- 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.
TAGS: PHYSICS, MATHEMATICS, TOE, GUT, E8
Hi Sabine, you've written an excellent summary.
ReplyDeleteThe coupling constants are there -- they're all one. Then they run.
Well, Thanks to you for patiently answering my questions :-)
ReplyDeleteI am missing the coupling to gravity, not the gauge couplings. You've defined it into the fields and I keep looking for the Planck scale. It is confusing because it is useful for the question of derivatives in the action. The action is dimensionless, the gauge fields enter via F^2 meaning A has mass dimension (4-2)/2, whereas the fermions enter as \psi D \psi meaning they have mass dimension (4-1)/2. Usually you can't just add them.
btw, why did you order the names in the acknowledgements in reverse alphabetical order? So Peter is the first? Best,
B.
Hi bee,
ReplyDeleteThis blog is one of my favorites.
I really like your posts and summary's...I haven't commented yet because they've always kept me thinking. I was wondering if you're a teacher? If not, you'd make a really good one.
My nickname is also b .
Hi b, thanks for the nice words. If I got you thinking, I rate that as success :-) No, I'm not a teacher. I occasionally feel like a preacher, but I don't particularly like it. Best, B.
ReplyDeleteSmolin's "Ring of Truth"
ReplyDeleteI might of missed that one in the book. What page was that?:)
The complexity of the table of elements by Mendeleev is a far cry from this topic, but imagine being able to place the particle constituents within the framework as Mendeleev did with new elements.
This indeed would be a success.
Hi Plato... ooohm, given that the 'search inside the book' option at amazon is missing, I actually don't know. Hmmm. I think he used it repeatedly. Somewhere in the first some chapters, about unification, theories making new predictions? Or something about scientific revolutions, changing our view of the world? Apologies, it's been more than I year I read the book, I can't recall. Maybe it actually wasn't from Lee's book *scratchhead*...
ReplyDeleteRandom browsing resulted in ... "But we are also fairly sure that we do not yet have all the pieces. Even with the recent successes, no idea yet has that absolute ring of truth." p. 255 (US hardcover). Sounds like a reference to a previous mentioning, so at least you can now be sure it's the right book I was referring to. Best,
ReplyDeleteB.
It's a very nice paper indeed, and a lot of people with the relevant skills [not me!] should work on it to see how far it can be pushed.
ReplyDeleteHaving said that, I'd just like to echo Bee's
"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."
Yes, the problem is that in gravity one does not have just any old O(1,3) bundle, it's a special one. [Failure to understand this simple fact is part of the reason that a lot of string theorists don't understand general relativity properly.] So the process of symmetry breaking not only has to produce O(1,3), it must at the same time explain why that sub-bundle miraculously turns into the bundle of orthonormal frames. Unless that is built in from the start, which would be cheating......
bee:
ReplyDeleteWith the acknowledgments, I thought about ordering them according to how much people had helped, but that seemed too political. So I was going to go alphabetical, but it occurred to me that these people had been dealing with their alphabetical placement their whole lives. So I tried to help balance this a little by reversing the order.
Units... The unit issue is trickier than particle physicists usually deal with because the metric or vierbein is floating around too -- one needs to work out the units in curved spacetime. It doesn't make sense for the coordinates to have units, since they're just labels. The length unit (-1 mass dimension) is more sensibly in the vierbein, or the metric if you like, [g_ij] = -2. Since the partial derivative, d_i, is now also dimensionless, this means the connection is dimensionless as well,
[d_i] = [A_i] = [om_i] = 0
as is the curvature [F_ij] = [d_i A_j] = 0. (If you hate the idea of a dimensionless connection, we could contract it with the vielbein, but then we'd have to take this back out before calculating it's curvature -- so that's probably a bad idea.) Since we use the metric to raise indices, [F^ij] = 4, and the volume form has [sqrt(g)] = -4, so the "F^2" action term in curved spacetime is dimensionless. For the fermions, the action term in curved spacetime is
sqrt(g) psi e D psi
and since [e]=1 and [D]=0 we have [psi] = 3/2. So this does encounter the problem you're complaining about, as our superconnection is A+psi. I'm not sure if this is bad though, since the A's are 1-forms and the psi's are Grassmann numbers -- I'll have to go see how these dimensions get juggled in the BRST approach before I have a good answer for you.
plato:
Did you have a look at the periodic tables in the paper I wrote? Also, I initially opened the paper with a nice quote from Poincare that I stole from you, but my reviewers deemed such a flourish inappropriate -- party poopers.
Garrett, very pretty.
ReplyDeleteWhen I saw equation (2.4) I of course thought of the neutrino mixing matrix under the "tribimaximal" assumption, which of course is a cross generational thing. See the matrix in equation (3.2) of Koide's paper, and references, or my recent note on the charged lepton masses.
However, it didn't seem to me, at a first quick read, that this matrix is related to how the generations work in your model. I wonder if, in one of your other versions of this model, you ended up with that matrix having something to do with the generation structure.
Carl:
ReplyDeleteThat is a great clue. I remember looking at Koide's mass matrix idea several years ago, but somehow I had missed this correspondence. This rotation does in fact relate to how generations are related through triality. I'm not sure yet exactly how the masses are going to come out, but the pieces (like this one) seem to be falling into place nicely. Thanks for the hint -- I'll have another look at the Koide paper and yours.
george:
The symmetry breaking question is the same as for others. The cosmological constant is a little better in this model, because it needs to be there. But, for the most part, this model just unifies the existing problems.
Hi Carl, Hi Garrett:
ReplyDeleteIndeed, I too was reminded of that matrix. Though I wouldn't get over excited about it. It's not really hard to get. It popped out in some rather simplistic model that I've been working with a while ago, but the details (mass splittings) didn't. I thought about this again last year, after I read that paper by Kovtun & Zee. If I recall that correctly they say somewhere the mass splitting isn't a problem. I asked them about it, but couldn't clarify this. If somebody could let me know? I.e. in my model it turned out one gets this matrix only if two of the masses are identical. More generally, the mass-splittings are related to the deviations from the tribimaximal mixing, and this relation turned out to be fatal (i.e. in conflict with data). Best,
B.
Hi George, Hi Garrett:
ReplyDeleteThe cosmological constant is a little better in this model, because it needs to be there. But, for the most part, this model just unifies the existing problems.
I'd say this is a matter of perspective. Your model doesn't work without the CC to begin with. However, you have chosen the action by hand. If you'd want to claim this is an advantage, you'd have to argue there is no way to do it without a CC. Saying there is a way to it with a CC isn't sufficient.
Also, correct me when I'm wrong, but the way the CC is related to the VEV is not usually present, and I'd call that problem worse. I.e. the CC is evidently not the order of the Higgs VEV, and one can't just ignore this mismatch. In your case this is not the vacuum energy problem - which is commonly ignored - but actually the constant appearing in the Lagrangian. Best,
B.
Hi Other,
ReplyDeleteThanks for your comment, you expressed that more clearly than I did!
Hi Garrett,
Reg. the dimensions of the fields/ and or coordinates. Sure, one can shift them elsewhere, I just find it very confusing. Let me know if you find an intuitive explanation for the gauge field + fermion question. I'm still hoping I'll wake up one morning thinking 'Oh THAT was what he did!'
Best,
B.
Thanks Bee for the Clarifications on Ring of Truth.
ReplyDeleteI think that "sense of ringing" can be profound on different people, of course depending on How Lee supposedly mean that terminology to be expressed. How people receive it. How they remember it?
Ultimately, I know that deep down all of us want to be based on a good foundation "to progress from" when we are dealing with the world. Finding the basis of that truth is very important. Philosophically, as well as mathematically and with just plain dealing with reality, what ever that is?:)
I think Lee was talking about "paradigmatic changes" in regards to Kuhn.
This comment has been removed by the author.
ReplyDeleteHi Garret,
ReplyDelete"The weights of these 222 elements corresponding to the quantum numbers of all gravitational and standard model exactly match 222 roots out of the 240 of the largest simple exceptional Lie group, E8."
What I am seeing in my mind ultimately needs to be seen in relation to the complexity of the E8 "dimensional complexity." So it would have to ultimately lead to the elements in question.
Now, if you had looked at Grace satellite( I know your busy) and how it is measuring the planet, there is a response to the gravitational consideration by their tether?
So observationally, "densities of the elements," have a affect on that tether? Oui Non?
I am glad to see progression on the E8. Wonderful information. As a layman, I needed to see how your model would have been attached to the reality we live.
Seeing the "vibrational nature of all elements," E8 would serve for me and help me to understand that elemental structure being defined to it's roots. Escher and Poincaré?
Mendeleev when he developed his table, allowed for "prediction to be inserted" within the confines of the nature of that elemental table by it's "signatures."
Your "particles must fit within the complete rotations." So you are saying this. From a universal prospective, ultimately the 4 dimensional expression had to be reached. So gravity "is included" all the way down?
Ultimately, I am seeing the gravitational inclination of each of these elements within the context of your model. But I am a layman, so those testing your model have to do a good job of tearing it apart:)
Yes, party poopers indeed. But hey, your in the arxiv now. :)
bee:
ReplyDeleteYes, at first I considered the large value of the cosmological constant in this model to be a worrisome bug. But now this idea is in agreement with current theories of a large cosmological constant at high energy (ultraviolet fixed point) running to the tiny value we experience at low energies. So the bug now looks to be a feature.
There may be other ways of understanding it, but if you want to have the realization of 'Oh THAT was what he did!' then you'll have to have a dream about how these fermionic fields could emerge as BRST ghosts.
Two quick questions:
ReplyDelete1. What is the loophole in the Coleman-Mandula theorem used in this construction? note that the theorem allows constructing theories where internal and spacetime symmetries are unified, as long as those theories are free.
2. When packaging bosons and fermions together, at least one set of fields will have the wrong spin statistics relation. In addition to violating unitarity etc., this definitely is not what is going on in the standard model.
Hi moshe,
ReplyDelete1. Yes, the Coleman-Mandula theorem assumes a background spacetime with Poincare symmetry, but this theory doesn't have this background spacetime -- with a cosmological constant, the vacuum spacetime is deSitter. So this theory avoids one of the necessary assumptions of the theorem, and is able to unify gravity with the other gauge fields. On small scales though, Poincare symmetry is a good approximation, and on those scales gravity and the other gauge feels are separate, in accordance with the theorem. (I'm not the first person to dodge C-M this way.)
2. The cool thing about the exceptional groups is that the femions come out, algebraically, with the correct spins -- so they do satisfy the spin statistics relation as Grassmann fields.
other:
ReplyDeleteThe spin connection and gravitational frame are both parts of the E8 connection.
Hint 2: Garrett, for the benefit of string theorists, I was wondering what you thought of the reduction from the heterotic group, as discussed by kneemo.
ReplyDeleteA comment to Bee and a question to Garrett:
ReplyDeleteBee,
the su(3) structure of the baryons was found independently by Neeman and Gellmann, but Neeman released his work a little earlier than Gellmann.
Garrett,
what is the mathematical identity of a connection containing both, Grassmanian and non-Grassmanian objects? is this construction of yours a direct sum? Am I missing something?
Hi Almida:
ReplyDeleteYes, thanks. I know. I just tried to keep it brief and referred to details in footnote [2] to Stefan's earlier post.
Best,
B.
kea:
ReplyDeleteAre there benefits for string theorists now? I didn't know things had gotten THAT bad for them. ;)
almida:
A connection composed of Grassmannian and 1-form elements is called a superconnection, or a Z2 graded connection. It may be more familiar to physicists as a BRST extended connection. And no, I didn't make it up -- though I did make up this one.
Just googled "BRST extended connection" and the only results I got point to Garret Lisi's papers/talks.
ReplyDeleteOn the other hand, the term "super connection" is indeed well known and simply denotes a superfield expansion. However, in that case, the fermion fields are all multiplied by the corresponding power of the Grassmann variable \theta so that all the terms in the sum have the same spin dimension.
What Mr. Lisi is doing is complete nonsense as he simply adds fields of different spins.
Hi Anonymous:
ReplyDeleteIf you're referring to my earlier comment you're twisting words in my mouth. This was not what I said. I a priori have no problem with adding fields of different spin in some algebra, provided the addition is well defined. If one can do this for n-forms, why not also for spins and vectors? What I am confused about is if you treat them so similarly, how can you nevertheless get them to appear differently in the equations of motions (thus my comment about the different order of derivatives in the Lagrangians). This seems possible, but the construction Garrett currently has just looks to me too ad hoc to be really pretty. Might be an observer independent statement, but I'd like to see some more investigation of this point. Best,
B.
besides this, your comment
ReplyDeleteJust googled "BRST extended connection" and the only results I got point to Garret Lisi's papers/talks.
Is pathetic and confirms (again) my concerns about the negative influence of the internet on scientific discussions.
-B
Anonymous, your google-fu is poor, try again. This kind of extended connection, combining Grassmann number and 1-form fields valued in a Lie algebra, is common in the BRST literature, though it often goes by different names.
ReplyDeletebee:
ReplyDeleteI agree that the action is assembled by hand, in order to be in agreement with the standard model. This does cry out for a more elegant description. But the "modified BF theory" action I wrote down isn't so bad. This action, including gravity, gauge fields, and fermions, only has one derivative, so I'm not clear on what's making you unhappy?
I'm not unhappy. I just don't like it. It's supposed to be a TOE, so if you say "that the action is assembled by hand, in order to be in agreement with the standard model." it just doesn't sound particularly compelling to me. But what me or you do or don't like doesn't say anything about whether it could actually be a description of nature. Best, B.
ReplyDeletebee:
ReplyDeleteOK, good, I agree with your dissatisfaction. I think the "true" action will be slightly different, but don't know what it is yet.
OK, so I used your google search and founf this paper:
ReplyDelete"Modified Maurer-Cartan condition and fiber bundle structure of the gauge theory of gravity".
By Nakamura, Kikukawa and Kikukawa, Published in Prog.Theor.Phys.91:611-624,1994.
So, I downloaded the KEK scanned version of their paper from SPIRES to see the definition of BRST extended connection.
Indeed, as I'd expected it is just a kind of a superfield expansion in terms of Grassmann variables \xi^A.
The explicit expression for the connection is given by eq. 3.33 of their paper.
However, this expression is very different from what you call "extended connection". Note that in eq. 3.33
all the terms in the first line in the sum inside the parenthesis are bosonic. Indeed, since both d\xi^D and \partial_D are fermionic, their product is bosonic and therefore the sum makes perfect sense.
In contrast, when I look at your formula (1.1) I see nothing like the BRST extended connection I've found in the above paper. In fact, your formula in 1.1 makes absolutely no sense as the terms in the sum have different dimensions.
I'm very surprised that Smolin did not say anything to you about this nonsense.
Actually, just looking at eq. 1.1 in your paper I keep wondering what happened to the Lorentz spinor index of the fermion fields.
ReplyDeleteYou wrote that u=U^AT_A but U^A also has another index. Where did it go?
Bee said "...and confirms (again) my concerns about the negative influence of the internet on scientific discussions."
ReplyDeleteI think there is a real problem here. When physics blogs started, they were a tremendous boon for people like me who are working in isolation --- at last I could feel that I was a member of a community, and could get all the latest news etc. But gradually I find that reading most blogs [not yours!] is actually having a very bad effect on me. It's not just the nastiness [though that is bad enough, it just reflects reality]. It's the dogmatic way people talk. Even in face-to-face discussions over lunch, very few people use words like "nonsense" as freely as they do in a lot of blogs. Also, it worries me that young people may get the wrong impression from reading some of this stuff. I'm not referring so much to Lubos Motl's stuff, which is so extreme and vicious that anyone can see that they ought to take his words with a very large grain of salt; I'm thinking of blogs like Peter Woit's, with his anti-Landscape propaganda. I really wish that people would stick to technical comments and avoid trying to cultivate a "everyone knows that that is nonsense, snicker snicker" kind of culture.
Anonymous:
ReplyDeleteRather than following either of the two references cited in my paper for a clear and conventional definition of "BRST extended connection," you have deliberately chosen a paper that confuses the issue. The point of the paper you cite is to describe the Grassmann valued fields of a BRST extended connection as 1-forms in the vertical part of a principal bundle. This idea is consistent with the established fact (that I did not make up!) that Grassmann numbers may consistently be treated similarly to 1-forms, including the formal addition of the two in a superconnection. However, treating Grassmann fields this way has been tried many times before and it doesn't quite work right, though I agree with the sentiment that motivates this approach.
If you wish to understand the conventional definition of what a BRST extended connection is, try looking in the review paper I cited, in which it is called a "generalized connection" and defined on p70. Or, if you're unhappy with that paper for whatever reason, I'd be happy to provide other references. But I suspect your desire is not to understand.
Anonymous:
ReplyDeleteThe double indices are described on page 9 of my paper.
Dear Mr. Lisi,
ReplyDeleteAfter looking at the definition on page 70 of the reference you suggested I don't see how this is related to your eq. 1.1.
In eq. 4.49 of the reference quantity C(\xi) is not a Lorentz spinor. Instead, it's a left-invariant 1-form. In addition, the Grassmann variable c=c^aT_a is a ghost and therefore does not transform as a Lorentz spinor, even though it is anticommuting.
I don't understand how the double indices on page 9 help me understand what happened to the Lorentz spinor index of the fermions in your eq. 1.1.
If this index in contracted, then how is it contracted? I think that this is a very simple question to which you should be able to give a quick answer instead of referring me to page 9.
Tank you!
Hi Other:
ReplyDeleteI agree with you that there is a really problem here, but it doesn't originate in the blogosphere, it just becomes more apparent. Also, it depends on who you go to lunch with, I've certainly heard worse than 'nonsense'. The worst being somebody mentioning person X - his new paper or latest talk - and everybody just laughing. This usually doesn't happen on blogs though. We've had a previous post on the problem of online communication, in case you're interested, I do in fact think that online communication poses a significant challenge to constructive argumentation. The medium and the problem is fairly new, and unfortunately people are not sufficiently aware of the issue. In a certain sense, the internet seems to amplify bad habits. Best,
B.
Ah, anonymous, maybe I gave up on you too quickly!
ReplyDeleteAs you correctly point out, the C on page 70 of van Holten is a Lie algebra valued Grassmann number, C = C^A T_A. This is formally added to a Lie algebra valued 1-form, H = H^A T_A, to create what I am referring to as a BRST extended connection, A = H + C.
So, how can I be crazy enough to call C a fermion, which (as a spinor) should algebraically be in the fundamental representation space? That is a beautiful thing about the exceptional Lie algebras -- some of their basis Lie algebra elements, T_A, behave algebraically as basis elements of a fundamental representation space! The spinors we need are built into the Lie algebra structure of the exceptional groups! This is explained on page five, and used throughout the rest of the paper. It's a key idea that unlocks everything. Sadly, I was not the first to realize E8 has this structure -- I got the idea from He Who Shall Not Be Named, and it's been known to mathematicians for a very long time.
Dear Mr. Lisi,
ReplyDeleteI'm not sure if I follow your line of thought. You write that the u quark is u=u^AT_A. What are the T_A's in this case? Do they run over the generators of the color
SU(3)? But then, u is in the adjoint of SU(3). The u quark is both a fundamental of SU(3) and a bi-spinor of the Lorentz group, so ignoring the SU(2), it's gotta have those two indices at least. So far, I see only one index - "A" standing for the gauge group. So, again my question is, where is the Lorentz spinor index?
I'm sorry but I did not find an explanation to my question on page 5.
The other thing I wanted to ask is about the avoidance of Coleman Mandula by not being Poincare.
ReplyDeleteDo you suppose that spacetime is 4+1? And that you can pick up 3+1 quantum field theory from 4+1 classical statistical mechanics? And can the non Poincare symmetry be be tied to Euclidean relativity? And finally, does this give a decent explanation for the utility of Wick rotations?
Dear Mr. Lisi,
ReplyDeleteI guess, my question is obvious once you look at eq. 1.1 and compare the expressions for g and u. For g_i^A one clearly sees both the gauge group index - "A" and the Lorentz vector index - "i". On the other hand, u^A carries only the gauge group index but the Lorentz bi-spinor index is somehow not there. How is it contracted?
Bee, do you know the answer? Am I asking a stupid question?
anonymous
ReplyDeleteThe quarks, such as u = u^A T_A, are in the fundamental 3 of su(3), and are in the adjoint of g2. The su(3) subalgebra of g2 acts on the quarks, under the g2 adjoint, as the su(3) acting on the fundamental 3. This is described, in complete detail, on pages 5 and 6. The fermions, including the quarks, are in the fundamental spinor 4 of so(3,1), and they are in the adjoint of f4. When f4 and g2 are combined as parts of e8, the quarks are in the adjoint, with u = u^A T_A an element of e8, and the so(3,1) and su(3) subalgebras of e8 act on the quarks, under the e8 adjoint action, as if they were in the fundamental spinor 4 and color 3. This remarkable fact is what the paper is about. The entire standard model fits in e8 this way.
The i index of g_i^A is a 1-form index. There is no similar index for the fermions. They are Grassmann numbers valued in the algebra, and A is their algebraic index, corresponding, ultimately, to the basis elements, T_A, of e8.
This algebraic equivalence is not obvious, but it is true.
carl:
I treat spacetime as 3+1. However, with a positive cosmological constant, the vacuum solution is de Sitter spacetime, which has so(4,1) as its symmetry algebra. At short distances, the so(4,1) algebra looks like the Poincare algebra. Since you're a Clifford algebra guy, I can describe these algebras in excruciating detail. Using Cl(3,1) basis vectors, ga_a, and bivectors, ga_ab, their algebra under the anti-symmetric product is the same as the algebra of Cl(4,1) bivectors, so(4,1) = Cl^2(4,1) = Cl^{2+1}(3,1). The Poincare algebra is the same, except that [ga_a, ga_b] = 0 for the Poincare algebra (translations commute), but [ga_a, ga_b] = 2 ga_ab in so(4,1), and therein lies all the difference. Poincare symmetry is the number 1 assumption of the C-M theorem, and by not satisfying this assumption we dodge the implications of the theorem and unify gravity with the other gauge fields using this so(4,1) = Cl^{2+1}(3,1) spacetime algebra of bivectors plus vectors.
To the degree that spacetime is locally flat -- which is to a large degree -- the Coleman-Mandula theorem applies, and gravity is separate from the other gauge fields. But, at very high energy, it's all E8.
"The i index of g_i^A is a 1-form index. There is no similar index for the fermions."
ReplyDeleteIn which representation of the Lorentz group are g_i^A and which index "i" or "A" denotes this Lorentz representation?
Garrett: He Who Shall Not Be Named
ReplyDeleteheh heh, that's funny.
Donald (H. S. M.) Coxeter :)
There are two reasons that having mapped E8 is so important. The practical one is that E8 has major applications: mathematical analysis of the most recent versions of string theory and supergravity theories all keep revealing structure based on E8. E8 seems to be part of the structure of our universe.
And to think strings is being saved by your adventure? No, they see this in another way. :)
5. Regular polytope: If you keep pulling the hypercube into higher and higher dimensions you get a polytope. Coxeter is famous for his work on regular polytopes. When they involve coordinates made of complex numbers they are called complex polytopes.
Looking at the "false vacuum" to the "true," it seems to me, as if, "the complete rotation" much like the belt trick, could have so much complexity in the "E8 overall design."
ReplyDeleteGreg Egan's animations are always very helpful :)
Quite a few of you have spent lots of your professional lives trying to understand representations of reductive groups, so I won't say too much about why that's interesting. A little of the background for this particular story is that it's been known for more than twenty years that the unitary dual of a real reductive group can be found by a finite calculation.
I hope good science people never tire of the layman's pursuit for understanding and good foundational beginnings?
All of this is way too advanced and technical for me, but the animation is pretty. :-)
ReplyDeleteanonymous:
ReplyDelete"In which representation of the Lorentz group are g_i^A and which index "i" or "A" denotes this Lorentz representation?"
That is not a good way to think about it, but if you must... The A is the Lie algebra index, and the i is the 1-form index. Under a coordinate transformation, 1-forms are invariant, but their mathematical expression changes, as do tensors with this i index, by a Lorentz coordinate transformation matrix. Spinors are also invariant under a coordinate transformation, and their expression may change due to the change of variables, but they do not have any indices that get contracted with a transformation matrix. Spinors are collections of scalar fields with respect to coordinate transformations. Spinors do have indices that change under a change (Lorentz rotation) of the gravitational frame (vierbein), but this is a separate consideration from coordinate transformations, and happens in the A index. I realize this is confusing, but this is the conventional description of spinors in differential geometry. Do not confuse coordinate transformations with physical transformations of the frame.
plato:
Coxeter had it right.
And every physicist begins life as a layman.
Hi Anonymous:
ReplyDeleteBee, do you know the answer? Am I asking a stupid question?
Well, no, you are asking the natural question. In fact I've asked very similar questions. But I sense a certain amount of hostility in your comments, and I would appreciate if you could at least consider what Mr. Lisi says makes sense instead of being preoccupied by the conviction it is nonsense. I don't feel really qualified to talk algebra, it's somewhat outside where I've worked the last 12 years, but let me give it a simplistic try (Garrett, please correct me if I am crudely wrong).
You're starting with a large algebra with two hundred something dimensions. You can label elements in there with indices whatever you wish. Now you can write down all stuff that is in the algebra. So far that doesn't have anything to do with LORENTZ vectors, spinors or whatever.
Next thing is to notice the elements of the algebra act on the algebra itself, and to understand how they do so. It's here where the exceptionality of E8 plays a role that allows you to get the fundamental representation (usually for the fermions) togehter with the adjoint (for the gauge fields). The representation of the algebra induces a representation of the group. Now Garrett shows that you can decompose the elements such that some of them act on others in the way you'd expect for generators of lorentz trafos. Work needs to be done to show this transformation behavior has something to do with the rotations/boost in a base manifold - that's the point I and Other above have mentioned. So far there is no base manifold present whatsoever, thus no vector (coordinate) index.
I think what you are asking with the indices is how you get these transformations to say something about vector/spinor character. For this identify the SO(3,1) with the tangential space, and analyze which field transforms how under the SO(3,1) generators. Some of them do like vectors do, some like fermions do. There's no reason all of them need to have the same behaviour. That's nice. However, getting the gauge fields and the fermions together in this way comes now back to my problem with the action. If you are looking for the gauge-covariant derivative then you need the gauge fields in the connection. If you treat the fermions similarly, they should appear in the derivative the same way if the action was just E8 invariant. That however is not the cased for the SM, thus one needs to come up with some other way to get an action. I don't think you can blame that on symmetry breaking, because that wouldn't address this issue.
Does that help somehow?
Best,
B.
bee:
ReplyDeleteYes, this is an accurate description.
Let me sketch (roughly) an answer to your questions about the fermionic part of the action. If we start out with a connection that is just 1 forms,
A = H + C
And a modified BF Lagrangian that is invariant under arbitrary gauge transformation of the C part of the Lie algebra,
L = B F + B_H*B_H
in which B = B_H + B_C is a Lagrange multiplier field that lives in both (H and C) parts of the algebra, and the curvature is
F = d A + A A
= (d H + H H + C C) + (d C + H C + C H)
Then we can replace the C part of the connection with BRST ghosts, Psi, and write the new effective action for this theory as
Leff = Bg D Psi + F_H*F_H
in which
D Psi = (d C + H C + C H)
is the Dirac derivative, Bg are BRST anti-ghosts (anti-fermions and frame), and the H part of the curvature is
F_H = d H + H H
The C C part you mention goes away because C is replaced by ghosts, and there's no ghost-ghost (Psi Psi) term in the effective action.
This is my guess as to what is going on, and it's in a previous paper I wrote. However, in the current paper I just started with a superconnection, A = H + Psi, without justifying why, since the BRST path I choose to get there may not be what other people choose.
P.S. I'm impressed and flattered that you're spending so much time on this while simultaneously running a conference.
Hi Garrett:
ReplyDeleteGiven that I interpret correctly what you write above, the part that bothers me is not the CC in F, but the dCdC in the Lagrangian (I guess I could live with a CCCC). From what you wrote above, you've just put the second dC into what you call Bg - similarly to what you do in the paper, but why should that be?
P.S. I'm impressed and flattered that you're spending so much time on this while simultaneously running a conference.
German organization pays off, the conference is mostly running on its own and very nicely so. Best,
B.
bee:
ReplyDeleteYes, you understand, great. And good question. I'm not sure, but I think the modified BF action may be chosen by nature because the BF action is very natural -- it just says, classically, the curvature vanishes. And it uses a Lagrange multiplier to do it. Now, why, if it's so nice, is there a modifying term that breaks symmetry? Hell if I know.
If I ever organize a conference, it will be ruled by chaos.
I must be missing something. You have a TOE. Didn't you just win physics?
ReplyDeleteI mean, why isn't this, like, front-page news? Why aren't academics all across the world going "wow, we're done! Better drop what we're working on now because the TOE has been found, and all the fun is there!"
Clearly there's some reason that this isn't as revolutionary as it seems to be?
Thanks,
-Domenic-
Domenic:
ReplyDeleteAs with any new theory, it might be wrong. One needs to be cautious about these things. Also, there are parts of it, such as Sabine and I have been discussing, that aren't perfectly well understood. And even I don't feel confident enough in this theory to make solid predictions with it yet; though they will certainly come out as it develops. Nevertheless, it looks pretty good, more people are working on it than just me, and you can expect articles about it to start showing up in the popular press in a week or two.
And yes, I am staring your sarcasm straight in the eye. ;)
Domenic:
ReplyDeleteI've been around long enough to see a lot of people run about waving ToE's that didn't work. This makes me somewhat jaded, so I guessed you were being sarcastic (as I might be, towards someone claiming a new ToE), but you're young and might of been asking these questions honestly. Either way, my answer is the same. But I apologize for mislabeling your post as sarcasm if it wasn't.
Also, I like your last blog post.
Hi Garrett,
ReplyDeleteD might have been both, asking honestly, and with a healthy dose of sarcasm.
Great the way you don't lose your rag easily, clearly you are not on a short fuse - even if 'jaded'.
Both The Post, and the interaction or debate in the comments section have made interesting reading.
I mean, why isn't this, like, front-page news? Why aren't academics all across the world going "wow, we're done! Better drop what we're working on now because the TOE has been found, and all the fun is there!"
ReplyDeleteClearly there's some reason that this isn't as revolutionary as it seems to be?
I just want to mention that the paper has been on the arxiv now for about 48 hours.
Hmm, I thought I posted a reply, but it hasn't shown up yet... trying again.
ReplyDeleteYeah, I was asking honestly, and was more... exaggerating than being sarcastic. I mean, I was missing something, so I just kind of exaggerated my views to make it clear what I was missing. :)
Bee, with respect to Garrett's paper about "... E8 ... over a four dimensional base manifold ...",
ReplyDeleteand
your remark "... Kaluza-Klein theory ... 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 ...",
and
"... the exceptional Lie-groups ... 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 ...":
Instead of just putting E8 and 4-dim spacetime together by hand,
what about
taking the 248 dimensions of E8
and
combining them with a 4+4 = 8 dimensional Kaluza-Klein Spacetime similar to that described by N. A. Batakis in Class. Quantum Grav. 3 (1986) L99-L105, which was successful with respect to the Standard Model gauge bosons and was not generally accepted because of difficulties of adding fermions.
Perhaps the boson-fermion structure of E8 could solve those difficulties of the Batakis model.
By combining the 248 E8 dimensions with the 8 Batakis Kaluza-Klein spacetime dimensions,
you could form the 256 dimensional Cl(1,7) Clifford algebra,
whose structure could be used as a guide for naturally constructing a Lagrangian that could reproduce the successes of the Standard Model and Gravity based (as Garrett does) on MacDowell-Mansouri.
Tony Smith
PS - Maybe I should have said that structurally the 248 dimensions of E8 break down into
128 Spin(16) half-spinor fermionic dimensions
and
120 dimensions of the Spin(16) adjoint bosonic dimensions,
and
the other 8 dimensions (Batakis Kaluza-Klein spacetime) of Cl(1,7) could correspond to the 8-dimensional root vector space of the Spin(16) Lie algebra, which is the same root vector space as that of E8.
In short,
256 Cl(1,7) generators = 248 E8 generators + 8 root vector space dimensions,
and the 8 root vector space dimensions correspond to 4+4 dimensional K-K spacetime.
Hi Tony,
ReplyDeleteThanks for picking up that line of thought! Indeed, I had suggested something similar to Garrett (and at least two other people), but nobody seemed to be really excited about it. If possible I would even be more extreme and just take E8 (as a group) and deform it/break the symmetry, identify part of it as the 4-d spacetime with the appropriate remaining local Lorentz + gauge symmetries, instead of tying it to an additional space. Admittedly I don't know if that makes any sense. I would find it interesting, because such a large number of extra dimensions could allow one to address the hierarchy problem that lurks somewhere in the background, and so far nothing in that respect can be said whatsoever. Either way, I am just babbling around, and I myself am not excited enough about the model to work myself into all the nasty details of E8. One reason for me writing this post though is my hope to inspire people to look into the matter, cause I am curious to see what comes out of it. Best,
B.
Tony (and bee):
ReplyDeleteI'm very fond of Kaluza-Klein theory, but I don't think it's a necessary part of this E8 theory. I could change my opinion on this in the future, but right now I don't have a reason to.
Also, I'm pretty sure the E8 Lie algebra is not a subalgebra of Cl(8). However, I'd not be surprised if E8 were a subalgebra of Cl(16).
The idea of starting with just a non-compact E8 manifold, and letting part of it go wavy and be our spacetime base manifold, is very pretty. I believe this construction is called a Cartan geometry. The idea is worth thinking about, and I have a little, but I don't think it will work because I don't think there's a large enough subgroup of E8 to pull it off -- though it may work in some way I haven't considered. Thinking about constructing a Cartan geometry with E8 as a subgroup of a larger group is also interesting, but as nice as it would be if true, I don't think Cl(8) has E8 as a subalgebra.
Garrett, you say that you are "... pretty sure the E8 Lie algebra is not a subalgebra of Cl(8). However, [you would] not be surprised if E8 were a subalgebra of Cl(16). ...".
ReplyDeleteA set of 128 Cl(16) half-spinors accounts for 128 of the 248 generators of E8,
and
the Cl(16) bivectors form (but with the Spin(16) Lie bracket product instead of the Clifford product) the other 120 of the 248 generators of E8,
so
you can say (if it is OK with you to use two types of products, the Lie bracket and the Clifford) that E8 is contained in the structure of Cl(16),
and
then, due to 8-periodicity of real Clifford algebras,
Cl(16) = Cl(8) (x) Cl(8)
where (x) denotes tensor product,
so
you should be able to write all the E8 generators in terms of Cl(8).
Tony Smith
PS - Of course, if you are willing to be free with using various product rules, you might also be able to fit E8 inside Cl(8) itself.
I'm still not convinced you can bypass Coleman Mandula that way.
ReplyDeleteAppealing to DS might evade it near the gravity regime, but from the point of view of the effective theory near standard model scales, you will still have highly suppressed but still apparent unitarity violating terms.
If it was that easy, we could always bypass it, for any theory (since presumably we live in a world with nonvanishing cc)
Tony:
ReplyDeleteI'm not willing to alter the products, because then we'd not be using the algebra we're claiming to use. By your parenthetical remarks I see you acknowledge this opinion of mine -- and I appreciate that, and thank you for playing by this rule.
So, is it possible to embed E8 in Cl(16), associating twice the antisymmetric Clifford product with the Lie bracket? The so(16) = Cl^2(16) part works just fine, but I'm not sure if the 128_S+ spinor works.
If it does, then I agree we could associate E8 elements with elements of two copies of Cl(8) as you describe (but not with one, unless (as you suggest) we mess with the products). Tangentially: I do think so(16) is a subalgebra of Cl(8).
Hmm, I suspect Sabine's going to pull the plug if we inundate her blog with an algebra discussion -- unless she's interested, maybe we should continue via email.
anonymous:
ReplyDeletePlease don't take my word for it. The first person I know of to point out this loophole in Coleman-Mandula was Thomas Love in his 1987 dissertation. There is also a discussion of this loophole in this recent paper by F. Nesti and R. Percacci: Graviweak Unification. Or you can go to the source and look at Coleman and Mandula's paper, in which their condition (1) for the theorem is "G contains a subgroup locally isomorphic to the Poincare group." The G = E8 I am using does not contain a subgroup locally isomorphic to the Poincare group, it contains the subgroup SO(4,1) -- the symmetry group of deSitter spacetime.
Dear Bee, Garrett Lisi's paper has been removed from hep-th as the main archive and reclassified as gen-ph, thanks God.
ReplyDeleteI am amazed how uncritical and stupid things you're ready to write about this manifestly crackpot paper. Couldn't you sometimes approach science not by asking whether something is attractive, unattractive, cool, etc., but whether it is right or wrong? I have never seen you thinking in this way.
Lumo expelled: "Dear Bee, Garrett Lisi's paper has been removed from hep-th as the main archive and reclassified as gen-ph, thanks God."
ReplyDeletePerhaps You, Lumo, had something to do with this? Judging by Your absolute certainty about everything, may we assume that Your "thanks God" referred to YourSelf? Perhaps Bee is too humble to regard herself as a living God?
It seems that a lot of papers are going to be re-classified in this way, including such classics of crackpottery as
ReplyDeletehttp://arxiv.org/abs/hep-th/0007206
Dear Lumo, It appears someone has been removed from Harvard and reclassified as a crackpot.
ReplyDeleteIt is now absolutely clear that Lumo is an obstacle to science. An arrogant psychopath that cannot see outside the box but still powerful enough to shut-down voices that don't fit his rigid line of reasoning.
ReplyDeleteDear Lubos,
ReplyDeleteI am amazed how uncritical and stupid things you're ready to write about this manifestly crackpot paper. Couldn't you sometimes approach science not by asking whether something is attractive, unattractive, cool, etc., but whether it is right or wrong? I have never seen you thinking in this way.
That might be related to the fact that you've never seen me to begin with. Either way, thanks very much for your valuable comment with high scientific content. I think I've explained my criticism on Garrett's work sufficiently in the post and the comments above. I don't know in how far it is relevant to the content of the paper whether or not it's classified as high energy physics.
Everybody else:
As far as this blog is concerned, I am God and I enjoy a big deal hovering with the cursor over the 'delete this comment' button. So let me remind you once again that I don't tolerate anonymously made insults.
Best,
B.
As far as this blog is concerned, I am God
ReplyDeleteI note that Bee has elevated herself from "Her Majesty" now to "God".
Also that this points out a direction to a new theology where God is local, not universal.
It seems that a lot of papers are going to be re-classified in this way, including such classics of crackpottery as
ReplyDeletehttp://arxiv.org/abs/hep-th/0007206
I personally can't believe that there is actually a paper with Lumo's name on it that claims that "the vacuum selection problem of string theory might plausibly have an anthropic, cosmological solution".
How did I miss that?... I thought that I'd read this paper! After all these years of purely dogmatic and unjustified contempt for the observed reality... Lumo has gone anthropic!?!
anthropic, cosmological solution
Let's get one thing straight though... anthropic selection isn't a "cosmological solution", rather, it's a cop-out on one... losers.
*waiting for "god's" hammer to strike me dead*... ;)
Garrett
ReplyDeleteYour E8 Lie algebra decomposition on page 18 seems to contain a non-compact form of E6, namely E6(-14), with 5-grading g_0=so(1,7)+R+iR, dim_R(g(-1))=16 and dim_R(g(-2))=8.
E6(-14) arises in the coset space associated with non-BPS, Z=0 orbits in symmetric N=2 d=4 supergravity. See table 3 of Ferrara and Gunaydin's hep-th/0606209. In such supergravity models the exceptional Jordan algebra describes the charge space of an extremal black hole.
Who knows, maybe your model has a supergravity dual. It's worth a look.
With respect to Coleman-Mandula (particularly with respect to fermions) it is useful to consider
ReplyDeletewhat Bee said
"... 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 ..."
and
what Steven Weinberg said at pages 382-384 of his book The Quantum Theory of Fields, Vol. III (Cambridge 2000):
"... The proof of the Coleman-Mandula theorem ... makes it clear that the list of possible bosonic symmetry generators is essentially the same in d greater than 2 spacetime dimensions as in four spacetime dimensions:
... there are only the momentum d-vector Pu, a Lorentz generator Juv = -Jvu ( with u and v here running over the values 1, 2, ... , d-1, 0 ), and various Lorentz scalar 'charges' ...
the fermionic symmetry generators furnish a representation of the homogeneous Lorentz group ... or, strictly speaking, of its covering group Spin(d-1,1). ...
The anticommutators of the fermionic symmetry generators with each other are bosonic symmetry generators, and therefore must be a linear combination of the Pu, Juv, and various conserved scalars. ...
the general fermionic symmetry generator must transform according to the fundamental spinor representations of the Lorentz group ... and not in higher spinor representations, such as those obtained by adding vector indices to a spinor. ...".
In short, since E8 is the sum of the adjoint representation and a half-spinor representation of Spin(16),
if Garrett builds his model with respect to Lorentz, spinor, etc representations based on Spin(16 consistently with Weinberg's work,
then
a beautiful aspect of Garrett's model is use of the fermionic and bosonic aspects of E8 so that Coleman-Mandula is satisfied.
Tony Smith
Dear Arun:
ReplyDeletethis points out a direction to a new theology where God is local, not universal.
With the appropriate connection, it should be possible to parallel transport local godness. Global constructs are often inappropriate to our local problems, thus my dissatisfaction with static, eternal and universal laws. Books that were written some thousand years ago shouldn't been taken too literally when applied to our present situtation.
I don't want to offend anybody who believes in one or the other God, so let me point out that the above remark of mine was meant as a joke.
Hi Island,
Since the more appropriate title would actually be Godess, I have to admit that hammer sounds too male to me and isn't quite my tool of choice. Either way, however interesting Lubos' early anthropic excursions might be, I'd appreciate if we could stick to the topic here. Thanks,
B.
Almida, if you forget all the crude comments that Lubos Motl speaks in, his physics is very robust.
ReplyDeleteThis is not a pro string theory ranting. The paper by Lisi is rubbish. I like Bee, but I am extremely surprised that this guy was invited to perimeter to give a talk.
Anonymous said: The paper by Lisi is rubbish. I like Bee, but I am extremely surprised that this guy was invited to perimeter to give a talk.
ReplyDeleteAnonymous, your declaration of Garrett's paper as 'rubbish' without giving any arguments whatsoever is not particularly convincing. I have explained above in detail what my opinion is on various aspects of his model. I did not invite Garrett for the purpose of giving a talk, but I offered that he could give a seminar during the duration of his stay. Also, I did not invite him because of his recent work. In fact, as he might recall, when we meet in Morelia I was pretty much convinced his theory is doesn't make sense because of an issue that he however was able to resolve later. One way or the other, I would never judge on a person based on a 20 minutes presentation, or a model they have worked on for a while.
I also want to point out that as a postdoc at PI I myself am responsible for the use of my visitor grant.
Best,
B.
This paper firmly establishes loop quantum gravity as a true competitor to string theory. All known particles and interactions have been incorporated into the theory, just like string theory, but without arbitrary constants.
ReplyDeleteAlso, like string theory it may take 15-20 yrs to understand and develop the theory, depending on the results from the LHC.
I wonder if we can "test" this theory against string theory? We won't compare fundamental assumptions but numerical results. If both E8 and Strings yield similar results then E8 is just as good as strings, given present experimental evidence.
Hey Anonymouses: It's getting really confusing in this comment section. Could you be so kind to please at least enumerate yourselves? Better, show creativity and create a random pseudonym, e.g. what about charm quack or D-brain.
ReplyDelete@ the last Anonymous: could you please clarify your statement "This paper firmly establishes loop quantum gravity as a true competitor to string theory"? So far I don't see how Garrett addresses the problem of quantizing gravity at all, and I also I don't see how the Lie-algebra and its decomposition that he provides is specifically bound to LQG. Best,
B.
I must admit that I'm really confused at this point. LM would have us believe that GL's paper is "obviously" a crackpot effort. Yet reading his critique, his first comment about "bursting into laughter" shows that he doesn't know elementary algebra. On the other hand, it does seem hard to believe that Coleman-Mandula can be so easily circumvented. So can anyone tell me: what is the mainstream view of this paper? I don't mean mainstream as in "haw haw, he's not from Harvard, must be a crackpot", I mean, is there something obviously wrong with this paper, obvious I mean to people who understand simple algebra and particle physics? Something that might justify the very drastic re-classification, which to my eyes borders on the scandalous?
ReplyDeleteOther:
ReplyDeleteThis theory is currently running the gauntlet of public scrutiny. There are a lot of very good people looking at it. After two days in public, I think it's safe to say it's good enough to be wrong -- and an open mind girded with healthy skepticism is appropriate. Clearly, a string theorist who moderates hep-th doesn't like the paper -- I don't find that surprising, or particularly condemning. The idea introduced in this paper may be wrong, and turn out not to be a true theory of nature, but it's not obviously wrong, and I think it's got a shot at developing into a fully successful theory, or I wouldn't work on it.
Bee:
An ony moose hasn't chimed in with a comment on why this paper establishes LQG as a competitor to strings, so I will. The primary LQG program is a search for a successful method for describing a background independent, quantum theory of a connection -- the connection of gravity. What this E8 theory does is include all fields in a large connection. So, combined, LQE8 theory or whatever one wants to call it would be a competitive ToE, vying with strings. Success would require both this E8 theory and LQG to work, and work together, which I am first to admit is a long shot. But the idea is young.
Bee commented a bit about "... why [ Garrett's ] paper establishes LQG as a competitor to strings ...".
ReplyDeleteHere is another point of view in favor of mutual support among LQG and Garrett's E8 structures:
Garrett says in his paper:
"... e8 = f4 + g2 + 26 x 7
... The 26 is the the traceless exceptional Jordan algebra ...
the central cluster of 72 roots in Figure 4 is the E6 root system, which acts on each of the three colored and anti-colored 27 element clusters of the exceptional Jordan algebra ...".
As John Baez said on sci.physics.research on 16 Jan 2001 ( with the 26-dimensional traceless subspace of the 27-dimensional exceptional Jordan algebra being denoted H3_0(O) ):
"... The quotient of Lie algebras e6/f4 is a vector space that can be naturally identified with H3_0(O). That's really cool!
But the quotient of Lie groups E6/F4 is what matters for the spin foam models, and this is a bit "curvier" - it has a natural metric that's not flat.
They are closely related, however: e6/f4 can be viewed as a tangent space of E6/F4. ...".
John then quoted my earlier post saying that I "... would like very much to be told how such a construction goes, because in my opinion such an H3_0(O) spin foam model should lead to, not just quantum gravity, but a Theory Of Everything. ..."
and
John replied to that saying:
"... Shhh! That's supposed to be secret. :-)
Yes, of course something like this is my goal, but I'm not eager to count my chickens before they are hatched, nor take my ideas to market while they're still half-baked. ...".
So, although unification of LQC spin foams with E8 containing copies of 26-dimensional H3_0(O) may be (as John indicated) not yet fully baked,
it is an idea that knowledgeable people such as John Baez have as a goal,
and
that indicates that Garrett's work is worth pursuing and has a reasonable chance of paying off by unifying it with LQG and so reaching the goal described by John Baez.
Tony Smith
"if you forget all the crude comments that Lubos Motl speaks in, his physics is very robust...."
ReplyDeleteLumo is a fanatic guy. He is blinded by his arrogance, and his reactions to people that don't fit his square are childish. All these are obviously reflected in his view and understanding of physics. I therefore would not reccomand studying physics (and essentially anything) from this guy.
Dear Bee, you wrote the following to anonymous:
ReplyDelete"Anonymous, your declaration of Garrett's paper as 'rubbish' without giving any arguments whatsoever is not particularly convincing."
You must know very well that this justification of your attitude is unjustifiable. I wrote a rather detailed analysis of this crackpot paper and you don't care about it either.
For you, the discussion is simply not about checking things whether they work or not, it is not about arguments. You have an agenda and you told us quite clearly what the agenda is. Was it really you who invited the crackpot to PI? Wow. Amazing.
Your description of that paper is completely uncritical, it is just downright dumb. For example, you write "This is without doubt cool: He has a theory that contains gravity as well as the other interactions of the SM."
He doesn't have any unifying theory. His superambitious statements are based on roughly 10 major misunderstandings, to be sketched again below. If 1 out of these 10 misunderstandings appeared in a paper of a physicist who is viewed as a serious one, he or she would be deeply embarrassed.
But in certain other circles, 10 of them is not a problem.
First, it is impossible to unify bosons and fermions in one field, unless one has a fermionic symmetry (usually supersymmetry) that relates them. Lisi obviously doesn't have anything like that so his formulae adding bosons and fermions together are just like when you add temperature and time or apples and oranges - it's just elementary misunderstanding of elementary school science.
A related thing is that one can't unify fields with integer and half-integer spins, for analogous reasons.
Another level of this thing is that one simply can't interpret gravity as a bulk field, which means that one can't unify gravity with gauge forces in his childish way. Again, the relevant excitations transform differently with respect to the Lorentz group. Diffeomorphism group in 4D is not a special example of Yang-Mills theory of any kind. They're just not isomorphic.
There are many wrong technical statements, for example the E8 fundamental representation doesn't lead to 3 families under that decomposition.
Then there is a lot of more subtle errors, but all of the major ones show that the author simply doesn't know much about physics of spacetime. He might have learned roots of E8 but physics can't be reduced to roots of E8. Roots of E8 are just about one type of internal indices that fields can carry.
But fields in physics also carry Lorentz indices, statistics, and they interact according to formulae that are extremely important and cannot be ignored like in this paper.
Anyone can scream "I have found a theory of everything" but it doesn't yet mean that he is automatically a contributor to high-energy physics.
Also, I am displeased by the kind of hateful uninformed people who make up your readership. What is almida, for example? Why do you harbor this garbage here?
"gravity as a bulk field" should have been "gravity as a bulk gauge field"
ReplyDeleteThe lumo asks "what" is almida.
ReplyDeletealmida is the web name of a person with great sensitivity to malicious people, and who can identify psychopaths from afar. Besides, almida is educated in spacetime physics at least at the level of lumo's education. And finally, almida does not hate Lubos Motl. He feels pity for him.
"Almida", you know very well that what you write is completely absurd. You can't have any education about "spacetime physics" because you're a complete idiot who even doesn't know how this part of physics is referred to - certainly not "spacetime physics".
ReplyDeleteThanks for feeling a pity for me, and if you really want to help me ;-), please shut up and make all the remaining aggressive organized crackpots from your community - Sarfatti, Smith, Woit, Hossenfelder, Smolin, and numerous others - shut up because I find it really annoying. Thank you very much.
Hi Sabine,
ReplyDeleteI know you are a Goddess :) but if I may, moderation now would be a good time. Do not let an interesting discussion section sink into this. And please remove my comment afterwards.
Lumo:
ReplyDelete(1) "First, it is impossible to unify bosons and fermions in one field, unless one has a fermionic symmetry (usually supersymmetry) that relates them. Lisi obviously doesn't have anything like that so his formulae adding bosons and fermions together are just like when you add temperature and time or apples and oranges - it's just elementary misunderstanding of elementary school science."
BRST symmetry.
(2) "A related thing is that one can't unify fields with integer and half-integer spins, for analogous reasons."
This is part of the Lie algebra structure of E8.
(3) "Another level of this thing is that one simply can't interpret gravity as a bulk field, which means that one can't unify gravity with gauge forces in his childish way. Again, the relevant excitations transform differently with respect to the Lorentz group. Diffeomorphism group in 4D is not a special example of Yang-Mills theory of any kind. They're just not isomorphic."
You misunderstand the geometry of principal bundles. The diffeomorphims group is not a subgroup of E8 in this theory -- the spin connection and frame are included as parts of the E8 connection. The gauge theory of gravity, as described by the MacDowell-Mansouri theory, is unusual, but it works. Gravity was introduced as a bulk field in four dimensions. You may think Einstein was wrong in this, as do some others, but I do not.
(4) "There are many wrong technical statements, for example the E8 fundamental representation doesn't lead to 3 families under that decomposition."
I'm working in the adjoint of E8. For a mathematical review of the how the 3 family structure works in this theory, through triality, this is a good reference: .
The rest of your comment is filled with unjustified opinions, and you're welcome to those -- I won't talk you out of them. But your 10 reasons (which you apparently count in base four) are very simple misunderstandings on your part that are addressed above.
There are real concerns and unknowns that cast this theory in doubt, such as the symmetry being broken by hand, and it may turn out this theory doesn't work, but not for any of the reasons you've given so far.
Hi Garrett, Hi Tony,
ReplyDeleteRegarding my question about LQG, I was trying to disentangle fact from wishful thinking. I was just pointing out that so far there is no reason to get over excited.
Hi Christine,
You have a point there, but I want to see whether Garrett wants to reply. Since he's somewhat more West, I'd want to wait until he's up. Best,
B.
Ah sorry, our comments crossed...Good morning Garrett... you're up early :-)
ReplyDeleteBee:
ReplyDelete"Regarding my question about LQG, I was trying to disentangle fact from wishful thinking. I was just pointing out that so far there is no reason to get over excited."
I agree.
Lubos: I wrote a rather detailed analysis of this crackpot paper and you don't care about it either.
ReplyDeleteI didn't read it because thinking about what you write has repeatedly proven to be an enormous waste of time. In addition to this, if I try to open the website I always get forwarded to a random junk site. As usual, you evidently didn't read more of what I wrote than that one sentence which you dislike. As long as you're asking for quantum numbers in a representation, you can have 1/2, 1 and 2 together in there, for much the same reason as you can have various q in the multiplet. The challenge is to show that this specific quantum number has something to do with space-time symmetries, and if you'd care to read what I've written above in the post and comments, you'd have noticed that this is exactly the point that I dislike. Also, I'd think that in this case there should be operators relating fermions with bosons, pretty much as in supersymmetry, and I don't know why there's no 3/2. Questions I've asked Garrett before, so here they come again - Garrett, can you enlighten me?
You write: "Diffeomorphism group in 4D is not a special example of Yang-Mills theory of any kind.", I wrote "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.".
I don't give a shit whether you are 'displeased' by my readership. In those cases where I've followed comments on your blog, the fraction of 'hateful uninformed people' was pretty close to 1.
make all the remaining aggressive organized crackpots from your community - Sarfatti, Smith, Woit, Hossenfelder, Smolin, and numerous others - shut up because I find it really annoying.
At least I've progressed to being an organized crackpot. Yeah, I guess it can be kind of annoying, if people have an opinion that differs from yours.
Best,
B.
Hi Other:
ReplyDeleteSomething that might justify the very drastic re-classification, which to my eyes borders on the scandalous?
Notwithstanding the scientific question behind specifically Garrett's paper, it's a rather common thing to happen. There's a nice quotation that is attributed to Schopenhauer (but not sure whether he really said that)
Alle Wahrheit durchläuft drei Stufen. Zuerst wird sie lächerlich gemacht oder verzerrt. Dann wird sie bekämpft. Und schließlich wird sie als selbstverständlich angenommen.
(Every truth goes through three stages. First it is called ridiculous. Then it is attacked. Finally it is accepted as self-evident.)
Though this is probably the better known version
Four stages of acceptance:
i) this is worthless nonsense;
ii) this is an interesting, but perverse, point of view;
iii) this is true, but quite unimportant;
iv) I always said so.
(J.B.S. Haldane, Journal of Genetics #58, 1963,p.464)
That is to say people defend their believes and even if they run out of arguments, they still stick to them (progress is not made from conference to conference, but from funeral to funeral - forgot who said that).
Most often however, the conservatists are right.
Best,
B.
"Also, I'd think that in this case there should be operators relating fermions with bosons, pretty much as in supersymmetry"
ReplyDeleteA BRST symmetry (which is a kind of supersymmetry) relates the fermions in this theory with bosons. This is not a supersymmetry between the fermions and the physical gauge fields, but between the fermions and the bosons they have replaced as ghosts. I can provide references if you like -- there are two good references for understanding this stuff listed in the paper. I didn't include a section on this because I'd written about it before and it was cleaner to just introduce it as a kind of superconnection.
Bee:
ReplyDeleteThanks for finding the history of " the four stages of acceptance." I suspected this theory was in the middle of them, but I didn't remember the exact reference to what the stages were called.
I've been looking forward to comments appearing like "You can see this same idea in my paper of 198..." ;)
This comment has been removed by the author.
ReplyDeleteHi Garrett:
ReplyDeleteThanks for the clarification. Well, gauge symmetry by itself does also relate fermions with bosons. I still don't understand though whether or not in the root diagram you have operators raising and lowering spins, and your answer with the ghosts does not really address this. If not, why not? I mean, is there a chance you could lay out the difference to susy in a way that fits into this comment section? Sorry if that's a stupid question.
Best,
B.
Bee:
ReplyDelete"whether or not in the root diagram you have operators raising and lowering spins"
The left and right chiral parts of the gravitational spin connection raise and lower the spin quantum numbers of the fermions in interactions.
Or maybe this wasn't what you were asking.
The roots correspond to QFT particle eigenstates, and we can associate quantum raising and lowering operators with each of these, as in canonical QFT.
"is there a chance you could lay out the difference to susy in a way that fits into this comment section?"
I could give a brief review of the BRST technique here, but it would be kind of a pain, so you'd have to ask me twice. I'd just transcribe the description of the BRST technique on my wiki.
Garrett,
ReplyDeleteMaybe it's also worth to take a look at the papers of the "obscure" Mohamed El Naschie. One of his last papers is called "High energy physics and the standard model from the exceptional Lie groups". They are almost all published in the journal "Chaos, Solitons & Fractals".
J.C.
Hi Garrett:
ReplyDeleteThe left and right chiral parts of the gravitational spin connection raise and lower the spin quantum numbers of the fermions in interactions.
[...]
The roots correspond to QFT particle eigenstates, and we can associate quantum raising and lowering operators with each of these, as in canonical QFT.
? I am not sure if that was the question I asked. My question is: you have fermions and bosons in your diagram, are there operators acting on the one resulting in the other. Your second answer would have been what I'd expect. Then the question is, why isn't there a spin 3/2 particle?
Thanks for the link. My question wasn't so much referring to the mathematical basis, but on a more practical level, what is the difference to SUSY phenomenolgy in terms of allowed interactions etc. Can you say anything about that without knowing how to break the symmetry?
Best,
B.
The fact that lumo fails to understand that his criticism of Garrett's work is vacuous, obviously points on his low qualification as a physicist. His automatic rejection of any original idea out of the context of ST, surly implies that he is fossil-minded. We all must face it: blinded by arrogance and narcissism, lumo will never admit his limitations.
ReplyDeleteGarrett,
I do not understands all aspects of your non-conformist work but it looks to me compelling and original. My suggestion to you, don't tie it to LQG; it certainly stands on its own. And, as you already said, time will tell its relevance (or irrelevance) to the real world.
Justin:
ReplyDeleteSweet, stage (iv). ;)
Do you have a link to one of his papers available online, or if not, could you email one to me? I don't have a repository of dead trees near me, only live ones.
I am very interested in related work. I am far from the first person to suggest E8 for a ToE, I just hope I've carried the idea a little further.
Garrett,
ReplyDeleteUnfortunately I don't have access anymore to the journal via www.sciencedirect.com. If you go to that website and type in "naschie" in the "Author field", you'll see all papers he has published so far...
J.C.
Bee:
ReplyDelete"you have fermions and bosons in your diagram, are there operators acting on the one resulting in the other."
Ah. No.
"Then the question is, why isn't there a spin 3/2 particle?"
Because it's not in the E8 Lie algebra.
"My question wasn't so much referring to the mathematical basis, but on a more practical level, what is the difference to SUSY phenomenolgy in terms of allowed interactions etc. Can you say anything about that without knowing how to break the symmetry?"
I can't say much about that, no. As you've pointed out, I only have a guess (backed by a consistent calculation) that these fermions could be the BRST ghosts of former bosons, and I have to impose the symmetry breaking by hand to get this to work.
I can say that the current particles in the theory do not have superpartners. So if sparticles show up at the LHC, that's a setback for this theory, and some support for strings (although string theory has backed away from predicting sparticles -- it's odd that they get to have it both ways). I could easily add another copy of E8 to get superpartners, but I don't wish to complicate things beyond necessity.
*sigh* I guess I just don't understand it. You've all the sm particles in a single algebra that's spanned by some few elements, how can you not have an operator relating fermions to bosons? didn't you explain me how one constructs these things adding vectors to each other etc? I guess I can live with the 3/2 not being in the algebra, since you don't really have a graviton either, but the frame field?
ReplyDeleteBest,
B.
Bee:
ReplyDeleteThe action has symmetry breaking terms in it such that this Lagrangian is not invariant under a gauge transformation taking the fermionic algebraic elements to bosonic ones.
Yes, you can add bosons to fermions in these diagrams to see what fermions come from that interaction.
The action has symmetry breaking terms in it such that this Lagrangian is not invariant under a gauge transformation taking the fermionic algebraic elements to bosonic ones.
ReplyDeleteOkay, thanks. So, you've made the action such that this doesn't occur. But lets go back to the root diagram without the dynamics, prior to symmetry breaking. You say
Yes, you can add bosons to fermions in these diagrams to see what fermions come from that interaction.
Can you get bosons from the fermions? In either case, how does the 'addition' look like? Can you give me an example for a boson 'added' to a fermion? How do the quantum numbers change? Do you have some kind of multiplets in this case? And again, what is the difference to supersymmetry? If you don't have supersymmetric partners, this kind of operation must stay within the SM particles, and it's not clear to me how so. Best,
B.
"Can you get bosons from the fermions?"
ReplyDeleteYou can do this in the algebra, and thus in the diagrams, but that doesn't happen in the action -- the Lie brackets between two fermions doesn't appear in the BRST extended curvature. However, there are terms in the action for fermions and anti-fermions to interact and give bosons.
"In either case, how does the 'addition' look like?"
It's just vector addition in eight Euclidean dimensions, but it works in two because it's a linear projection. You know, you draw an arrow from the origin to a particle, then move the base of that arrow over to another particle, and see where the arrow head ends up. Yes, this works in all those diagrams, and gives particle interactions -- it's kind of embarrassing, but grade school kids are going to be able to do this.
"Can you give me an example for a boson 'added' to a fermion?"
Try adding a gluon (blue circle) to a quark, or adding a W+ (yellow circle) to a left chiral fermion (the yellowish ones). Only some will work of course -- the ones corresponding to standard model interactions.
"How do the quantum numbers change?"
Addition.
"Do you have some kind of multiplets in this case?"
Yes, the ones we're familiar with.
"And again, what is the difference to supersymmetry? If you don't have supersymmetric partners, this kind of operation must stay within the SM particles, and it's not clear to me how so."
Yes, the interactions stay within the standard model -- the E8 Lie algebra is closed. Some of the additions don't land on a root, so there's no interaction there. All the additions of bosons to fermions correspond to the interactions in the standard model and gravity. This is a standard graphical description in representation theory -- so it's no surprise things work this way, but it's still very cool.
Garrett: Only some [interactions] will work of course -- the ones corresponding to standard model interactions. [...]
ReplyDeleteBee: "Do you have some kind of multiplets in this case?"
Garrett: Yes, the ones we're familiar with.
[...] the interactions stay within the standard model -- the E8 Lie algebra is closed.
Provided that you identify the quantum numbers with the SM + gravity. How can it possibly be you can have the SM interactions without breaking the symmetry by hand. The SM + gravity is a product of specific groups, E8 isn't. Usually the bundle is trivial over the manifold, and doesn't affect TM. 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?
Best,
B.
Bee:
ReplyDeleteThey all mix, and we fix that with the symmetry breaking.
..as Baez pointed out the e8-triality decomposition and the fact to identify the symmetry lie-algebras of the SM as sub-algebras of e8 are not new, so the idea to define a vector bundle associated to a 'physical' representation of E8 and single out 'physical' connections is maybe not quite so 'sophisticated' as it might seem at first sight. However I still wonder, and this argument was formulated in several versions by several commentators, if the constructed associated bundle carries a subbundle which is isomorphic to the bundle which is associated to the bundle of orthonormal frames by the standard representation of O(3,1) and furthermore, how do the other structures, i.e. the involved connections, reduce to that bundle. A similar question possibly applies to all the other physically relevant subgroups involved. Still I am not sure if this is actually what is 'physically' needed, since one could imagine that for instance the requirement for a reduction of the 'total' connection to eventual subbundles, namely that to a given ('physically meaningful') subgroup H (for instance O(3,1)) and its Lie-Alebra h there is a decompositon e8= h + g so that Ad(h)g is contained in g, could not be satisfied in most cases, even if there would be reductions of the total bundles to certain subroups. This could be even not physically expected, however it should be the case in some sort of 'low energy' limit, given the validity of GR and the SM. So in short, the variation of the 'total action' should lead to a total connection so that at least in the low-energy-limit one has reductions of the total bundle to appropriate subbundles with 'simultaeneous' reducing connections, AND in addition, these reductions should reproduce the known physical laws, i.e. those of GR and the standard model, that is, should also reproduce a Levi-civita connection of a certain metric on the base, at least in some classical limit. I am not quite sure if Lisi Garretts arguments do imply or even intend this, I have to admit.
ReplyDeletea.k.:
ReplyDeleteYes, a reduction of the structure group of an E8 principal bundle is an accurate description of what's going on in this paper. I have not worked out the global, topological aspects of this reduction, but am only working locally, with the Lie algebras and representations. I do expect questions regarding the topological admissibility of this reduction to be fascinating, but... baby steps...
Dear Garrett: Thanks for the clarifications. I know I've asked all that before, and you've explained all that before. Since it's the longer story behind the opinion I've put forward in the above post, I was just hoping it might be a useful dialogue. On a deeper level, my dissatisfaction with your model is simply that it doesn't explain why gravity is special, and that's something I would have hoped for from a TOE. Best, B.
ReplyDeleteBee:
ReplyDeleteI share this dissatisfaction; but this theory is not yet fully developed, and the story of how gravity is special in this theory will improve.
Sabine said: "There's a nice quotation that is attributed to Schopenhauer (but not sure whether he really said that)"
ReplyDeleteSurely a more apt quote would be "Mit der Dummheit kaempfen Goetter selbst vergebens"?
I still think it's scandalous that a paper can be re-classified like this. I'm betting that GL received no word of explanation. And if indeed this was done at the instigation of LM, well, it's abundantly clear that it was the result of malice and, frankly, scientific incompetence on his part. The arxiv really *is* in need of reform....
A much more apt quote, especially true in today's times of the internet and blogs, is from Einstein himself:
ReplyDelete"Die Herrschaft der Dummen ist unüberwindlich, weil es so viele sind und ihre Stimmen genauso zählen wie unsere"
Fortunately the Arxiv is not ruled by majority vote of the general public, but rather by educated scientists who know better.
Dear Bee, Dear Garrett,
ReplyDeletethis was helpful indeed, thank you for this great post. may I hope either of you could elaborate on
"You know, you draw an arrow from the origin to a particle, then move the base of that arrow over to another particle, and see where the arrow head ends up. Yes, this works in all those diagrams, and gives particle interactions -- it's kind of embarrassing, but grade school kids are going to be able to do this."
How does this work? I can not resist to hope that despite my lacking knowledge of intricate detail I might be able to understand a TOE!
Hi,
ReplyDelete"First they ignore you, then they laugh at you, then they attack you, then you win." is commonly attributed to Gandhi.
Fortunately the Arxiv is not ruled by majority vote of the general public, but rather by educated scientists who know better.
To misquote Monty Python - "No-one expects the Arxiv Inquisition"
Bee, if this comment is too technical, please feel free to delete it,
ReplyDeletebut it is an effort to answer your concern about "why gravity is special" in Garrett's model,
so I am submitting in case it might be useful in that regard.
Garrett's Table 9 lists physical interpretation of the 240 roots of E8,
but physical interpretation of the last three rows ( of the form x_i PHI ) with 3x6 = 18 elements\ seems to be "not perfectly clear".
Those 18 roots, projected into 3-dim, look like the 21-3 = 18 vertices of the 3-dim root polytope of Spin(3,4)
which is a 12-vertex cuboctahedron plus a 6-vertex octahedron.
To see what Spin(3,4) means physically,
consider a lower-dimensional analagous structure:
Spin(2,3) / Spin(1,3) = 4-dim anti-deSitter Space Translations (maybe better called rotations)
so
10-dim anti-deSitter Spin(2,3) is the basis of MacDowell-Mansouri Gravity =
= 6-dim Lorentz Spin(1,3) + 4-dim anti-DeSitter Translations
Applying that picture to Spin(2,3):
18+3 = 21-dim Spin(3,4) / Spin(2,4) = 6-dim Conformal Space Translations
so
21-dim Spin(3,4) =
= 18-dim Conformal Spin(2,4) + 6-dim Conformal Space Translations
Since Spin(2,4) = SU(2,2) = Conformal Group,
it can also be used as a basis of MacDowell-Mansouri-type Gravity,
as Mohapatra did in section 14.6 of his book Unification and Supersymmetry).
Therefore,
just as Spin(2,3) gives Gravity in 4-dim Minkowski spacetime,
you can
use Spin(3,4) to get Gravity in 6-dim Conformal spacetime,
and
that 6-dim signature (2,4) Conformal spacetime has (through twistors) a non-linear interpretation as 4-dim signature (1,3) spacetime.
In short,
Garrett's 240-222 = 18 left-over E8 generators may fundamentally describe Gravity
(plus some degrees of freedom probably related to Higgs, which may show how and why Higgs mass is connected to Gravity mass).
If that scenario works, it may give an answer Bee's "... dissatisfaction ... that it doesn't explain why gravity is special ...",
assuming that the inclusion sequence
Spin(7) in Spin(8) in F4 in E6 in E7 in E8
seems to be a natural sequence showing that Spin(3,4) inside E8 is "special".
Further,
the signature (3,4) may (to quote Garrett's paper) "... shed light on how and why nature has chosen a non-compact form, E IX, of E8. ...".
Tony Smith
Hi Tony,
ReplyDeleteThanks for your explanation. Though the details elude me I think I know roughly what you are saying. However, you don't even address the reason for my dissatisfaction. The 'leftover generators' might be used to describe gravity in whatever type, but there is no reason why they have to. Finding a scenario that works, as you put it, isn't going to answer the question. It's an artificial construct, and nothing in your explanation even gives a hint as to why gravity sits in this part, or how come it is so much weaker than the other interactions. I mean, hey, lets just call some other projection gravity and identify it with the tangential part of some other manifold.
Also, as we've noticed above, all of this only works with the appropriate symmetry breaking. Of course, you'd expect from a TOE that it literally mixes everything with everything, but one has to get rid of this, and esp. identify the Lorentz-Sector with the Tangential space. You say "Those 18 roots, projected into 3-dim, look like...". Yes, but E8 isn't a direct product of the projections much like a sphere isn't a product of two circles. That mixing one has to avoid, and the way it's presently done is an ad hoc choice of the action. See the exchange that precedes Garrett's above comment "They all mix, and we fix that with the symmetry breaking." That's just to clarify my above statement.
Best,
B.
Tony:
ReplyDeleteI have the same sorts of notions dancing around regarding these new frame-Higgs fields, and plan to spend some time going through the various possibilities. It's a good mystery. I'm slow and careful at figuring these things out, so this will be how I spend the next winter months. Thanks for sharing your ideas on it, they'll enter into the mix.
Another ony moose:
Choose any of the diagrams in the paper, draw a vector from the center of the diagram (the origin) to one of the particles, then draw another vector from the origin to a different particle. Add these two vectors (as described in the comment you quoted), and see if the result lands on a third particle -- if it does, that's the result of the interaction.
Today's show is brought to you by representation theory, and the number 8.
Hi Bee, Garrett, (and all the Other, Anonymous, friends)
ReplyDeleteI just saw today this nice discussion and, since I was cited, I want just to clarify.
In 'non-exceptional' graviweak unifications like the ones of my last two papers, one extends the internal gauge symmetry, and the extended vierbein acquires a VEV that provides the emergence of lorentz symmetry and gravity.
In other words, the extended vierbein gets a VEV breaking some generators and its goldstones give masses to the some gauge fields.
This happens to the unwanted generators, and also to the gravitational spin connection, because the background breaks also local lorentz (this is in Palatini spirit).
This is the reason why gravity is so weak: the graviton (fluctuation of the vierbein) gets kinetic terms only after the spin connection is decoupled. Its kinetic term should be normalized with M_Planck, while the other gauge fields have a dimensionless coupling.
Now, in Garrett's cool work, the frame field is inside the connection itself, therefore its VEV should be something like a Wilson line. However, in this line of reasoning, something that puzzles me is that the frame field goes together with the Higgs (e\phi) so I wonder how they can get VEV at different scales.
Other puzzles include (as I wrote to Garrett) how triality can emerge from the action, or in other words how the 8_v generation can couple correctly with the higgs and gauge fields... I suspect that one should include it by hand, as I suspect that should be done with the SU(4) breaking (that probably should happen at a different energy scale.. )
cheers,
Fabrizio
Bee, my apologies for not "... address[ing] the reason for [your] dissatisfaction ... It's an artificial construct, and nothing ... even gives a hint as to why gravity sits in this part ... all of this only works with the appropriate symmetry breaking ... mixing one has to avoid, and the way it's presently done is an ad hoc choice of the action ...".
ReplyDeleteWhat sort of construct would you consider to be natural ?
What about something like noting that E8 is an octonionic thing with an 8-dim root vector space,
and
introducing (based on our physical spacetime being 4-dimensional and/or the Standard Model SU(3)xSU(2)xU(1) being rank 4) a preferred quaternionic 4-dim subspace
and
seeing how the root vectors appear to be organized with respect to the preferred quaternionic subspace
and
then saying that the fully-mixed-up E8 octonionic stuff only lives at high energies (maybe Planck)
and
that the reorganized quaternionic stuff is frozen out at low temperatures to form what we see in our low-energy experiments ?
Of course, the details of how octonionic to quaternionic freezing out affects the E8 root vectors is not worked out here,
but
if it were to work out consistently with experimental results,
would you consider it to be a natural construct ?
Tony Smith
Hello Fabrizio,
ReplyDeleteYour description is accurate. With the frame-Higgs, they share the Higgs vev which results in a very large cosmological constant at high energy. The strength of gravity I also expect to be very large at this unification scale. At first this prediction of the theory seemed terrible to me, but now I'm thinking these large values will run down to the tiny values we experience at low energies. This thinking is based on the recent work of Reuter and your colleague, Percacci, on asymptotic safety. Please tell me if you think this line of wishful thinking is hopeless.
Hi Tony:
ReplyDeleteas I said above, I don't know very much about the octonions etc, I'm not much of an E8 specialist. What you write above sounds indeed much more attractive. Reg your question what I would find natural see the above comment. Ideally, I would of course want to come out that our space-time is 4 dimensional, I mean, it's supposed to be a theory of EVERYTHING :-) But I guess to start with I'd be fine with assuming it.
Yes, of course I'd want to have a dynamical mechanism for symmetry breaking with a symmetry that is restored at high (Planckian) energies. As I've said above, if you want to keep things as simple as possible, the obvios choice for the manifold is E8, no? What's funny about the gravity sector is the Lorentzian signature, so I'd try to break the symmetry by declaring three generators as boosts. Can one take the E8 bundle over the manifold, find a dynamical way to deform the manifold to lower symmetry such that 4 dim give standard space time w/ the 6 Lorentz generators providing the trafos in it?
However, whatever way I turn it, the problem with the squared fermion derivatives in the action doesn't vanish if one does so, so I don't see how one can avoid fixing the action by hand.
Best,
B.
Hi Fabrizio,
ReplyDeleteThanks for your comment. Yes, as I've also written in my post (the thing on top of this page), the CC problem is imo a problem that needs to be addressed. However, since it seems to be attached to Garrett's choice of the action, I see a chance that it might be possible to circumvent this problem with more investigations. The problem I see is not so much a potentially large 'natural' value of the CC or Higgs VEV, since we always have that, but how to generate the hierarchy between them. It's not clear to me whether the present construction allows this?
Best,
B.
That's Schiller.
ReplyDeleteHi Garret,
ReplyDeleteit may well be that the CC runs to smaller values a low energy, but
you are asking about a dynamical generation of a small CC in this particular E8 theory... This exceeds my capabilities, since my understanding of the Reuter-Percacci work is at best very partial. In particular I imagine that the running to low energy may depend on the couplings allowed, therefore being different for different theories. In particular I think they are trying now the running (toward UV) assuming terms R+R^2+R^3+... up to R^7... Ideally an effective theory description should include all of them, or have a mean to control their truncation. But somebody else should answer this point.
Bee:
I was not concerned about the largeness of the CC, that as you say may possibly be adjusted by changing the action.
I was concerned with the fact that you need a VEV <e phi>=M_PL while you need a VEV of <phi>=m_W. Here I see two problems: one is the hierarchy problem that you address, the other is a technical one, how can you give two VEVS to the same field (the 1-form component in the 4_lorentz x 4_weak sector)?
It seems that this field, call it E_mu^{m a} should first split in <e_mu^m> phi^a at energy M_PL, with phi being still dynamical, and only at lower scale, let <phi> to be nonzero. (dimensions can be arranged correctly I hope, resulting in a large dimensionless <e> and <phi>=m_H).
Also a question to Garrett (please pardon me if it was already asked, it does not seem to me): I am not used to Mansouri gravity, but I don't understand how you define the hodge dual using the connection only, usually you need a basis of one forms, a frame, though dynamical. Is this correct?
The point is important since the hodge gives rise to the gravitational coupling, etc... here you seem to have a volume-form that always depends on the higgs field, while one hopes to have different _dynamics_ for them.
Time to sleep for me, and to think more carefully about these issues.
(And maybe try to enter into Lubos' nightmares through the fabric of spacetime :)
cheers,
Fabrizio
Fabrizio expressed "... concern.. with the fact that you need a VEV ...[around]... M_PL while you need ...[another]... VEV ...[around]... m_W.
ReplyDeleteHere I see two problems: one is the hierarchy problem that you address, the other is a technical one,
how can you give two VEVS to the same field ...".
Since Garrett has a lot of stuff in his model, and in particular two scalar things phi and PHI (see his Table 9),
maybe when he works through in more detail over the winter it might appear that
one of them (say phi) might be the electroweak Higgs with VEV around M_W
and
the other of them (say PHI) might be related to some sort of generalized GUT and a VEV roughly around M_PL.
For example, Kolb and Turner say in he Early Universe (paperback edition Addison-Wesley 1994) Kolb and Turner say (at p. 526):
"... SU(5) GUT ...[has]... at the very least one complex 5-dimensional Higgs. The 5-dimensional Higgs contains
the usual doublet Higgs required for W-Boson SSB ...[which]... must acquire a mass of order of a few 100 GeV
and
a color triplet Higgs ... which can also mediate B,L [baryon,lepton] violation. The triplet component must acquire a mass comparable to ... M = 3 x 10^14 GeV ... to guarantee the proton's longevity, ...".
It may be, even if minimal SU(5) might be inconsistent with some currently accepted proton decay experimental analyses,
Garrett's model might be enough different from minimal SU(5) to get a longer proton lifetime that is consistent with those results,
while also being similar with respect to having two levels (with very different VEV) of Higgs.
Tony Smith
Fabrizio:
ReplyDeleteThe M-M formulation of gravity does not require the Hodge. It only uses the connection and an algebraic factor, gamma, to take an so(4,1) element to its dual, which is often called the "algebraic Hodge dual" by others. This is very nice, though a strange way of doing GR. However, the Hodge dual -- and pulling the frame, e, out of the connection -- is needed to construct the action (by hand) for the non-gravitational gauge fields. I think this is ugly, but is needed to match the standard model action in curved spacetime.
Sleep well.
Hi Fabrizio:
ReplyDeleteI was concerned with the fact that you need a VEV e phi=M_PL while you need a VEV of phi=m_W. Here I see two problems: one is the hierarchy problem that you address, the other is a technical one, how can you give two VEVS to the same field (the 1-form component in the 4_lorentz x 4_weak sector)?
I meant the same with the hierarchy problem, how do you get the two VEVs to be so different if they are so tied together. I'd just have said in Garrett's model it's two aspects: the one is getting the hierarchy scales in the first place, the other one is the technical one that you mention. Either way, there remains the general problem with the parameters in the standard model. Does Garrett's approach allow to compute them? Or at least to reduce their number? Sorry, but it's supposed to be a theory of EVERYTHING, so putting every parameter in by hand or through the symmetry breaking doesn't look like a big progress to me. I know I am being very not nice here, and I kind of feel like I should apologize, since it's not a very well understood model so far. So, it might be possible all these things work out nicely, I am just saying... (kicking...)
Best,
B.
Kea said: That's Schiller
ReplyDelete:-)
"Verflucht sei, wer sein Leben an das Große
Und Würdge wendet und bedachte Plane
Mit weisem Geist entwirft! Dem Narrenkönig
Gehört die Welt"
("Accursed, who striveth after noble ends,
And with deliberate wisdom forms his plans!
To the fool-king belongs the world.")
[Schiller, The Maid of Orleans]
Bee:
ReplyDeleteThere are many more parameters in the standard model than there are symmetry breaking parameters in this theory; so as things develop there are going to be predictions, right or wrong.
"Dem Narrenkönig Gehört die Welt"
ReplyDelete"könig"? But Czechia is a republic!
:-)
Anyway, just a very elementary observation, hope it helps. Bee is worried about the fact that the bundle of orthonormal frames is *not* characterized by its structure group O(3,1), so just getting O(3,1) out of a theory is necessary but not sufficient. In fact the bundle of frames has an extra structure, namely an R^4-valued one-form (a 1-form *on the bundle manifold). If GL can produce this thing then he will be on his way.....note by the way that I keep saying O(1,3) and not SO(1,3). Well, I would guess that in fact this version of E8 contains either Pin(1,3) or Spin(1,3) or one of the related groups....when you take all this to the group level from the algebras, you would need to settle which one it is.....finally I think you need a better explanation of Coleman-De Luccia. Nobody believes that the discovery of a positive cosmological constant immediately invalidated CM, do they? Finally, you would be performing a service to the community by protesting strongly but politely about the re-assignment of your paper. At least you ought to be able to force them to tell us (publicly) the alleged grounds for it. Good Luck (also abbreviates to GL....)
Oops, I meant Coleman-Mandula. Coleman did too many things....
ReplyDeleteother:
ReplyDeleteIf you're asking if there's a frame (aka tetrad or vierbein) in this theory, the answer is yes.
The Coleman-Mandula theorem: condition (1) is that there needs to be a Poincare' subgroup. There is no Poincare' subgroup in this E8 theory. By this I mean a stronger statement than you might think: there is no choice of parameters in this theory for which there is a Poincare' subgroup. Avoiding this condition, in this way, has been used by many others, from 1982 onwards, to get around this theorem. I have given these references in the comments above.
If you think you can prove the results of the Coleman-Mandula theorem while weakening condition (1), then by all means write that up and publish it, because it would be a stronger theorem. If you think I'm lying about there not being a Poincare' subgroup in this theory, then I encourage you to look for it. I'd be happy to help you with questions as you looked.
I mean no disrespect to Coleman -- we exchanged emails long ago and he helped me when I was a grad student. It's just that this theorem does not apply in this case.
other indicated that "... a positive cosmological constant ..." did not "... invalidate... CM [Coleman-Mandula] ..."
ReplyDeleteand
Garrett said that a "... condition ...[of]... the Coleman-Mandula theorem ... is that there needs to be a Poincare' subgroup.
There is no Poincare' subgroup in this E8 theory. ... The G = E8 I [Garrett] am using does not contain a subgroup locally isomorphic to the Poincare group, it contains the subgroup SO(4,1) -- the symmetry group of deSitter spacetime. ...
this theorem does not apply in this case. ...".
Steven Weinberg showed at pages 12-22 of his book The Quantum Theory of Fields, Vol. III (Cambridge 2000) that Coleman-Mandula is not restricted to the Poincare Group, but extends to the Conformal Group as well.
Since the Conformal Group SO(4,2) contains Garrett's de Sitter SO(4,1) as a subgroup,
it seems to me that
it is incorrect to claim that use of deSitter SO(4,1) means that Coleman-Mandula "... does not apply ..." to Garrett's E8 model.
However, it may be that an E8 model could be (for other reasons) consistent with Coleman-Mandula.
In an earlier comment here I quoted from Weinberg's book at pages 382-384 that the important thing about Coleman-Mandula is that fermions in a unified model must "... transform according to the fundamental spinor representations of the Lorentz group ... or, strictly speaking, of its covering group Spin(d-1,1). ..."
where d is the dimension of spacetime in the model.
As I said in that comment, E8 is the sum of the adjoint representation and a half-spinor representation of Spin(16),
so,
if Garrett builds his model with respect to Lorentz, spinor, etc representations based on Spin(16) consistently with Weinberg's work,
then
Garrett's E8 model may be consistent with Coleman-Mandula.
However, that may require Garrett to consider a 16-dimensional spacetime (the vector space on which Spin(16) acts) and then do dimensional reduction,
which should then be natural (it might even show Bee a natural symmetry breaking).
Tony Smith
PS - Lie superalgebras became popular in unified models (supergravity etc) because they had fermions contained within them in ways consistent with Coleman-Mandula.
MacDowell-Mansouri (anti-deSitter etc) was necessary to get gravity out of such Lie superalgebras as osp(n|2) whose Bose sector was o(n) + sp(2), where the o(n) was to give Standard Model type gauge groups and the sp(2) was the only thing around to give gravity.
MacDowell and Mansouri (and some others) realized that since sp(2) = so(3,2) anti-deSitter, they could actually construct gravity from it,
and there was a lot of interest for a while in such models.
So,
it was fermion content of Lie superalgebras, not MacDowell-Mansouri deSitter stuff, that made supergravity models consistent with Coleman-Mandula.
Tony, whatever you write is just so brutally wrong, this and countless other discussions would be a great source of amusement were it not so sad that the work of competent scientists is under attack by your kin, and well-intentioned laymen are victims. As an example for what comes out when laymen get involved, see eg. those discussions at physicsforums.com which are nothing but grotesque.
ReplyDeleteMore to the point.. here some examples, I just need to look at almost any random paragraph of what you write:
"As I said in that comment, E8 is the sum of the adjoint representation and a half-spinor representation of Spin(16).....
However, that may require Garrett to consider a 16-dimensional spacetime (the vector space on which Spin(16) acts) ....
"
Completely wrong... it's of course 10 dimensional space-time - space-time is based on quantities carrying vector, not spinor indices.
And so it goes everywhere. Upon randomly glancing over your opus, so rightfully rejected at the ArXiv:
http://www.valdostamuseum.org/hamsmith/2002SESAPS.html one thing that jumps right into my eye is:
" a 12-dimensional SU(3)xSU(2)xU(1) Standard Model symmetry group that is represented on the internal symmetry space by the structure SU(3)/(SU(2)xU(1) = CP2."
By the very definition of what a coset means, this posits that the electroweak SU(2)xU(1) woud be a subgroup of the color group SU(3). Nothing could be more wrong. I am afraid I don't want to read any further.
Why don't you just hold back and stop confusing any discussion on the web with those "ideas". The current topic is already bad enough.
This entire discussion reveals what sad shape academic physics finds itself in. The main actor professes his love for KK theory, something shown to be conceptually empty in the 30s by Pauli, while his enemies resort to personal attacks. Meanwhile, the work itself is nothing but the same old numerology that started with Gursey and is just as hopeless and empty now as it was then. There are no ideas that get next to the problem of matter. Furthermore, Tony Smith has already done all of this to death with a real idea to back it up (Wyler's), and he's ignored and banned. It makes me sick.
ReplyDelete-drl
Hi Garrett,
ReplyDeletethe algebraic hodge is indeed quite equivalent to the spacetime hodge, only you don't need explicitly the metric, but you need the frame. (Usually this is defined as epsilon^{mnrs}, on internal indices ).
I'm starting to suspect that the fact that you need to extract the frame from the connection, as you say, is probably the whole point.
In the connection you don't have the frame field, but only the frame-higgs H_mu^{m a}=e phi. If you introduce in the hodge the frame alone, it is like having to introduce the higgs alone.
I have a parallel in my works in graviweak unification.
The first, with Percacci, is based on the internal group SO(4,C). (It contains the lorentz plus a weak SU(2) group.) Now, SO(4,C) has automatically an invariant epsilon symbol (the internal volume form) so we were able to write a first-order action, that reduces to gravity plus weak interaction after symmetry breaking. This is in our paper Graviweak Unification. (The higgs must be supplemented as a standalone field, just to break the SU(2)_L and SU(2)_R symmetries).
In my second paper, alone, I use GL(4,C) and derive a whole standard model family and weak and strong interactions from GL(4,C)^4.
This is cute (ok not so 'exceptionally' cute.. :) but now however, there is no invariant epsilon symbol in GL(4,C), so it is difficult to write an action in the unified phase only with the frame field and the connection curvature, one needs an internal volume form.
I don't get yet what about this problem in your E8 action.
Fabrizio
Hi Guys,
ReplyDeleteLeaving all the E8's aside, could I come back to a question that was raised above about the 'interactions'? Garrett, you say
Choose any of the diagrams in the paper, draw a vector from the center of the diagram (the origin) to one of the particles, then draw another vector from the origin to a different particle. Add these two vectors (as described in the comment you quoted), and see if the result lands on a third particle -- if it does, that's the result of the interaction.
You could do the same thing for the SM, just add conserved quantities, if the result exists, then the interaction is allowed. The problem is, to have a conserved quantity you'd need to have an operator commuting with the Hamiltonian, and you have no Hamiltonian. Further, you have at this stage no Lagrangian, therefore no dynamics, so what sense does it make to speak of interactions to begin with? What makes you think drawing these arrows has something to do with an actual interaction? Further, how do you get 4 point interactions? If I recall correctly, there are none? What do you do with gluons, how do you couple the higgs? What do you do with gravity? If I'd think naively I'd have said it's not an abelian group, so there are A^4 terms. But that of course doesn't apply, because there's no Lagrangian. If you are claiming you'll get gravity renormalized similarly to how Fermi theory was rescued by the introduction of massive exchange particles, then I'd really like to know how you think you'll get away with that.
Best,
B.
I said that E8 was made up of adjoint and half-spinor reps of Spin(16), which would lead to consideration of 16-dim spacetime.
ReplyDeleteAnonymous said "... Completely wrong... it's of course 10 dimensional space-time - space-time is based on quantities carrying vector, not spinor indices. ...".
In fact,
The adjoint of Spin(16) acts on a 16-dim vector space candidate for spacetime.
Anonymous quoted my web site as referring to "... a 12-dimensional SU(3)xSU(2)xU(1) Standard Model symmetry group that is represented on the internal symmetry space by the structure SU(3)/(SU(2)xU(1) = CP2." ...",
and
Anonymous said "By the very definition of what a coset means, this posits that the electroweak SU(2)xU(1) woud be a subgroup of the color group SU(3). ...".
That is simply not true. In the model of Batakis in Class. Quantum Grav. 3 (1986) L99-L105, the SU(2)xU(1) is the little group of CP2 = SU(3) / SU(2)xU(1) = SU(3) / U(2) and the SU(3) is represented by global action on CP2 = SU(3) / U(2).
See the paper by Batakis for details.
Tony Smith
Dear Garrett, thank you for the explanations! And good luck with your research studies.
ReplyDeleteBee:
ReplyDeleteThe Lagrangian is eq (3.7) on page 26, or (3.8) if you like that better, and it's also in the summary on page 28. You're correct in your description -- the addition of quantum numbers shows which interactions are ALLOWED, but the action (Lagrangian) dictates which actually happen. The four vertex is allowed by adding four roots. I haven't calculated a Hamiltonian for this theory yet, but that's straightforward.
Sorry if I implied that ALL the root additions correspond to interactions, I'll try to be more careful in my wording. What happens is dictated by the Lagrangian. And, yes, there is one. But many (maybe even most?) of the interactions that can happen according to the Lagrangian correspond to those that are allowed by root addition, so it's not a pointless game.
Fabrizio:
I suspect the correct description will be somewhere between the two of ours, which is why I mentioned and cited your work in the paper. I do have some dissatisfaction with the way gravity is handled in this E8 theory, and expect it to change, so I'm not going to argue mine over yours. I'd be quite happy if yours, or yours and Percacci's, or Stephon's, were a closer description to the truth. I want to figure out how the universe works, and I'll use whoever's ideas I can in order to do it, and will readily discard my own past methods when they don't work so well. I try to cite generously, and it's not my intent to seize credit.
On the Hodge... I think we must have a disagreement of nomenclature or something. The "algebraic Hodge" does not involve the gravitational frame -- it's purely a mapping from Lie algebra elements to other Lie algebra elements. Similarly, the "spacetime Hodge" does not involve the Lie algebra -- it's a mapping from spacetime 2-forms to other 2-froms, using the metric or the gravitational frame.
In my paper, the "algebraic Hodge" operator is Clifford multiplication by the Clifford spacetime volume element, gamma. The "spacetime Hodge" requires pulling the frame out of the connection, which I agree is ugly as hell. It would be a little prettier if I used the frame-Higgs to build something that reduced to the spacetime Hodge times a scale factor, but I wanted to stick to conventional action terms as much as possible.
I currently like the frame-Higgs approach because it works well with the fermions in this E8 theory, but I'm flexible.
I hope that clarifies what I'm doing. But I'm still wondering what you mean by "algebraic hodge is indeed quite equivalent to the spacetime hodge, only you don't need explicitly the metric, but you need the frame." You must not mean the "gravitational frame," and by "equivalent" you must mean "similar in application"?
Hi Garrett,
ReplyDeleteI am confused because I was pretty sure that when you were here, you explained to me that all diagrams would be 3 point interactions, and I tried to tell you I don't see how the A^4 term could be missing if it's a non-abelian symmetry. I can't quite recall your answer, but maybe you do remember this discussion now? (Sundance was also present). Either way, if you've come to the conclusion you will have them, that's fine with me.
The Lagrangian is eq (3.7) on page 26, or (3.8) if you like that better [...] What happens is dictated by the Lagrangian. And, yes, there is one.
Yes, it's the one that you introduce, as you say by
"writ[ing] down an action agreeing with the known standard model and gravitational action"
with which it isn't much of a surprise that your approach gives the known standard model and gravity, and thus we are back to my disliking - we've had this exchange before, so no need in repeating the I-like-it/I-don't part. I just meant to clarify why I find this point crucial. You write down a root system, claim it depicts conserved quantum charges in interaction process, plus gravity. Just that without the dynamics, a Hamiltonian, a Lagrangian, a base manifold, you can't say anything about conservation, interactions, or relations to gravity. The most obvious choice to write down an action invariant under E8 symmetry doesn't work because, as I've pointed out above, it would give the fermionic part of the Lagrangian with the same order of derivatives as the gauge fields, even if you manage to break the symmetry nicely down into the SM one. What you do to cure the problem is to write down an action that gives the standard model - which I very much hope can be replaced by some better procedure. What it comes down to is that you've mapped the standard model to a nicely looking root system, and by counting the stuff in there, you've come to the conclusion there should be additional particles, much like in every GUT model. That's nice, but at this stage it doesn't quite fulfil my expectations for a TOE. Best,
B.
Bee:
ReplyDeleteSheesh, I brought up the subject, that day in the office, of replacing four-vertices with two three-vertices, as a crazy idea to ponder. I did it as something to think about, not as something necessary or that I had already done.
You know, as ugly as that little Lagrangian is, I can't help but love it a little. It describes the standard model and gravity in three terms! Oh, well, yes, it is a bit awkward, but maybe it will grow up and be prettier some day, as long as people keep kicking it.
Ah, okay, sorry for not being up to date. I'm glad you didn't mean to make it a feature, in the given context it didn't make too much sense to me.
ReplyDeleteas ugly as that little Lagrangian is, I can't help but love it a little.
Ah, sure, the own baby is alway the prettiest ;-) If a theory grows up, there will be a point where it starts disagreeing with your wish. I am always insulted by that. It's like I think if I made it, it should do what I want, and not have an opinion on its own. But maybe it's more important to watch your baby kicking, than watch other's kicking it. Best,
B.
Once more with emphasis
ReplyDeleteThis is all only phenomenology. No theory with spinors on background spacetime is ever going to work with gravity in forming a complete whole. Gravity IS background independence and is NOT the gauge theory of SO(3,1). It is NOT the spin 2 boson.
The issue is NOT to explain the SM gauge group. The latter works fine as it is, for the phenomena we see. The issue is to explain how space, time, and matter are related on a level BEFORE one posits matter on a background. There is no "asthenogravic" or "stereogravic" theory, and never will be. What must emerge is a framework of matter AS SUCH emerging simultaneously with spacetime, before the symmetries of interaction arise. This is the only hope of explaining weak symmetry breaking.
The problem with physics these days was completely nailed by Smolin. People do not spend any time thinking about fundamental issues, because they are too busy trying to survive in academia, where such thinking is career suicide (I should know). So, they get ideas in their heads that are simply wrong-headed, the wrong approach, and they know no history, so they don't even understand why what they are doing is wrong.
If you want to work in this direction, that is, futzing around with gigantic phenomenological Lagrangians, then why don't you help Chris Oakley with his work on creating finite results from QFT? That is infinitely more interesting and very necessary. Or, I have a set of 27 coupled non-linear differential equations that are beastly hard to solve. Or, just help Tony, who IMO, knows more about the relationship of Lie groups to particle phenomenology than anyone dead or alive. Have at it!
-drl
Hi,
ReplyDeleteFirst of all to Lunsford:
with not enough emphasis, I disagree on the use of 'phenomenology' as an insult. I believe we have no other mean to guess the next unification, but to use what phenomenology provides to guide us. This includes the standard model groups, masses and mixings etc. I studied string theory as a interesting intellectual challenge (I'm the one addressed so politely by Motl :) but being aware that no correct phenomenology was to be expected from it, exactly because of this reason. It's a pure top-down approach.
Garrett:
on the hodge: yes, you are right of course the algebraic hodge is independent of the frame, but it should boil down to the 4-form epsilon_{mnrs} below planck scale, when specialized to lorentz indices. In this sense it should become equivalent to the spacetime hodge (after lorentz is soldered with spacetime).
I think this is a requirement to obtain standard gravity at low energy.
Of course above the scale the dual is just the dual, and is the epsilon symbol I was referring to in the graviweak SO(4,C), so we were speaking of the same thing, sorry for confusing.
One can use the dual of SO(4C) and just avoid the spacetime hodge, as we do in graviweak unificatio - but we have a frame, and write the equivalent of the palatini (epsilonREE).
But you cannot do this because you don't have a vierbein (just the connection) and you are forced to pull that out of the connection (...so isn't this a destruction of the unification?). I was thinking it could be done without this.
On your comment, let me point out that in this criticisms I'm just trying to understand! Ambitious theories should be kicked more, not because of envy, but because we want them to work :)
I'm confronting mine to your approach, just because it is a concrete ground that I have to try to understand that basic point.
So I'll try to understand how your action boils down to gravity, and this entails sitting down to compute it, at least in principle.
...and now it's time to sleep again. It's not that E8 makes me sleep, it's that it IS a good dream.. :)
best,
Fabrizio
Several times Garrett has said something like the theory is not complete, but it is compatible with LQG, which could complete it. I don’t understand this. Could Garrett or Bee or anyone who does understand this please explain in very simple terms for the non-experts what is missing, and how LQG or some other theory might supply the missing pieces?
ReplyDeleteJim Graber
Hi Jim, sorry I can't be of help here. See my above comment about wishful thinking. Best, B.
ReplyDeleteJim said "...theory is not complete, but it is compatible with LQG, which could complete it. ... please explain in very simple terms for the non-experts what is missing, and how LQG or some other theory might supply the missing pieces?"
ReplyDeletenon-experts' can try to help themselves, in case the experts are busy. Here's my two cents:
LQG is basically a quantum theory of the connection.
(rather than a theory of the metric. a spinnet is a quantum state of a connection)
E8 presents a connection which describes both matter and geometry.
As it stands, it is not a quantum theory. The LQG community has accumulated experience in launching a quantum theory from a connection starting point.
Does that help at all? Or is it perhaps too obvious.
Fabrizio,
ReplyDeleteOf course referring to something as phenomenology is not an insult! Everything is phenomenology until it settles into a pattern with an overwhelmingly forceful organizing principle. The SM is not there yet - because Yang-Mills is massless in its natural state, the SM is "denatured" by SSB and that is the hallmark of phenomenology. Anything that posits yet another gauge group or yet another origin of the one we have is more of the same. That's not an insult. The real question is the origin in principle of weak symmetry breaking. A real theory will solve that problem.
-drl
in the intial blog post:
ReplyDelete"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."
I see lots of coupling constants in the new PDF that seems to be a good addtion to the arXiv paper. Maybe he heard your complaints. Anyway, it adds a lot that is not in the paper. Here's the link:
http://relativity.phys.lsu.edu/ilqgs/
Well Lunsford,
ReplyDeleteI'm not particularly inclined to discuss philosophical issues, but it is interesting to understand the approach.
The point is that 'organizing' is something very subjective: I may think that supersymmetry is very organized or terribly ugly; that doublet-triplet splitting is what explains the electroweak scale or that it is just an ad-hoc model; that strings are unifying or that on the contrary they produce a mess; that GR is the final geometric achievement, or that it is partial, an obstacle to the understanding of gravitation.
Probably there is no objective way to say one should be satisfied.
To make an simple example, I believe one should not think that the photon is massless because of gauge invariance, but the other way round: the world is gauge invariant because the photon has no measured mass!
So gauge invariance too is phenomenology in your definition. It's a nice 'principle', that links to a lot of geometry etc., but still phenomenological.
Given this, some may think that it is a waste of time to try to model phenomena at so different energy scales. This may well be true, but I haven't lost the hope to find a good framework that organizes (to me) existing groups, couplings and scales, and gives some predictions (his is what I tried in my work).
The process of choosing or building an organizing framework is highly subjective, while predictions are not. Therefore I admire Garrett's effort, and wait for predictions. :)
cheers,
Fabrizio
PS: And yes, electroweak breaking is a crucial point to consider.
Hi Marcus,
ReplyDeleteI see lots of coupling constants in the new PDF that seems to be a good addtion to the arXiv paper. Maybe he heard your complaints. Anyway, it adds a lot that is not in the paper. Here's the link:
http://relativity.phys.lsu.edu/ilqgs/
Sorry, I don't see what's added there? I just have to look at slide 2, where I see a frame field added to a gauge field, added to a fermionic field. Usually, these don't have the same mass dimension. That's the source of my confusion. If one changes these conventions, it would be nice to at least know how the fields are rescaled (see also our exchange above, right in the first some comments). Best,
B.
bee:
ReplyDeleteIt's not quite that bad. The gauge fields have 0 mass dimension, the frame -1, and the Higgs 1. So the mass dimension of the frame and Higgs cancel when they're combined as the frame-Higgs, and this can be added to the gauge fields and mixed together through gauge transformations. The fermions don't have 0 mass dimension, but the addition of fermions (Grassmann numbers) and gauge fields (1-forms) in the connection is a formal addition -- these fields don't mix. The mathematical name is that this is a Z2 graded algebra.
If we speculate that the fermions are the ghosts of pure gauge fields that they've replaced, then these original gauge fields should have 0 mass dimension -- but I don't think this implies the ghosts replacing them necessarily have 0 dimension. So, I think we're OK on dimensions, and don't need to write in dimension-ful coupling constants. The fermions as ghosts idea is controversial though, and I'm open to others.
Hi Garrett: I didn't say you're not okay with with couplings, I just said it's confusing to me. Best,
ReplyDeleteB.
A couple of questions for Garrett. I am not a physicist, so my understanding is very superficial.
ReplyDeleteDo the 248(?) roots of e8 correspond to particles, and do 3 roots a,b,c with a+b=c correspond to interactions.
Do these roots, as vectors in 8(?) dimensions correspond to 8 quantum numbers.
Is there an Abelian (additive) group homomorphism from the roots to $Z_2$ with bosons going to 0 and fermions going to 1, (assuming roots correspond to particles which are one or the other).
Is there any sense in which the Lie algebra or Lie group have a bosonic and fermionic `part'?
For which Lie algebras/groups does this treatment of bosons and fermions work. Thanks.
The classification boo-boo at ArXiv was corrected.
ReplyDeletePeter has some more to say about it.
In any case GL's paper is classified hep-th with cross-ref to gr-qc
Hi Marcus,
ReplyDeleteThanks for the pointer!
Hi Anonymous,
Your questions have been answered either in the post or in the comments above. Please be so kind to read before you comment. Thanks,
B.
another moose:
ReplyDelete"Do the 248(?) roots of e8 correspond to particles,"
Yes.
"and do 3 roots a,b,c with a+b=c correspond to interactions."
They correspond to ALLOWED interactions. But whether and how these allowed interactions happen is determined by the action. Though most of the allowed interactions between bosons and fermions do happen.
"Do these roots, as vectors in 8(?) dimensions correspond to 8 quantum numbers."
Yes, precisely.
"Is there an Abelian (additive) group homomorphism from the roots to $Z_2$ with bosons going to 0 and fermions going to 1, (assuming roots correspond to particles which are one or the other).
Is there any sense in which the Lie algebra or Lie group have a bosonic and fermionic `part'?"
I think this is the same question, and the answer is yes. The E8 Lie algebra splits into bosonic and fermionic parts. However, even though the distinction between what is a boson in the algebra and what is a fermion is easy to specify according to its quantum numbers, this is an ARBITRARY choice, enforced by hand in the action, so that this split happens. The algebra itself does not determine exactly how it should be split, but it provides a few such opportunities, one of which we use for the E8 Theory.
More technically, (I think) we're splitting E8 into the bosonic subgroup part:
SO(8) + SO(8)
and the fermionic part is the coset space,
E8/(SO(8)+SO(8))
But I'm doing this on the Lie algebra level and haven't worked it out topologically.
As you can tell, this sort of split, which we're doing by hand, can be done for almost all groups. But for the exceptional groups, the coset space is the fundamental representation space. (I think that's the correct description, but it might not be mathematically precise)
Garrett said that his E8 model works by
ReplyDelete"... splitting E8 into the bosonic subgroup part:
SO(8) + SO(8)
and the fermionic part is the coset space,
E8/(SO(8)+SO(8)) ...".
Actually, it is 120-dim SO(16),
instead of 28+28 = 56-dim SO(8) + SO(8),
that is relevant so that you have
the bosonic subgroup part 120-dim SO(16)
and
the fermionic part is the coset space,
128-dim E8/SO(16)
To see how SO(8) and SO(16) are related in more detail, consider
the graded structure of Cl(8)
1 + 8 + 28 + 56 + 70 + 56 + 28 + 8 + 1 = 256
and that by periodicity
Cl(16) = Cl(8) (x) Cl(8)
so
that the 120 SO(16) bivectors of Cl(16) should be made up of three parts:
28 SO(8) bivectors of the first Cl(8) in the tensor product
plus
28 SO(8) bivectors of the first Cl(8) in the tensor product
plus
64 = 8x8 tensor product of the two 8-dim 1-vectors of each the two Cl(8)s
to get the 28+28+64 = 120-dim SO(16) bivector algebra of
Cl(16) = Cl(8) (x) Cl(8)
Tony Smith
I'd like to make the suggestion that the exposition would be significantly improved if the derivation of the standard model Lagrangian from that in the text had more detail filled in. In particular, I'd like to see where the Yukawas and the Higgs self-interaction arise and what other interactions are present that are not present in the standard model. I'd also like to see a discussion of how the free parameters of the standard model arise.
ReplyDeleteRegarding the CM theorem, it is true that the S-matrix is not an observable in de Sitter space. However, it remains a very good approximate observable in the limit of small cosmological constant -- after all we compute S-matrix elements to describe what happens in accelerators. Thus, the procedure whereby one recovers these nontrivial scatterings such as we measure in accelerators without violating the CM theorem would be helpful to understand. It appears to me at a first glance that the symmetries between the Poincare part of the action and the rest are explicitly broken by the choice of the Lagrangian.
Like many others, I also don't understand how a bosonic symmetry can relate bosonic and fermionic variables. Perhaps rather than E_8, you mean some supergroup? Or do you claim that a quantization of your theory would violate the spin-statistics theorem? The fermionic fields that arise in BRST quantization are called ghosts for a reason -- they violate spin statistics, but also never appear as an external state, so it's ok. Your fields, however, are actual states in the theory.
Finally, could I make a plea for more modest titles? You don't have a theory of anything until you quantize something. Might I suggest something like "An E_{8+8} Classification of the Fields of the Standard Model and Gravity". (Which is not an endorsement by me that such a thing was done -- I do not have the time at the moment to work through the details of the paper.)
Aaron Bergman said "... I [Aaron] also don't understand how a bosonic symmetry can relate bosonic and fermionic variables. Perhaps rather than E_8, you [Garrett] mean some supergroup? ...".
ReplyDeleteSupergroups (or Lie superalgebras as used in supergravity models) are not the only things with both bosonic and fermionic variables.
The xceptional Lie algebras F4, E6, E7, and E8 also have this characteristic, as is well known. John Baez wrote about that at
math.ucr.edu/home/baez/week253.html
where he said "...
f4 ≅ so(9) ⊕ S9
e6 ≅ so(10) ⊕ S10 ⊕ u(1)
e7 ≅ so(12) ⊕ S12+ ⊕ su(2)
e8 ≅ so(16) ⊕ S16+
Here S9 and S10 are the unique irreducible real spinor representations of so(9) and so(10), respectively. In the other two cases, the little plus signs mean that there are two choices of irreducible real spinor representation, and we're taking the left-handed choice. ...".
That property of the Exceptional Groups was also the basis of Pierrre Ramond's paper hep-th/0301050 where he said:
"... they link spinors and tensors under space
rotations, which flies in the face of the spin-statistics connection. We discuss a way to
circumvent this difficulty in trying to generalize eleven-dimensional supergravity. ...".
Tony Smith
Spin-statistics. That's the whole point.
ReplyDeleteFabrizio;
ReplyDeleteGauge invariance is perfect as the principle behind Maxwell. If the SM were as perfect as EM, then Yang-Mills would survive intact as the basic idea, and not be adulterated by SSB. There would not exist a Cabbibo angle, there would not exist y5 projection operators in the Lagrangian. These things are, shall we say it? artistically clear.
Of course, Yang-Mills gets so close to being right, that it most likely *is* right in its unadulterated form, in the enveloping theory.
-drl
Like Aaron, I continue to be puzzled by the meaning of summing bosons and fermions, what is the statistics of the resulting object? commuting, anti-commuting, neither?
ReplyDeleteDear Moshe, you could express your opinion/knowledge about bosons and fermions more self-confidently and loudly. What you wrote would be enough for wise sensitive physicists who listen but it won't be enough on this forum.
ReplyDeleteIncidentally, I of course wrote the same #1 complaint about bosons and fermions on my blog. Carl Brannen from the same circles as Lisi wrote the first fast comments that it is common place in "geometric algebras" to do similar sums. As far as I could see, the papers containing this bizarre, conventionally sounding term, are of the same kind as Lisi's paper itself. Check it up.
Roger Highfield wrote a celebration of Lisi as a new Einstein in the Telegraph. Highfield is the same guy who wrote the rant "Einstein may have started the rot" in March. ;-) That's enough reason for me to prefer left-wing New York Times science section over such "conservatives" who jump on the first piece of porn that someone offers them.
I just noticed that my sentence in the comment from 5:36 PM, November 07, 2007
ReplyDelete"Might be an observer independent statement, but ..."
Should obviously have been
"Might be an observer dependent statement, but ..."
Doesn't string theory get fermions from E6 via orbifolds. Don't know why anyone would complain about the general idea that fermions can come from E6?
ReplyDeleteE6 is kind of just D4's adjoint (for bosons) plus a complex version of D4's vector (for spacetime) and a complex version of D4's spinors (for fermions). If you didn't need the complex part (and you do) then F4 would work. It's just a D4 Triality model. Once you are at E6, it is kind of natural to want to look at E7 and E8. (kind of quaternifying and octonifying the vector/spinors). What you might get from them is certainly something valid to look at; in fact you could get a Susskind-like bosonic M-theory and bosonic F-theory if you like things in string theory terms (as Tony Smith does in one view of his model). One can also think perhaps in other ways (more GUT-like, more Feynman-like, more Wolfram-like, more LQG-like); E8 almost seems to unify the unifyers and I think that's a great clue about E8's importance.
This was just posted on a political website, and as a layperson with a fascination with science, theoretical physics and string theory, and the quantization of gravity, I want to know if this is serious and worth paying attention to, or something not serious and not worth paying attention to.
ReplyDeleteMike
Dear Mike:
ReplyDeleteScience is done by arguments and discussions. These take time, while people exchange their points of view, form opinions, criticize, and possibly improve on weak points. This paper has been out since a couple of days only. It might take years to understand whether it might fulfil its promise - the early media attention is imho not very beneficial to the process. I think if you read this post and the comments you will get a very good picture of the different opinions.
There is no easy way to categorize things into hop or flop, thumb up or thumb down. If you have an interest in science in the making, you'll have to learn how to live with that.
Best,
B.
Hi moshe:
ReplyDeleteIn this theory the bosons are Lie algebra valued 1-forms, and the fermions are Lie algebra valued Grassmann numbers. They may be formally added, just as bosons and ghosts are formally added in the extended connection of the BRST technique. Because of the structure of the exceptional Lie groups, the fermions inhabit part of the Lie algebra such that, algebraically, they are spinor multiplets under the action of the bosons (including gravity), which inhabit the other part of the Lie algebra. Spin-statistics is satisfied. I hope this is clear, but if not, there are good references cited in the paper and higher up in this comment thread.
First of all, there are non-vector bosons in the standard model. But, more importantly, the BRST transformation is a grassman symmetry. See, for example above 2.24 on the reference you cite hep-th/0201124 or any other reference on the subject.
ReplyDeleteYour symmetry is commutative as it's described by a Lie algebra, not a super-Lie algebra. Such a symmetry cannot relate Grassman and ordinary variables. If I understand you, you are saying that because your matter fields are 1-forms, you are considering them as anti-commuting, but that's not true. For example, a gauge field is a one-form, but it is not the same thing as a Grassman field as you can see by the usual quantization of gauge theory.
I have a feeling I am watching history in the making...
ReplyDelete"Hi Anonymous,
ReplyDeleteYour questions have been answered either in the post or in the comments above. Please be so kind to read before you comment. Thanks,
B."
Sorry for redundant questions. I'm just a crusty old full professor and I'm not familiar with these new-fangled blog things.
Anyway, my questions stemmed from the talk, not this blog, but this seemed a good place to ask the speaker. And he was kind enough to answer my questions, including the follow-up questions I hadn't asked yet.
But to defend myself, I did quickly search the blog first to see if my questions had already been covered. Unfortunately, as I only realized later, the page I was on only had the comments and not your original post. Surely you were aware of this pitfall in your blog. But it is hardly reasonable to expect to expect your unwary guests to be aware of this pitfall, and it is most unreasonable to snap at them when they innocently fall for it. This scenario reminds me of Fawlty Towers.
More follow-up questions for GL.
During the talk, someone asked a question, saying that usually fermions lives at the vertices, whereas you have both kinds of particles on the edges only.
What was he referring to? Vertices of what (do the fermions normally live on)? Edges of what (do you have both kinds of particles living on)?
anonymous:
ReplyDeleteBe very careful, as the internets abound with sharp objects, and Bee is one of them. But you made it past the captcha, so I'll answer the question. ;)
The question you're referring to was asked by Abhay Ashtekar. He was referring to the edges and vertices of spin networks, as used in loop quantum gravity. In a spin network the edges carry representation labels for gravity and gauge fields, while the vertices (nodes) are labeled by numbers for the fermions. Since, in a sense, all the fields in this theory I presented are E8 gauge fields, including the fermions, this would mean all the representation labels would be on the edges. (I'm not a spin network expert, so someone please correct me if I've mangled this description.) I know of an LQG person who has an idea about reconciling these facts, but since it's not my idea I'm afraid I can't disclose it.
Dear Crackpot Garrett Lisi, the person whom you are threatening with sharp objects is a full professor, probably at the PI, and probably more than just an ordinary one, and you can perhaps intellectually reach his toes or less. I would expect a somewhat more respectful behavior from you here.
ReplyDeleteYou 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.
Your most recent comments about LQG are completely absurd because LQG is both bullshit as well as completely disconnected from your paper. What the hell does your crackpot paper have to do with LQG?
As far as I can say, you are a dishonest person who is constantly committing scientific fraud, selling your pile of shit as a meaningful physics paper.
Wow. This guy Lumo just accused you of threatening murder, fraud, and creating a "pile of shit" paper.
ReplyDeleteIf someone gets that emotional in a rebuttal to a paper about math it means you're probably on to something.
Best of luck, that was a fascinating read. Oh, and you might want to consider a restraining order, that guy sounds a tad unstable.
hi lubos,
ReplyDeleteis garrett paying you something to promote his paper or what? i (as many others i guess) would have really valued a harsh critique of the subject.
as it stands, the metaconclusion of the discussion for uninformed people is this: a) someone had a possibly bright idea that all but one abusive jerk discuss at length
b) the question of why a certain chech postdoc didn't find a permanent job in academia is answered.
is this really what you intend with these postings? if no, you should probably work on your iq a bit more to understand some basic facts about this world.
Dear anonymous,
ReplyDeletea jerk with iq problems is certainly a complimet for lumo. Nevertheless, I liked your post...
I came accross a paper written by Itzhak Bars in 1980 titled: "The exceptional group E8 for grand unification" (http://www.springerlink.com/content/3446623747256407/).
ReplyDeleteIts abstract says:
"A grand unified model based on E8 is described in which maximum unification within GUTS is achieved by classifying the fermions as well as the Higgs bosons in single and smallest possible representations. This provides a theory of flavor and of families and predicts , depending on the symmetry breaking chain, that the next three SU(5) families or the next two E6 families should be of V+A type and lie below 1 TeV."
The paper also appeared in the Phys. Rev. Lett. 45, 859 - 862 (1980) with a co-author and under a slightly different title: "Grand Unification with the Exceptional Group E8" (http://prola.aps.org/abstract/PRL/v45/i11/p859_1).
Already more than 25 years ago. :-)
Hi Justin,
ReplyDeleteI think John mentioned this paper to Garrett in the seminar, but he didn't have a reference. So thanks for that. However, the point of Garrett's work is not the Grand Unification, but getting it together with the gravitational part, i.e. TOE. That's why I keep insisting that from the bundle perspective the essential problem is identifying a Lorentz subgroup (after symmetry breaking) with the tangential space in the base manifold. Which doesn't happen 'naturally'.
Btw, Garrett, one more question that I know I've asked before: would you explain how you get the correct signature in SO(3,1)? Thanks,
B.
Hi Prof. Anonymous:
ReplyDeleteI'm just a crusty old full professor and I'm not familiar with these new-fangled blog things.
[...] to defend myself, I did quickly search the blog first to see if my questions had already been covered. Unfortunately, as I only realized later, the page I was on only had the comments and not your original post. Surely you were aware of this pitfall in your blog. But it is hardly reasonable to expect to expect your unwary guests to be aware of this pitfall, and it is most unreasonable to snap at them when they innocently fall for it.
Well, then.
Welcome to the blogosphere :-) If you survive the first week, you will realize how hilarious this comment is, so I will just echo it.
If you rate my reply to your comment number onehundredseventysomething asking you to "Please be so kind to read before you comment" as 'snapping at you' then you should be very grateful my imood presently doesn't show pmsy.
B. (unbalanced, untenured, and some other un-s)
Hi B.,
ReplyDeleteI'm aware of the difference(s) between GUT and TOE, but my point was merely that if it was already completely obvious and accepted that E8 is the underlying "blueprint" of the electromagnetic, weak nuclear, and strong nuclear forces, then already more research would have been available attempting to combine gravity with it.
To compensate for my earlier post, here is an even more "obscure" abstract from a paper titled: "A CHERN-SIMONS E8 GAUGE THEORY OF GRAVITY IN D = 15, GRAND UNIFICATION AND GENERALIZED GRAVITY IN CLIFFORD SPACES”
Abstract:
"A novel Chern-Simons E8 gauge theory of Gravity in D = 15 based on an octic E8 invariant expression in D = 16 (recently constructed by Cederwall and Palmkvist) is developed. A grand unification model of gravity with the other forces is very plausible within the framework of a supersymmetric extension (to incorporate spacetime fermions) of this Chern-Simons E8 gauge theory. We review the construction showing why the ordinary 11D Chern-Simons Gravity theory (based on the Anti de Sitter group) can be embedded into a Clifford-algebra valued gauge theory and that an E8 Yang-Mills field theory is a small sector of a Clifford (16) algebra gauge theory. An E8 gauge bundle formulation was instrumental in the understanding the topological part of the 11-dim M-theory partition function. The nature of this 11-dim E8 gauge theory remains unknown. We hope that the Chern-Simons E8 gauge theory of Gravity in D = 15 advanced in this work may shed some light into solving this problem after a dimensional reduction."
Source: http://gaupdate.wordpress.com/2007/03/23/c-castro-a-chern-simons-e8-gauge-theory-of-gravity-in-clifford-spaces/
It’s probably related to this one titled "On Chern–Simons (super) gravity, E8 Yang–Mills and polyvector-valued gauge theories in Clifford spaces" (http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=JMAPAQ000047000011112301000001&idtype=cvips&gifs=yes)
BTW: I'm on the opinion that this thread deserves at least 248 posts. :-)
J.C.
Great blog, Bee
ReplyDelete"You do not really understand something unless you can explain it to a string theorist."
justin:
ReplyDeleteYes, that's a good paper. These ideas have been considered many times before. But that paper does not include a description of gravity as part of E8, which is a significant difference. As with most ideas in science, the work I've done has relied on bringing together the work of many others. I just hope I've advanced these ideas a little further. (Ah, Sabine answered this... I should have scrolled down.)
bee:
As with so much in this theory, we get so(3,1) by hand. Since the approach we are taking is a "top-down inspired bottom-up approach," we build so(3,1) into a non-compact form of e8 that can accommodate it as a subalgebra. Now that it's built, I agree that it would be nice if there were a top-down justification for this, and I see hints of this possibility, but nothing solid enough to talk about.
My brain is now full thanks.
ReplyDeleteHi Justin,
ReplyDeleteI didn't mean to imply you were not aware of the difference, just meant to point it out. If I look at our today's visit stat (already at 2000 at 10 am), we're likely to get to the 248 comments you're hoping for.
Hi Marcus,
Thanks... Hope it's good for something. Lubos always raises the question in my why I'm doing that to myself, so I appreciate encouragement.
Hi Garrett,
Thanks for the clarification. I am not really sure your classification as an 'top down inspired bottom up approach' works out, since you don't have a 'top down' in your case (what is the theory - you only have a root diagram). Without the bottom up, there's nothing that can be called gravity in your model. As we've discussed above, without identifying by hand the SO(3,1) part to the base manifold's tangential bundle, there's nothing that provides this relation. So which 'top' does that 'inspiration' come 'down' from?
Best,
B.
So which 'top' does that 'inspiration' come 'down' from?
ReplyDeleteas audience, I would say the top is that geometry and matter are the same thing and should be described by the same math object---but there is a way to describe the geometry of a manifold which is more natural than the metric used in classical geometry dynamics---a connection is as good as a metric and better in some ways.
so IMHO the top idea is to describe both matter and geometry as the same thing by a principal bundle connection, because of the naturalness. And the rest follows on down from that.
(may be a paraphrase of something in the ILQGS seminar talk)
Hi Marcus,
ReplyDeleteas audience, I would say the top is that geometry and matter are the same thing and should be described by the same math object---but there is a way to describe the geometry of a manifold which is more natural than the metric used in classical geometry dynamics---a connection is as good as a metric and better in some ways.
so IMHO the top idea is to describe both matter and geometry as the same thing by a principal bundle connection, because of the naturalness. And the rest follows on down from that.
Well, as audience you are not very well listening. What I have repeatedly pointed out before is that the 'rest' does *not* follow from that. You are *not* done by taking an E8 principal bundle and the connection. A principal bundle needs a base manifold to begin with, and the 'geometric' part of the bundle needs to be identified with the tangential space. Garrett does that by hand. That is doable, but it's neither 'natural' nor 'top down'. The other problem with just taking the connection is the fermionic part of the action, you find that in the comments above as well as in the post. Thanks,
B.
I enter the fray with great trepidation. I am no physicist. I am, however, a student of logic and I do have a concern.
ReplyDeleteI have watched with considerable amusement the publication of a number of books over the years ('Holy Blood, Holy Grail', and 'The Hiram Key' come to mind) where the authors have a pre-conceived conclusion then jury-rig the presented facts to fit within the conclusion, basically "here is the specific bit of mental gymnastics you need to perform to fit this round peg into this square hole".
I originally wasn't going to say anything, but some of Bee's concerns early on in the thread kept bringing this to mind. However, when I read Bee's concern (and Garrett's response) as to whether this was a bottom-up or top-down approach that I saw the connection. You can't have it both ways.
I am not distracted by the physics itself (I've admitted up front I know nothing), I am approaching the bare logic.
Bee:
That's why I keep insisting that from the bundle perspective the essential problem is identifying a Lorentz subgroup (after symmetry breaking) with the tangential space in the base manifold. Which doesn't happen 'naturally'.
From a top-down approach, he's right. Unless you are dabbling in the fallacy of the undivided middle, you need a better justification than (paraphrased) "I had to use this because it's the only thing that would work". That leaves you wide open to the accusation of "numerology" - that you're jury-rigging your theory to fit within the E8 structure.
I've noted Garrett's posts acknowledging the the work of others concerning E8 and that his contribution is to bring gravity into it (if I've understood correctly).
However, if the only way of doing so is to engage in logical fallacy, then the point is moot.
Garrett, please prove me wrong (in the language of logic, please, I've got enough of headache already).
Bee, you are absolutely right! It was careless of me to say that. The rest does NOT "follow on down from that."
ReplyDeleteThat would be a top down approach, which is not what we were talking about.
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.
Hi Garrett,
ReplyDeleteFrom your earlier reply to Moshe: "Yes, the Coleman-Mandula theorem assumes a background spacetime with Poincare symmetry, but this theory doesn't have this background spacetime -- with a cosmological constant, the vacuum spacetime is deSitter. So this theory avoids one of the necessary assumptions of the theorem, and is able to unify gravity with the other gauge fields. On small scales though, Poincare symmetry is a good approximation, and on those scales gravity and the other gauge feels are separate, in accordance with the theorem."
Regarding the "small scales": small compared to what? what sets the scale in this theory? where does it come from? The whole notion of scale (and units in general) seems problematic here considering that your connection is the sum of two fields with different mass dimensions (as Bee mentioned earlier and you confirmed). Have you explicitly shown somewhere that in your model gravity and the other gauge fields become separate at "small scales"? (i.e. did you do a calculation that verifies this, and can we view it somewhere?)
(Sorry if this has already been asked and answered, it's hard to keep track of the contents of 200 comments...)
So which 'top' does that 'inspiration' come 'down' from?
ReplyDeletethis is such a good question, it needs repeated consideration.
if the theory taking shape bottomup really is topless, or imprecisely inspired, then why do I keep thinking "deSitter manifold" and "Cartan connection"? maybe it's just a leap of faith. does John Baez ever visit here at backreaction?
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.
ReplyDelete