you identify with SO(3,1) mixes both fermions, and bosons until you have fixed it with the symmetry breaking as well? That's my interpretation of Moshe's question above, point 1, being essitally my earlier question.

Best,

B.

Aaron Bergman said...

Also, if it just irrecoverably turns your stomach to be adding these Grassmann fields and 1-forms in one connection, then go ahead and consider them as separate fields, inhabiting different parts of e8. The physics is the same.

The addition is not the problem. It's the symmetry. You're transforming commuting fields into Grassman fields. The only way to do that is with a super-Lie algebra. To respond to one of the various anonomi, this is how string theory (or, more precisely supersymmetry) packages bosons and fermions together: through a super-Lie algebra.

To Eric, it seems clear that Garrett is working with Lie algebras, even if his notation leaves a lot to be desired. It is not true, however, that SU(4) is isomorphic to SO(6) as a group -- the correct isomorphism is to Spin(6).

As for C-M, I'm not sure it really applies here because the full E_{8+8}, even ignoring the Grassman vs. commuting stuff, is not a symmetry of the theory, at least if I understand (3.7) correctly.

garrett said...

eric:
The lower case indicates these are Lie algebras. Throughout the paper, I'm working at the Lie algebra level. I haven't worked out this group breakdown on the global, topological level yet -- but that raises some fascinating and difficult questions.

amused:
Please answer for me: which other part of the so(7,1)+so(8) subalgebra of the e8 Lie algebra is it that the so(3,1) doesn't commute with -- given the way I've broken it up? I believe answering this will help in understanding this issue.

Observer said...

"for example in Journal of Asian Mathematics" he says, and points to a revised and renamed version of the original "Chinese mathematicians prove the Poincare Conjecture!" paper Cao/Zhu published in that journal; a paper that, according to some, caused Perelman to abandon mathematics. Now that's one broken process if I've ever seen one ;-)

garrett said...

eric:
It's a description of how the Lie algebra breaks up.

bee:
The SM gauge transformations don't mix with the Lorentz transformation. For further clarification, have a look at the question I asked Amused.

aaron:
"You're transforming commuting fields into Grassman fields." No, I'm not. I'm replacing 1-forms, which anti-commute, with Grassmann fields.

Aaron Bergman said...

That's confusing two different gradings. The best example is the one-form in ordinary EM. It is a commuting field even though it is a one form, and to have a symmetry with a Grassman field, you need Grassman symmetry which is what is done in super-yang-mills.

Also, to be completely precise, you're not describing how the Lie algebra breaks up. What you are doing is describing how the adjoint representation decomposes when you pass to a subgroup.

Bee said...

Garrett:

The SM gauge transformations don't mix with the Lorentz transformation. For further clarification, have a look at the question I asked Amused.

Are you referring to this:

which other part of the so(7,1)+so(8) subalgebra of the e8 Lie algebra is it that the so(3,1) doesn't commute with -- given the way I've broken it up?

*sigh*

First, you said above they do mix ("They all mix, and we fix that with the symmetry breaking."). I thought I understood this, and tried to live with it. E8 is not a direct product of SO(3,1) with something. But isn't that what you need if you don't want them to mix? Best,

B.

chris said...

hi amused,

garrett explicitly stated that his lagrangean is explicitly a gauge fixed one, so no, it does not have E8 gauge symmetry all the way down even to the effective scale he writes his lagrangean.

if this makes sense - i honestly have no clue. but that is how he formulated it. so the point of the nontrivial SO(3,1) embedding could just be moot.

Eric said...

Garrett,
So when you write su(N) you are referring to the adjoint representation of SU(N) rather than the gauge group SU(N)? So, su(4) = u(1) + su(3) + 3 + 3-bar is just the breakdown of the adoint of SU(4) after a symmetry breaking?

garrett said...

aaron and bee:
There is no symmetry here that mixes the Grassmann fields and gauge fields. Chris reiterated for me, and I will re-reiterate: the full e8 symmetry is broken (by hand) by the action.

Whew, two and a half with one stone. I have to go for a bit -- back later.

Bee said...

Hi Garrett: Thanks. So your above comment again referred to after the symmetry breaking, but they do mix before the symmetry breaking? Sorry for repeating that, I just think there are still people who haven't understood that you are not just doing E8 gauge theory, but fix these details by hand.

Everybody: would you please, please do me the favor to read the comments and the post above before you reiterate part of the discussion?

Best, B.

Wes said...

Having read the entire blog (yes), I'm struck that so far there have been no comments about the predictions that could be made from this theory. My question to those of you who best understand this are:

1) What are the predictions that this theory can make that are most likely to be tested first.

2) For those of you that so strongly argue that the theory is flawed, does it predict something testable now that could be used to disprove the theory quickly?

Thanks

Jonathan said...

Wes,
I agree that I'd like to see a discussion of this. As an engineer, to me the old adage regarding the relative weights of theory and evidence is always the guiding light of these things -- no matter how elegant the theory.

Bee said...

Wes, Jonathan,

As I wrote in my post

To make predictions with this model, one first needs to find a mechanism for symmetry breaking which is likely to become very involved.

Further, there is the very general observation

He finds a few additional particles that are new, which are colored scalar fields.

Given that these couple strongly, their masses have (as usual) to be so high as to not yet have been observed, but maybe potientially LHC testable.
Best,

B.

chris said...

hi bee,

he has a gauge fixing term in his lagrangean. this is also how he can get 'ghosts' in the first place. and he repeatedly stated that this is done ad hoc. so as i get it, it really is just an effective theory, not the fundamental one and yes, certainly not a E8 gauge theory in the classical sense.

Bee said...

Chris: Yes, thanks. I just wanted to make that clear, once again, that's one of the reason why things work out, and it doesn't arise from the E8 root diagram, there are a lot of extra assumptions that go in here. I am still not convinced though that the gauge fixing can be sufficient to make it work. In what sense do you mean 'effective theory'? If you are referring to the 'top down inspired bottom up' remark, please see the comments above. - B

amused said...

chris,

If the action in the paper is obtained through gauge fixing then the C-M considerations can be applied to the original, unfixed action (whatever that is). The physics is independent of the choice of gauge fixing, presumably. So I would expect the C-M considerations to still hold.

Garrett,

"Please answer for me: which other part of the so(7,1)+so(8)subalgebra of the e8 Lie algebra is it that the so(3,1) doesn't commute with -- given the way I've broken it up?"

As Bee already mentioned, E8 is not the direct product of SO(3,1) with something. That's the relevant point for the C-M stuff.

David Gerard said...

A present for Dr Lisi from the cheap seats: http://uncyclopedia.org/wiki/UnNews:Surfer_dude_stuns_physicists_with_theory_of_everything

chris said...

gear garrett,

it is none of my business whether you decide to submit your paper to a journal, but i would like to respond to your very broad assertion of the brokenness of the peer review process.

up front: yes, i think it is broken in some ways. i do not want to indulge further, but basically all the brokenness is due to many papers being "under the radar" and therefore exposed to subjective treatment by sometimes a single referee. that can be harsh and everything, but ultimately, i don't know of any good papers in my field that didn't take that hurdle.

in your case, given the ongoing media attention, your paper is certainly on the radar. i guess that every serious journal would be well advised to treat your paper very carefully and properly, so i really think that you need not worry about ill treatment there. in fact, what is happening here is already kind of a peer review. and as you can see the spectrum of reactions is quite broad.

it is of course your choice and in the end we will see, if the paper gets cited or not. but let me just tell you that for most of us the peer review is nothing else than putting your money where your mouth is. if what you say is correct and relevant and if you welcome honest and serious critique, there should be nothing to fear.

chris said...

hi bee,

by effective theory i mean, that a gauge fixing term in the lagrangean can hardly be accepted as god-given (or nature given if you happen to be atheist :-)). garrett also said so in a previous post, so i guess this is his take on it, too. so there should be some dynamics (at some higher scale i suppose?) to provide this term.

as i said, i have no idea if this makes any sense.

chris said...

hi amused,

i am not here to defend garretts theory. but *if* you swallow that his theory contains this gauge fixing term that effectively comes from dynamics at higher scales, then you evade C-M at low scales by not violating it and at high scales by not being locally isomorphic to the lorentz group (above m_planck, which i guess would be close to m_lisi :-), i do not expect minkovski space to be a good approximation anymore, no?)

the big mystery to me is how you would get an effective gauge fixing term.

Your Friendly Moderator said...

How can I possibly say that clearer: Given today's status, Garrett's model does *not* naturally lead to a unification of the SM interactions with gravity (he has to chose the action by hand that contains both), it does *not* allow us to understand quantum gravity (since there's nothing said about quantization), it does *not* explain the parameters in the SM (since there isn't yet a mechanism for symmetry breaking), it does *not* explain the cosmological constant or its value (as said above, to claim there has to be one, it would be necessary to show there's no way to do it without one), it does *not* explain the hierarchy problem (and I see no way to do so), it does *not* explain why we live in a spacetime with 3 spatial and 1 timelike dimensions, it does *not* in my very humble opinion yet qualify being called a Theory of Everything.

At this point, I think this thread should be closed. I strongly recommend the blog owner to do so. While it may be obvious to the paper's author about how this all is covered, the paper obviously needs more authoring.

Observer said...

"How can I possibly say that clearer" By posting under your own name, perhaps, instead of pretending that it's up to you to moderate someone else's blog.

Your Friendly Moderator said...

observer telling your friendly moderator to post under a real name? Haha!

The bold text was a quote from Bee in case anyone hasn't read the previous comments.

chris said...

hi friendly moderator,

could you please enlighten us about why we have to stop discussing? has the pope put out a bull and we will be under his ban-ray if we continue enjoying ourselves here?

garrett said...

So long, and thanks for all the fish.

Bee said...

Hi All

It seems the discussion to this thread has reached the blog recurrance length. People have started to comment without reading the previous exchanges (or the post to begin with). Questions, answers, arguments and criticism on both sides have been repeated several times.

I see no point on continuing the discussion under these circumstances, and therefore I will close the comment section.

We might have a follow up posting in the next weeks, summarizing this thread, and opening a possibility to continue the discussion for those who are still interested.

Thanks to everybody for your contributions, it has been very interesting.

Best,

B.

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