Garrett Lisi's New E8 Paper
    By Tommaso Dorigo | June 28th 2010 08:34 AM | 103 comments | Print | E-mail | Track Comments
    About Tommaso

    I am an experimental particle physicist working with the CMS experiment at CERN. In my spare time I play chess, abuse the piano, and aim my dobson...

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    Social networks are a great help for this kind of news: a new paper by a FB friend does not go unnoticed (at least by me) as it once would. I learned today that Garrett Lisi (picture below), the surfer and theoretical physicist, has deposited another paper in the Cornell arxiv. And it looks as a significant addition to his previous studies of the E8 group. He explicitly calls it "a companion" to the previous article, "An Exceptionally Simple Theory of Everything".

    The new article is titled "An Explicit Embedding of Gravity and the Standard Model in E8". In it, Lisi shows explicitly how the algebra of the gauge fields (the forces of Nature) acting on fermions (matter) can be described by real matrices (arrays of numbers) which relate with elements of the very special group called E8. Why is this relevant for theoretical physics ?

    The unification of the mathematical description of the forces of Nature in a single structure is a Holy Grail for theoretical physics, ever since the Standard Model was discovered. The observation that the different strength of the four forces may be an accident of our living in a cold universe brought theoreticians to speculate that we see these forces as distinct, while in reality they are different manifestation of the same "unified field".

    In the past, attempts at achieving a unified picture of the forces by choosing the "natural" SU(5) group to derive the symmetry groups SU(3) and SU(2)xU(1) that describe respectively quantum chromodynamics and electroweak groups were thwarted by unwanted implications that were in unsolvable conflict with experimental observations. Lisi's attempt with the huge E8 group might well be plagued by similar problems, but the mathematics is sufficiently rich that a solution could be found for any apparent inconsistency.

    Rather than trying to make sense of the complex algebra contained in the bulk of the paper, it is interesting to read the conclusions that Lisi draws at the end of his new work:

    "If the geometry of our universe is fundamentally that of an E8(-24) principal bundle, this symmetry must be broken, with the frame-Higgs part of the connection attaining a vacuum expectation value, and the structure reducing to that of gravity and the Standard Model. [...] it predicts the existence of many new particles, including W', Z', and X bosons, a rich Higgs sector, and the existence of mirror fermions. The charges of these elementary particles [...] are presented in the Elementary Particle Explorer [...] It is a distinct possibility that some of these new particles may be detected at the Large Hadron Collider."

    The Elementary Particle Explorer! Let's go give a look! Wow. You can play with this tool and make weight diagrams for your theory of choice... Nice, although I do not exactly know what to do with the weights. I guess I have to go back to my notes of Group Theory! Anyway, let us continue with the conclusions (suitably amended):

    "Although the explicit embedding of gravitiy and the Standard Model in E8(-24) described here is incontrovertible, it raises many questions. Since a direct interpretation implies the existence of mirror fermions, which are not known to exist in nature, it will be necessary to understand how these particles obtain large masses or otherwise work out in the theory. Fortunately, the embedding in E8(-24) also predicts the existence of axions and other Higgs scalars, which interact with the mirrors. These scalars could also help explain the existence of the three generations of fermions - possibly as scalar-fermion composites- and their masses. [...]"

    So, it is always the same old story... We have not found supersymmetric particles nor mirror matter, so these things are either hidden from our view (by having a large mass for some strange occurrence), or the theory is wrong. Too bad, but still, a fascinating hypothesis.

    He concludes with a note of hope: there is work to do, but it is promising:

    "The explicit embedding of gravity and the Standard Model [...] described here provides a solid base from which to develop these ideas further. [...] From the solid foundation of these explicit matrix representations and their embeddings, we might get a better perspective on many of the deep open questions remaining."

    Good job. A ray of sunshine in the present landscape, and a boost to the hope to understand much more about Nature within our lifetimes!


    Comment by Lubos slagging this off in 5...4.....3....

    Great post Tommaso, i love the smell of cargo cult science first thing on a Monday morning.
    I'm going back to bed!

    I disagree John. This is not cargo cult science - it is science that takes advanced studies in mathematics to understand, for one thing. I used to be pretty good with group theory (my teacher in the PhD course was none less than Gian Giudice) and I still have problems with the math.

    In any case, you don't have to like the E8 idea. If you prefer other paths to the search of GUTs, you are entitled to follow your own path... But criticizing a work without giving any reason is sterile.
    It's a capital mistake to theorize, Tommaso, but if the person who theorises is a 100% crank who doesn't understand the difference between diffeomorphisms and Yang-Mills symmetries and the difference between bosons and fermions, such a "theorist" needs to be supported by one half of the Internet, right?
    Oh Lubos, you took a sentence by Sherlock Holmes and you attribute it to me now because I posted it in my blog ? You better find other rethorical tools.

    So why the heck were you posting it? You have actually said much more than Sherlock Holmes. He was talking about detective work and it is you who promoted similar mis-ideas about science. You can't really back off now.
    But extra dimensions is about blurring the difference between diffeomorphisms and Yang-Mills symmetries, and supersymmetry is about blurring the difference between bosons and fermions. Or so we learn in any divulgation book.
    (Perhaps the problem is that the real cargo science is the thing done by writting (and filming) divulgation, at least for divulgation of theoretical particle physics)

    Some people have blurred their whole brains.

    Well, indeed, extra dimensions may "blur" the difference between diffeomorphisms and Yang-Mills symmetry;  but Lisi's theory has no extra dimensions. And indeed, supersymmetry "blurs" the difference between bosons and fermions, but Lisi's theory has no supersymmetry.
    Moreover, the "blurring" by extra dimensions may allow us to obtain Yang-Mills symmetry from diffeomorphisms, but not vice versa.
    Lisi does not understand the difference between fermions and Bosons? well, he's not allone. there are all these supersymmetry cranks that don't get it either :-)

    Dear Chris, ...

    I don't know whether you're joking or not, but there are surely many idiots who say such things seriously. 

    So let me say the key thing here: supersymmetry is an actual symmetry that relates the properties of bosons and fermions. The mathematics and physics work. It's the only symmetry that can be added to the known symmetries and that can achieve this goal, as the Haag-Lopuszanski-Sohnius theorem (an extension of the Coleman-Mandula theorem) proves.
    On the other hand, none of the crackpot theories of Lisis and similar "physicists" respects the rules of supersymmetry, so they can't relate the properties of bosons and fermions. Whoever says that he has a theory unifying bosons and fermions but it has no supersymmetry is a crackpot.


    But, does Garrett explicitly rejects the possibility of reshaping his model to include supersymmetry?Or, has it been proved somehow that a such symmetry implies new contradictions in his model? If there is still some place for susy there, then it is not a big trouble.
    Now, I would focus ;-) on another kind of problems that happen when one puts the gauge and the diffeomorfisms in the same bag. For instance, how do you rotate away the mass eigenstates from the charge eigenstates, to get the CKM mixing? One of the motivations to discard kaluza klein was that the inner space was not able to reproduce the charges of the standard model. The other, more widely known, was the inability to reproduce chirality (and thus implying the need of mirror fermions)

    I think everyone has right to say his opinion.

    Hey Tommaso,
    Thanks for posting on the paper. The Elementary Particle Explorer is a lot of fun. You brought up SU(5) -- so here's a weight diagram from EPE showing how a proton, composed of two up quarks and a down of different colors, can decay into a positron and pi meson (up + anti-up) via one of the colored SU(5) X bosons. Also, it shows the weak mixing angle of about 38 degrees. Click on particles in EPE, you'll get the idea. These weight diagrams are useful because charges along all axes are conserved in interactions.

    Hi Garrett,
    yes, I sort of remember a bit of my group theory now. Keep up with good surfing (work seems to be going well I gather).

    "... but the mathematics is sufficiently rich that a solution could be found for any apparent inconsistency."

    - Tommaso

    Nicely said. (Sorry for my dry humor!) ;-)

    By the way, this sentence is pretty characteristic for the attitudes of the people who hate advanced maths and who are pretty bad in it: when it comes to maths, everything is about completely irrational emotions....

    If they don't like a theory or its author and something else and it depends on maths, they would say that math sucks - it's just hot air, math is evil, and so on. If they like it, it's "sufficiently rich can can solve anything".

    However, the richness of mathematics is not about solving any apparent inconsistency. Richness mathematics is about exactly saying which of them can be solved and which of them can't, and about many unexpected statements about these statements.

    It"s very clear to anyone who actually understand maths that these "apparent" problems simply can't be solved within this "GraviGUT"-like framework. They're not apparent problems but real problems and maths is rich enough that it allows us to generally prove that any approach to physics that is remotely similar to this GraviGUT approach is dead at birth.
    By analogy to Dr Lisi's use of the rich structure of E8 to represent known particle physics, Dr Woit applied Clifford algebras towards the same end in his paper , e.g. page 50:

    "There is a long tradition of trying to use the rich structure of the Clifford algebras to classify the particles and symmetries of fundamental physics."

    And page 51 he concludes:

    "... the most basic geometry of spinors and Clifford algebras in low dimensions is rich enough to encompass the standard model and seems to be naturally reflected in the electro-weak symmetry properties of Standard Model particles."

    These ideas are buried 50 pages deep into a technical paper, have not been widely popularized/hyped. He can afford to be more cautious because he is more secure financially, and thus doesn't need to attract research grants or students to attend courses to help pay the rent. Probably to attract funding and student interest, Dr Lisi's E8 ideas were presented in a more enthusiastic fashion and were subsequently reported in the popular media, like the first string theory revolution back in 1984 (which captured my imagination as a teenager studying physical science, until I read Feynman's evaluation of string theory in the 1988 book Superstrings by Davies and Brown, p. 194: "… I do feel strongly that this is nonsense! … I don’t like it that they’re not calculating anything. I don’t like that they don’t check their ideas. I don’t like that for anything that disagrees with an experiment, they cook up an explanation ...").

    While it is obvious even to morons with an IQ of 50 that mathematics is needed to describe and predict the quantities which are the basis of physics, it's does not automatically follow that mathematics is anything more than symbolic accountancy. E.g., Feynman had argued back in 1964 against mainstream applications of calculus to physics as being the ultimate basis of reality because a continuously curved spacetime is simply an approximate, statistical model for real fundamental forces produced by the exchange of discrete field quanta:

    "It always bothers me that, according to the laws as we understand them today, it takes a computing machine an infinite number of logical operations to figure out what goes on in no matter how tiny a region of space [because there is an infinite series of terms in the perturbative expansion to Feynman's path integral] ... Why should it take an infinite amount of logic to figure out what one tiny piece of spacetime is going to do? So I have often made the hypothesis that ultimately physics will not require a mathematical statement, that in the end the machinery will be revealed, and the laws will turn out to be simple, like the chequer board with all its apparent complexities." – Richard P. Feynman, The Character of Physical Law, November 1964 Cornell Lectures, broadcast and published in 1965 by BBC, pp. 57-8.

    It's true that ignorant morons may object to advanced QFT mathematics simply because it's so abstract. However, as Feynman's argument shows, there are intelligent reasons for questioning the fundamental validity of even well established mathematical techniques used ubiquitously in particle physics...

    Feynman would probably turn in his grave.

    Yes, but 2 pi or 4 pi?

    clockwise or counter-clockwise ?

    Hey, check out this post from Jacques Distler. I guess we can all (including Lisi) forget about Lisi's work now. OK?

    I am astonished that Garrett Lisi still creates great expectations. Blog physics seems to have its own standards for what it is interesting and these standards have very little to do with the content. There are probably hundreds of articles about unifications involving E_8 and no-one notices them. The original model was full of elementary errors. In the recent model the prediction for three generations is lost and unless one is ready to believe in heavy mirror fermions the model reduces to some wishful thoughts. Why this surfer dude stimulates such a fuss? Is the reason that he speaks native American english and US desperately wants its own Einstein?

    We've already had our own Einstein, he was called Einstein, Einstein.

    Mr. Blake. Einstein is gone, but there can come a second Einstein. All wait for him. He is needed.

    A second Einstein?  Einstein 2?   That sounds clunky so I will simply call him The Deuce.
    Well actually, the accurate name of an Einstein successor would be "Zweistein"

    Wait for "him"? You're so politically incorrect, Ulla. It should be "her", and she is a lesbian from the black Africa.
    I hope you will respect her then:) Someone particular you had in mind? There are few women on top.

    No, I said him. I think he is not an US guy. And not Czech, lol.

    I had no news of Dambisa Moyo being lesbian. Nor physicist, now I think about.

    Why then mention her?

    With all this about E8 and Clifford Algebras, considering that Brauer and Weyl connected Algebra to Atoms, has anyone generalized the Brauer-Weyl construction to include quarks, rather than just electrons ? One might ask where they get a quaternion algebra for each electron so they can make a Clifford Alg by direct products. Perhaps this is simpler than trying to go for the whole field theory.

    Peer-reviewed, to be published in a respectable journal:

    Not peer-reviewed, not published:

    The input of a respectable theoretical physicist:

    I'd also post the laughs of great physicists when this subject came about, if there was an available video of a seminar by Ginsparg on the arXiv that I attended a while back.

    Distler and Garibaldi formalized these laughs by proving that Lisi-like theories cannot have chiral fermions. I think that summarizes it all, in one polite sentence.

    Bonny Bonobo alias Brat

    Hey guys, are you aware of any new flavours for us to take a mexican hat off to yet? And is there any chance of you or someone you know updating this lowly Wikipedia article for us chimps and bonobos? You  talk a lot about idiots and IQs which is all very amusing, but I personally would be very grateful if those qualified could share their obviously immense knowledge on the subject a bit more in public places like this, or is Wikipedia beneath you?<?xml:namespace prefix = o ns = "urn:schemas-microsoft-com:office:office" />

    Weak interaction    

    From Wikipedia, the free encyclopedia

    <?xml:namespace prefix = v ns = "urn:schemas-microsoft-com:vml" />

    This article needs attention from an expert on the subject. See the talk page for details. WikiProject Physics or the Physics Portal may be able to help recruit an expert. (February 2010)


    This article includes a list of references, related reading or external links, but its sources remain unclear because it lacks inline citations. Please improve this article by introducing more precise citations where appropriate. (February 2010)


    This article cites its sources but does not provide page references. You can help to improve it by introducing citations that are more precise.





    Quote “According to the electroweak theory, at very high energies, the universe has four massless gauge boson fields similar to the photon and a complex scalar Higgs field doublet. The gauge bosons are associated with a SU(2) ×U(1) gauge symmetry. However, at low energies, one of the Higgs fields acquires a vacuum expectation value and the gauge symmetry is spontaneously broken down to the U(1) symmetry of electromagnetism. This symmetry breaking would produce three massless Goldstone bosons but they become integrated by three of the photon-like fields through the Higgs mechanism, giving them mass. These three fields become the W+, W– and Z bosons of the weak interaction, while the fourth gauge field which remains massless is the photon of electromagnetism”.


    Quote “Although this theory has made a number of predictions, including a prediction of the masses of the Z and W bosons before their discovery, the Higgs boson itself has never been observed. Producing Higgs bosons will be a major goal of the Large Hadron Collider at CERN”.


    My article about researchers identifying a potential blue green algae cause & L-Serine treatment for Lou Gehrig's ALS, MND, Parkinsons & Alzheimers is at
    Dear Helen, I have personally made thousands of edits to Wikipedia

    and about one-half of them are about physics (you have to go through the whole history). So that included having written hundreds of physics stubs that became full-fledged and useful articles as time was going by.

    In my opinion, the article about the weak interaction you included above is pretty good. Someone added the label "in need of an expert" above which doesn't mean that the article has serious issues. In expert disciplines it usually means that a layman has misunderstood it, or fails to believe it, and hopes that another "expert" will qualitatively modify the article so that it will say something that the layman likes.

    Of course, in some cases the complaints are justified, but it's no rule.

    But otherwise the article about the weak interaction does sound technical because the issue it describes is technical. There's no sense in making some superficial explanation of the concepts over there - because there would be the same laymen's explanations in thousands of similar articles. The article is only about the things that are specific for the weak interaction - and not just some popular presentation of physics - and these issues are inevitably less comprehensible for the non-experts.
    Distler and Garibaldi formalized these laughs by proving that Lisi-like theories cannot have chiral fermions. I think that summarizes it all, in one polite sentence.

    ...too quick summary. They only apply to E8.

    In fact the following published, peer reviewed, paper proves that a GraviGUT idea is perfectly sensible:


    You get a whole SM family of fermions (chiral) from a SO(3,11) gauge group, extending the gravitational and SO(10) gauge groups. This is the only, unique, orthogonal group with chiral spectrum.

    Dear Fabrizio,

    you, Percacci, Lisi, and others are confused about some basic objects in GR, and moreover, you use terminology to fool yourself. You may call SO(3,1) "gravitational Lorentz group" but it's just a fact that it's still just a local, pointwise group in all your models, and such Yang-Mills groups cannot replace diffeomorphisms that mix different points.

    In GR formulated with a metric tensor, the local gauge symmetry are the diffeomorphisms. In GR formulated with vielbeins, it's the semidirect product of diffeomorphisms and the local Lorentz group. But there is no formulation of gravity that would only have the local Lorentz symmetry. You simply can't eliminate the diffeomorphisms - and you never get diffeomorphisms from any Yang-Mills-like group in the bulk.

    This is a basic point that kills this whole program and hundreds of useless and nonsensical papers you keep on writing. You may "get" fermions that transform as some chiral reps of some group but the Yang-Mills group will never induce any gravity.

    Best wishes

    You simply can't eliminate the diffeomorphisms - and you never get diffeomorphisms from any Yang-Mills-like group in the bulk. This is a basic point that kills this whole program (...)

    No, too quick Lubos, you are mistaken: diffeomorphisms are there, exactly in (semi)direct product with the local gauge group. Nothing exotic here. Diffs+gauge, as in gravity with a vierbein.

    Dear Fabrizio,

    you can't fix it either way: if the diffeomorphisms are there, there cannot be any SO(11,3) symmetry because such a symmetry mixes the 1+3 coordinates for which the diffeomorphisms exist, as you claim, with 10 additional coordinates for which the diffeomorphisms don't exist - so there can't be any symmetry in between them.


    Lubos, no. Get a standard review on Cartan Gravity (e.g. the one by Hehl, or Blagojevic book, or Percacci recent review which is clear).

    Simply put, and speaking of gravity alone: diffeomorphisms act on spacetime (and functions on it), the gauge group acts in internal indexes only, not on the coordinates!

    There is no compulsory link between these two invariances, and so you can disentangle them and consider other groups, as hinted by the fermion spectrum. For example the maximal SO(3,11).

    Dear Fabrizio,

    I am not telling you these things in order to argue or in order to find out about something I don't know. I am trying to help you to avoid elementary errors.

    Not only there is no "compulsory" link between the two qualitatively different kinds of symmetry: there is no possible field-theoretical union among them. Gauge symmetries can't be combined with diffeomorphisms - or other spacetime symmetries - and extended into a broader group such as your SO(3,11). This result is known as the Coleman-Mandula theorem or, if Grassmannian generators are allowed, the Haag-Lopuszanski-Sohnius theorem.

    In your case, one doesn't need any complex theorems because you're proposing a very specific identification of the groups that is manifestly incorrect even without those theorems. There can't be any symmetry mixing "coordinates" that are equipped with spacetime dimensions with thse that are not - because this is the most serious, any-symmetry-breaking difference between two coordinates one can ever think of. So there can't ever be any theory with the symmetry SO(3,11) if 3+1 of those 14 directions in SO(3,11) correspond to spacetime coordinates while others don't.

    You can't even make a more general theory where the original symmetry would be spontaneously broken. The only way how diffeomorphisms in 3+1 and 10 dimensions could be treated differently is compactification but with compactification of the 10 dimensions, the SO(10) symmetry would be clearly broken as well.

    I am telling you an absolutely trivial issue. Please try to think about it. I have read the reviews by all the authors you mention and all of them are equally confused about these elementary things as you are. Perhaps, you may think that this "shared confusion" among 3 or 4 people makes the mistakes less silly except that it does not. All of your are making elementary errors.

    Best wishes

    Hi Helen,

    I do not understand. What is wrong about the highlighted text ?

    Helen has just copied and pasted the Wikipedia article including all of the formatting, as inter-formatted through the Microsoft Office. Is that so hard to see it? What's wrong with you? ;-)
    Again I am astonished by your quick confused comments.

    Nobody is uniting Diff and gauge. Who told you this? Reread my post. Which I rewrite again for last time:

    There is no mixing between SO(1,3) and Diffs in standard (Cartan) gravity, so you can simply extend SO(1,3) to a larger group and leave the Diffs as they are, acting on our standard FOUR spacetime dimensions.

    There is no error. Thanks for the good intentions.


    Dear Fabrizio,

    you're just trying to fool yourself. You say that "there's no mixing between SO(1,3) and Diffs in the standard (Cartan) gravity". But this is manifestly incorrect. The SO(1,3) group is the group that rotates the directions of the tangent space at any point which is locally (at each point) - or globally, assuming a Minkowski background - indistinguishable from a diffeomorphism. For example, on the Minkowski background, the global Lorentz transformations are just a subgroup of the group of diffeomorphisms.

    You can extend a vierbein (4) to a vierzehnbein (14) but with the vierzehnbein, you can't construct any laws of physics (e.g. no Lagrangians respecting the SO(3,11) symmetry) in 4 dimensions. To construct laws of physics, one needs not only the vielbein but also its inverse - to be able to lower indices and to contract the lower indices from derivatives. But a 4 x 14 matrix is not invertible. Only square matrices may be invertible and square matrices induce an isomorphism between the auxiliary indices and the genuine spacetime vector indices; their number must therefore coincide and they may be locally viewed as parameterizing the very same space. There is surely no freedom in changing the symmetries acting on a vielbein and the symmetries that can act on spacetime independently. All these symmetries boil down to spacetime symmetries and you can't pretend that spacetime has two different dimensionalities at the same moment.

    Percacci also tries to fool himself by claiming that the SO(3,11) symmetry may still be there but it's "always broken". But this is a completely unphysical comment. If one can't restore a symmetry by any actual configuration allowed in the theory, not even approximately and not even in principle, the symmetry physically doesn't exist. If we denied this point, we could also say that QCD has an SU(390) color group instead of SU(3) but it is always broken to SU(3) by requiring 387 color components of the quark to be zero. But one can't actually make them zero by any genuine physical mechanism.

    Note that spontaneously broken symmetries - in the conventional sense - always constrain the theory by equally strong conditions as the unbroken symmetries. For example, they imply the existence of the Goldstone bosons. If there were an SU(390) color symmetry for quarks or an SO(3,11) "GraviGUT" symmetry, the broken generators would have to produce some signatures - which they clearly don't in the real world so the symmetries can't exist in the usual meaning of the "existence of symmetries".

    Also, one can't say that the "unbroken phase" is the point where the vielbein is singular. By definition, the vielbein can't be singular because it's supposed to be inverted all the time. In other words, the vanishing value of the vielbein is infinitely far away on the configuration space. So it would be "infinitely difficult" to reconstruct such a symmetry if it were there. That's just another way of saying that the symmetry would be unphysical.

    Clearly, you just want some "bookkeeping device" that organizes the SO(10) GUT spinors and Lorentz spinors into spinors of a bigger group. But it's a completely flawed program. It is a complete coincidence that the fermions are "spinors" in SO(10) GUTs. In SU(5) GUTs, they're the fundamentals o-plus tensors, 5 and 10, and in E6, they're fundamental 27. Also, other particles such as gauge bosons clearly don't fill any SO(3,11) multiplets. So you're just being misled by the fact that the word "spinor" appears at two places. But the character of the spinors and all other things are fundamentally inequivalent and one can't build any meaningful story just on the word spinor's being at two places.

    If you remember some string theory, there are many ways to see how different the "spinorial" character in spacetime and the internal SO(10) space is. For example, take the E8 x E8 heterotic strings. A single E8 has the 248-dimensional rep, transforming as 120+128 under its SO(16) subgroup. The 128 is the spinor. It's produced e.g. by quantizing the 16 left-moving fermionic degrees of freedom on the world sheet in the periodic-antiperiodic sector, if you adopt the fermionic description of the extra heterotic degrees of freedom (32 extra fermions). And indeed, you may also get spacetime spinors (which appear in the heterotic string, e.g. for the E8 x E8 gauginos) by quantizing the psi^mu spacetime RNS fermions on the world sheet in the periodic sector.

    However, there is a qualitative difference between the 32 heterotic fermions lambda^a and the 10 RNS fermions psi^mu: the latter appear in the world sheet supercurrent - they have world sheet superpartners X^mu - while the former don't. Even more importantly, and this second point is related, the 32 heterotic fermions producing the E8 x E8 group are left-movers while the psi^mu fermions are right-movers (and only right-movers may enter the world sheet supercurrent, which is why it's related to the first difference). There can't be any symmetry that transforms purely left-moving fields into purely right-moving fields. There's no "phase" in which something like that would make sense. The only thing one can try is to "hide this qualitative difference from his own eyesight". But the difference is physical, crucial, and makes any interpretation in terms of SO(3,11) or SO(25,1) invalid.

    This was just an example. In the heterotic case, everything is explicit and one may say exactly what is the big difference between the SO(10) directions - extended to SO(16) directions - and the spacetime directions. They're left-movers and right-movers, respectively. But they're qualitatively different in any description of physics with the same low-energy limit.

    Best wishes

    Wew, not the infinite blog, Lubos! Anyway, crucial errors:

    The SO(1,3) group is the group that rotates the directions of the tangent space at any point

    Noo: it rotates some internal stuff that has in principle nothing to do with spacetime tangent space.

    ...which is locally (at each point) - or globally, assuming a Minkowski background - indistinguishable from a diffeomorphism.

    No again: the local and global Lorentz are very different grops. The local (gauge) lorentz is well separate from diffs.

    The global Lorentz is our experimentally verified group of invariances, but it exists only at low energy, and only in minkowski background of course.

    You can extend a vierbein (4) to a vierzehnbein (14) but with the vierzehnbein, you can't construct any laws of physics (e.g. no Lagrangians respecting the SO(3,11) symmetry) in 4 dimensions. ...
    To construct laws of physics, one needs not only the vielbein but also its inverse

    Oh, nice, here is an other point: in so called first order formalism you do not need to invert the vierbein. Read the paper the lagrangians are there.

    At low energy only, GR needs of course an invertivle 4x4 vierbein, and it's exactly the nonzero part of the 14x4 matrix. This part is the good old vierbein providing the background (e.g. Minkowski).

    String theory (digging my memory) has a completely different setup: spacetime spinors are generated in a different way.. constraints from worldsheet dynamics enter dramatically, and it may well be that its spacetime Lorentz symmetry can not be extended (from SO(9,1)). This may in fact be a limitation of strings. But I have not much more capabilities to discuss strings... it is just a different world.

    It is in fact remarkable that a simple extension of Palatini GR can have an enlarged Lorentz symmetry.


    No, Fabrizio, there exists no regime, not even in principle, in which the internal indices of the vielbein are unrelated to the spacetime directions.

    It's the very point of the vielbein that it provides one with the isomorphism between the two spaces. If it doesn't - e.g. if it is non-invertible or even zero - then it's not a vielbein and it can't be used to construct any physical laws that could be connected with the realistic ones. So there's no way to "unbreak" your (non-existing) SO(3,11) symmetry; the symmetry is pure fantasy i.e. rubbish.

    All your latest confusions have already been debunked and re-explained above and you didn't write anything new besides your repeated old misunderstandings.

    No, Fabrizio, there exists no regime, not even in principle, in which the internal indices of the vielbein are unrelated to the spacetime directions.

    This is exactly what happens in first order formalism. But since I already pointed to respectable books and reviews, I can not make better job here, and this conversation has to come to a stop. Hopefully been useful (also) to others.


    Dear Fabrizio, any respectable review of the Palatini first-order formalism agrees - and, if possible, explicitly states (see below) - that the frame field or tetrad or vielbein defines an isomorphism between the tangent bundle of the spacetime manifold M, TM, and the trivial bundle M x R^d where "d" is the manifold's dimension.

    For example, look at page 1 of this review of Ashtekar's variables, so that you won't criticize me for using proper science such as string theory:

    "Let M be the four-dimensional manifold representing spacetime. We consider (at least locally) a frame field, or tetrad, e to be a vector bundle isomorphism between the tangent bundle TM and the trivial bundle M× R4 (a trivialization): e : TM→M× R4"

    It's a complete misunderstanding that they can be unrelated. You may imagine that these indices are unrelated before you actually start with any physics, but once you do physics, the very purpose of the tetrad is to define the isomorphism between the two spaces you call "unrelated". To incorporate fermions, it's convenient to have an orthonormal basis of the tangent space, but it's still the tangent space that the "internal" indices are labeling.

    Your approach is clearly physically vacuous.

    Another physicist could also identify or merge color indices with Lorentz spinor indices, or flavor indices, or anything like that, and claim that there's an enhanced group SL(36,C) because SL(2,C) of the Lorentz group is extended by 3 colors and 6 flavors. But it's clear that the actual physical theories won't respect the SL(36,C) symmetry, in the very same sense as no realistic theories will respect your hypothetical SO(3,11) symmetry. The kinetic terms only care about the SL(2,C) part of the group, not the SU(6) part or SU(3) part.

    The symmetry is simply not there in the correct laws of physics that can actually reduce to the desired physics and the symmetry just can't be there. You must know very well that you can't write any action or other self-consistent laws of physics that would allow a realistic spacetime and that would have such an SO(3,11) symmetry. So why are you talking about such a symmetry if it is clearly a nonsense?

    One more comment about spontaneous vs explicit symmetry breaking.

    If I have a QCD with 390 colors, the example I used above, I can also add a "Higgs field" h_{cd} where c,d are color indices, and use it to multiply all quark fields - i.e. replace "q_c" by "h_{cd}.q_d". Then c,d may be color indices going from 1 to 390 and if h_{cd} is equal to diag(+1,+1,+1, 0, 0, 0, ..... 0, 0, 0), I get SU(3) QCD back, of course. The 387 wrong colors are decoupled (all their terms are multiplied by zero) and you won't be able to prove whether they exist or not.

    But the field h_{cd} used to "break" the symmetry from SU(390) to SU(3) has no kinetic terms in the action: we say that it is "non-dynamical". It doesn't propagate in spacetime and the physical objects have no control over its values. That's why the "breaking" induced by such a "field" which is not really "field" is not a spontaneous symmetry breaking but an explicit symmetry breaking. Any explicit symmetry breaking may be obtained from a (fictitiously) symmetric theory by adding non-dynamical fields.

    SO(3,11) is no exception. You may invent this (non)symmetry and argue that the physical laws have this symmetry - except that if you want to break it to SO(10) x SO(3,1), you have to use "Higgs fields" that are non-dynamical, which is the reason why the symmetry breaking is explicit and the symmetry was not really there in any regime. On the other hand, just like in the case of SO(3,11), you may equally claim that the physical laws "respect" any symmetry that contains the SO(10) x SO(3,1) subgroup. It will be equally meaningless and physically wrong as your statements.

    I am not saying anything difficult or controversial here.

    Bonny Bonobo alias Brat
    Sorry that was post was rather large and I don't understand where the capitals came from, I need to get a grips with this editor. But the point I was trying to make is there seems to be a lot missing in Wikipedia and I thought you guys might be able to fill in these gaps for people like me who are trying to understand QFT and weak interactions etc.

    Thanks Lubos for your comment in which you say "The article is only about the things that are specific for the weak interaction - and not just some popular presentation of physics - and these issues are inevitably less comprehensible for the non-experts". I know its a big task as I am a non-expert but I do think it would be more comprehensible without these obvious gaps.
    My article about researchers identifying a potential blue green algae cause & L-Serine treatment for Lou Gehrig's ALS, MND, Parkinsons & Alzheimers is at
    Bonny Bonobo alias Brat
    Sorry, I was getting a bit frustrated. I followed your link Lubos and can see that you really have been a big contributor to Wikipedia. Please accept my apologies. I'll keep plodding away.
    My article about researchers identifying a potential blue green algae cause & L-Serine treatment for Lou Gehrig's ALS, MND, Parkinsons & Alzheimers is at
    Dear Helen, you surely don't have to apologize to me! I always happened to agree that such things are important although they're not infinitely important and many people have good reasons why they focus on more technical things. Serious science often has to work without external distractions - although in some cases, this statement becomes an excuse not to help to spread knowledge....

    Despite the significant differences we have in opinions over science and society, Tommaso has also contributed to the proliferation of knowledge although Wikipedia is not necessarily his tool of choice.

    Hi Helen,

    yes, your post was really, really way too long. A link would have sufficed... The problem with such long cut-and-paste style of comments is that the relevant text is lost. I was unable to make sense of your question...

    You are right, wikipedia needs improving. I think Lubos said it well, it is important but nobody has infinite time. I am quite willing to take on specific questions on weak interactions, even the dumbest or smartest ones, and try to explain things in a understandable way in a separate post. Please shoot and I will try to keep the promise.

    Lubos - why on earth does one want to "unify" fermions and bosons ?

    Dear Joel, ...

    we live in one world, and therefore all physical objects and phenomena are consequences of one "set" of laws of Nature. So any dis-unity we observe is really a matter of incomplete understanding. 

    This general statement has been supported by numerous fundamental examples in the history of science - the unification of energy and heat by Joule, the unification of space and time in relativity, the unification of electricity and magnetism, the unification of energy and frequency in quantum mechanics, electromagnetism and weak interaction, perhaps electroweak interactions and the strong ones, probably complete stringy unification of foces. Unification of bosons and fermions is just a part of it.

    Even developments in the 1930s and 1940s are crucial to the unification of bosons and fermions at some general level. Before the 1930s, there would exist completely different formalisms and logical systems for bosons and fermions. Fermions would be understood "just as particles" that happen to have wave properties in quantum mechanics; bosons would be understood as "just fields" (e.g. electromagnetic field). Pauli was the main person who showed that both types are really described as particulate excitations of the same kind of "quantum fields" that only differ by one sign (anti/commutator) for bosons and fermions. The formalism was unified - but the precise dynamical rules for the terms in the Lagrangian etc. were still independent and un-unified. Could the properties of these two types of fields be linked much like electricity is linked to magnetism in relativity?

    Serious physicists didn't really "want" to unify bosons and fermions as an independent goal. The necessity of unification was imposed upon them when they tried to construct string theory that included fermions, see

    That's how any kind of supersymmetry was actually first constructed by Pierre Ramond in the history of Western physics. This construction was generalized from 2 dimensions to higher dimensions, including 4 dimensions (famous model by Wess and Zumino). People started to study it because this fascinating possibility suddenly emerged from the maths, although no one has really predicted it by philosophical arguments. In the Soviet physics, SUSY was mostly discovered as a mathematical curiosity - but on the contrary, the people who did so didn't quite have the same technical appreciation for the realistic phenomenological theories of particle physics. Only when dust settled, people realized that SUSY is both mathematically sound and pretty as well as capable to describe completely realistic models.

    How does SUSY emerge in string theory?

    If the string is allowed to look like a spacetime fermion which transforms as a Lorentz spinor, there have to be world sheet fermions that nontrivially transform under the Lorentz group as well. The only way how they can co-exist with the bosons is if there's another kind of gauge symmetry aside from world sheet diffemorphisms, namely the worldsheet supersymmetry.

    The natural stringy constructions with world sheet supersymmetry also imply spacetime supersymmetry. Supersymmetry is so shockingly nontrivial that it would have been thought impossible  - at least for interacting theories - in the past if you asked someone about the possibility. And no one "invented" it deliberately. But if it can exist - and it can - it is surely a nontrivial player in the unification of reality.

    Cheers, LM

    Well said, Lubos, and a nice endorsement of Quantum Field Theory.   
    we live in one world, and therefore all physical objects and phenomena are consequences of one "set" of laws of Nature. So any dis-unity we observe is really a matter of incomplete understanding. 
    Do bosons ever masquerade as fermions?
    Dear Hank, thanks for your compliment. And interesting experiments (attempting to detect photons that followed the Fermi-Dirac statistics) - although very unlikely ones to produce a surprise, even a priori haha. I think that in 4D, even Lorentz symmetry can be partly relaxed and the spin-statistics theorem by Pauli remains in full force. ...

    But the experimenters wanted to observe something that is even crazier: having the same type of particles, photons, that suddenly acquire antisymmetric wave functions. That's really silly. The statistics is such a defining feature of the electromagnetic field that photons that wouldn't follow it would really have to be "different particle species". So even if they observed the "crazy, shocking" result they wanted to observe, the interpretation wouldn't be in terms of photons that "play by different rules for a while". It would be about a discovery of a new particle species that is sometimes emitted together with photons.

    Hey, don't forget Cooper pairs!!

    Sorry, I don't understand why I shouldn't forget about them in this discussion. ...

    There is nothing special about Cooper pairs. Cooper pairs are 100% bosons, much like any bound state of particles including an even number of elementary fermions. All bound states  with an odd number of elementary fermions are fermions themselves. This rule is completely universal, it is an easy consequence of the addition of spins (or multiplication of the signs from the statistics), and it surely doesn't contradict anything I wrote above, does it?
    If photons could be presented as pairs of 2 particles which are fermions, these fermions could be or would be fermions, but photons would not. This exactly corresponds to the comment in my last sentence that in such a case, a new particle species would be observed.

    There are other, more complex reasons why photons can't be bound states of 2 particles in the same bulk spacetime.

    On the other hand, Cooper pairs "are not" electrons. They're pairs of electrons and pairs of electrons are different objects than individual electrons themselves. You can create Cooper pairs out of electrons. But you can also create Z-bosons out of quark-antiquark pairs. It doesn't mean that they're the same objects. A Z-boson "is not" a quark-antiquark pair. In the same sense, a Cooper pair "is not" just electrons. The change to produce a Cooper pair may look "less qualitative" than the production of a Z-boson but it's still important to know whether you talk about individual electrons or their conglomerates. The spin and statistics surely does depend on the number of fermions inside and I have never indicated otherwise.
    Still, photons can be seen as closed strings, can it? And in some way a closed string is formed by joining two open strings. Should these strings be fermions, or bosons?

    Yes, and with SUPERSYMMETRY as tool. Obviously crackpottish :) :) Pure fantasy?

    Tommaso, why this long thing? I thought the comments should be kept short???

    Dear Alejandro,

    in heterotic string models, photons are closed strings. In braneworlds, (more confusingly) in F-theory, they are open strings. In heterotic M-theory, they're gauge fields on the boundary of the world.

    Open strings can be both bosons and fermions. There are infinitely many states they can be found at. Closed strings can be both bosons and fermions. There are infinitely many states in their Hilbert space, too. One of the bosonic ones is a photon.

    A closed string can be produced by a merger of 2 open strings. It can also be produced in a merger of 3 open strings; 56 open strings; or 1918 open strings plus 2010 closed strings. There is nothing fundamentally special about any interaction. You surely can't say that a closed string is the same as 2 open strings, much like it's not the same thing as 1918 open strings plus 2010 closed strings.

    (There is a useful isomorphism between a closed string Hilbert space and the tensor product of two open string Hilbert space - because left-movers and right-movers each behave as an open string Hilbert space. But this isomorphism is clearly much more complex than the thing you asked about.)

    The only constraints in such reactions are conservation laws. Energy conservation restricts the masses of the particles you may create in a collision. Charges have to be conserved. The spin and/or statistics conservation implies, among other things, that a single particle produced in a collision of particles or strings with K fermions among them will be a boson if K is even and a fermion if K is odd.

    You seem to dream about an answer that open strings are always "particles of F type" and the only thing they can do is to create a closed string which is always a photon. But that's not the case. There are infinitely many species of open strings, infinitely^2 species of a closed strings, and infinitely^infinitely many types of reactions how these things may react and produce any final state from an initial state - as long as the conservation laws are respected.

    Cheers, LM

    Hey Lubos, I sort of understood your last comment... Either I or you are making progress ;-)

    "Hey Lubos, I sort of understood your last comment... Either I or you are making progress ;-)"

    And with some extra progress, you can surely also figure out which answer is correct. At any rate, congratulations! Time to open champagne for you, me, and Eric. ;-)

    One more comment about the Cooper pairs: they look more like a composite of 2 electrons because the electrons are very weakly coupled: they're bound by a phonon-induced interaction. The stronger the coupling is, the less "clear" it is how to divide a composite object to the elementary building blocks.

    I can't post with my account now - it says that the field is empty although it's not. I had to post the longish comment above twice.

    Hey Lubos, I sort of understood your last comment... Either I or you are making progress ;-)


    I can't believe what I'm reading! Did Tommaso actually say that? This warrants a celebration! Let's pop those corks from those champagne bottles! lol ;-)
    Lubos .. Thanks for that extended reply and the link. Just wondering if you have a favorite approach in case supersymmetry is not the way Nature goes ? For example, if the Standard Model might be "fixed" in a manner where the particles can be explained, and is already "unified" in its own way - like something underneath the Standard Model - more fundamental - from which one can construct Dirac Algebra, which after all is just direct products of quaternions. Something Minimalist -a strategy that often seems to work well in Physics.

    Dear Joel, as Einstein said, theories should be as simple as possible but not simpler. The kind of ideas you propose seem to be simpler than possible.

    Believe me, I have studied the very question you asked now for roughly 20 years, often from the most well-known towers of the world, and the answer is that the only "minimalist" condensation of the Standard Model that work and that are known are linked to grand unification (unif. of nongravitational forces and multiplets of fermions), supersymmetry (the only unification of bosons of fermions), and string theory (the only big framework that includes gravity and QFTs).

    In particular, comments such as "Dirac algebras are direct products of quaternions" are not helpful in any sense. Fundamentally speaking, they're not true and the repetition by the likes of John Baez can't change this fact.

    The quaternions are a very specific algebraic structure with non-commutative multiplication and SO(3) isometry; they can be inserted to matrices to produce USp(2N) symplectic groups. But the multiplication has to play some physical role for us to say that the quaternions actually appeared in science; the multiplication rules for i,j,k can't be modified, either. In this sense, the Dirac algebras are only relevant for quaternions if the representations are pseudoreal - and this kind of maths has been understood for quite some time. It is unreasonable to expect any substantial progress coming just from statements that "there are quaternions over there".

    The kind of science where one randomly identifies a mathematical structure with an aspect of observational reality and spends the rest of his life by constructing proofs that it has to be the case is what I call crackpot science, by definition. Real science appreciates that most such identifications are wrong. And indeed, it abandons them if they don't work. Only those that work - have some nontrivial evidence supporting them, and are consistent with the observations are the appropriate level of accuracy - may be investigated further.

    LM said:
    "The kind of science where one randomly identifies a mathematical structure with an aspect of observational reality and spends the rest of his life by constructing proofs that it has to be the case is what I call crackpot science, by definition."
    So is this a definition for string theory? Exactly what connection to the real world does ST have? Cheers.

    Aggressive idiot,

    string theory agrees with all qualitative and many quantitative properties of the real world which includes the validity of quantum field theory, fermionic and bosonic excitations with spin 0, 1/2, and 1, Yang-Mills theories that can be broken or confined, renormalization group effects, the existence of gravity described by general relativity, organization of matter species into families. In this sense, it is a complete theory of everything.

    By the way, concerning my phrase, string theory is the best example in all of science where the warning is not satisfied. String theory was not constructed to describe ambitious physics at all. It was born as a technical attempt to describe a very partial feature of the strong nuclear interactions. It was accidentally discovered 5 years later that string theory also inevitably includes spin-2 massless particles that inevitably happen to interact as gravitons from quantized general relativity, and, later, that it includes everything else to agree with the real world as well.

    No one wanted this in advance. These facts were forced upon the scientists when they calculated things more closely. So in this sense, string theory is the best example of un-prejudices science that the history of science can offer us, and I insist that whoever tries to deny this self-evident fact is an imbecile.



    There exists an alternative to string theory. I do not mean LQG or the likes. It is possible to use in Hilbert space operators with eigenvalues that have a higher number field dimension than the number field over which the inner product is defined. Thus instead of quaternions one could use octonions or 16-ons (which are not sedenions!). The 2n-ons have the nice property that they include the 2m-ons in their lower m dimensions. This makes the arguments of extended unitary operators to storage places for the actions of multiple fields. The eigenfunctions of such operators can than form the base of the micro dynamics that takes place inside particles. In macro dynamic sense this approach allows curvature in the manipulation of items. This is the source of general relativity and electro-magnetism. The influence of this manipulation on observations becomes apparent for dimensions n>1. Thus starting from the quaternions. On micro scale it reveals the reason of the existence if special relativity. It also clears the relation between observed space and several notions of time.

    this is all treated in If you find it all gibberish then I like to hear it.


    If you think, think twice
    That E8 heterotic string theory and Lisis E8, can you shortly explain the difference according to the dimensions, math etc. so that a non-mathematician can understand. Also with relation to octonions. Thanks.

    Dear Ulla, there is exactly one similarity: E8, the largest exceptional Lie (continuous) group - appears in both cases - and an E6/SO(10)/SU(5) subgroup of this E8 is used in the same way as in grand unified theories from the 1970s.

    There is no direct relation to octonions although E8 has many substructures linked to octonions - for example, the group of octonions' automorphisms, G2, is contained within E8, and its centralized in E8 is F4 which can be interpreted as a group of symmetric 3x3 Hermitean octonionic matrices (with a symmetrized product) of some kind.

    The rest of the text are differences:

    1) E8 x E8 heterotic string was discovered in 1985; Lisi's E8 theory was invented in 2007. The word "discover" refers to something that actually exists in Nature; "invents" means that people constructed it from arbitrary pieces.
    2) The heterotic string works, Lisi's theory doesn't. This will be discussed in detail below.
    3) The particle spectrum of the heterotic string - or any other string - is infinite and contains arbitrarily massive particles; Lisi only has an old-fashioned theory with a finite number of species
    4) Heterotic strings propagate in 9+1 large dimensions; Lisi in 3+1 dimensions.
    5) The E8 x E8 group may be derived from an algebra of the remaining 16 dimensions of the heterotic string (the total is 26 dimensions), it arises from a 16-dimensional torus and only 2 shapes of the torus, and therefore 2 groups - SO(32) and E8 x E8 - are possible; Lisi's E8 cannot be derived from anything, it is just postulated because the author had seen other contexts where E8 is useful such as the heterotic string.
    6) The heterotic string has spacetime supersymmetry which is the only valid way to unify bosonic and fermionic multiplets; Lisi has no spacetime supersymmetry, so all his statements about the unification of bosons and fermions are bogus.
    7) There are two E8 groups in the heterotic string - in the M-theory picture, each E8 lives on one of two boundaries of a "layer" of 11-dimensional spacetime (the Standard Model is fully embedded in one of them, the other is a hidden sector); there is only one E8 in Lisi's theory
    8) The heterotic string may produce the right spectrum of fermions (representations, spins etc), including multiple generations, when compactified on a Calabi-Yau space; Lisi can't produce generations and it also can't produce chiral (left-right asymmetric) fermions, among many other fundamental features of the fermionic spectrum of the Standard Model
    9) The heterotic string theory includes gravity and a graviton which is obtained as an excitation of a string, just like all other particles, but it is an excitation that isn't contained in the E8-excitations; Lisi incorrectly believes that the graviton itself may be a part of the gauge multiplet (but diffeomorphisms are surely not a special case of Yang-Mills symmetries, and he can never get the diff symmetry)
    10) The heterotic string allows us to calculate all the scattering amplitudes of all particles at any energy, and their low-energy limit agrees with general relativity and the Standard Model when the Calabi-Yau space is properly chosen; Lisi's theory doesn't allow us to calculate anything that goes beyond the known results of the Standard Model and general relativity, and it seems obvious that it can't agree with any of these approximate theories, anyway
    11) String theory, including heterotic strings, actually solves all the known problems of quantized general relativity such as the previously non-renormalizable divergences and previous information loss; Lisi's theory has nothing to say about quantum gravity whatsoever, and in fact, none of its features has been ever tested in the context of quantum theory (the tests would fail)
    12) Heterotic strings are serious science first written down by 4 serious physicists, including one Nobel prize winner; Lisi's theory is just a meme promoted by people who hate maths and physics but like the idea that someone tells them that they may learn how the fundamental physics works by surfing the ocean and knowing no maths.

    About producing generations, can a gauge theory produce generations out of its group?. Generation symmetry does not commute with poincare symmetry, does it? MNS and CKM tell us that we need rotate the eigenvectors of mass to produce the eigenvectors of families.
    So in the field theory limit, I would expect that any generation symmetry produced by string theory should "ungauge", producing instead a global symmetry, flavour.

    Thanks, it was fair of you although we have our 'battles'. In fact Lisis presentation of the E8 is impressive, but I cannot understand how it could work, maybe as some picture at a much higher level of complexity?

    Can you then expand your explanations to also include the theories of TGD and T. Banks? You know Banks. The theories are very much alike. In what way do E8 heterotic strings differ from their math?

    There are hierarchy instead of hidden dimensions?

    Dear Ulla, apologies, but I don't see any similarities between Tom Banks's theories and TGD, or any of these and E8 x E8 heterotic strings.

    Lubos - Atiyah, Bott & Shapiro "Clifford Modules" classifies Clifford Algebras and they are indeed direct products of quaternions (split-Q and division-H) . I am not theorizing - just asking where Brauer & Weyl get those quaternions from (which are "associated with" electrons) to build up n-electron atomic configurations, and also one might ask where Feynman gets a complex phase "associated with" particles. One might expect quanta to be defined by an algebra that contains both - complex octonions would be a logical candidate. There are lots of ways to generate complex octonions, providing lots of distinct quantum oscillators - one can even get something that looks quite "generational" with +++, +--, -+-, --+ in the even subalgebra (in 16 component frames). If that +--, -+-, --+ business is even remotely related to generations of fermions, that would argue for it being theoretically interesting. I gather from Feynman's QED that Dirac provides the polarizations. Somehow one needs to put the octonion oscillator into space so if we consider two oscillators we pluck out a quaternion from each and make a Clifford Algebra which can carry the dynamical 4-vectors. In other words - octonions do not define a particle 'in space' but only just "consistent with space". Once you take the direct product of quaternions, it forgets the octonion details and is like points in space - which is what physics is always dealing with. A muon looks like an electron in this Clifford Environment, but Octonions can distinguish them. That might explain why fermion generations are such a mystery.
    I would not attempt to glorify this hand-waving by calling it a theory. I am just philosophising about how QED and the Standard Model might make deeper algebraic and octonionic sense. Just to adhere to Brauer & Weyl, I am ignoring bosons, which seems to violate the spirit of Pauli treating them on the same footing, as you mentioned above. It does however raise an interesting question about whether the fundamental algebra ought to be Alternative. If so then one is stuck with complex octonions and one could rule out OxHxC (G.Dixon), or putting octonions in matrices to make supersymmetric algebra. I have no idea of whether Alternativity is a criterion for the Design of the World, so I can not rule out supersymmetry. Enuf handwaving ! But I would say that Octonions are quite "unified" all by themselves. If it defines quanta then it is 'already quantized', and the signature is all Minkowski all the time, so there are no 'extra dimensions' - which one might consider a highly desirable feature.
    It makes more sense if the 16 component complex octonions are 'tetrahedral' rather than vectorial - then it is consistent with 3-space, though it does not define a solid body as that would violate Lorentz. The natural oscillator phase is cos(theta) + (oabc) sin( theta), where (oabc) is any permutation-association of {o,a,b,c} for 120 possibilities and double that for signature +--- to -+++, so these are all duality oscillations, and all like complex numbers. A simple oscilator being like 1 + o + a(bc) + o(a(bc)) - the rest of the components being zero. One might imagine dragging this thing around the possible paths in Feynman's analysis of the Double Slit and doing complex arithmetic to get the interference pattern, but to get real QED one needs to justify Dirac algebra. I can't tell whether it is a crackpot idea yet - your warning is well taken, and thanks for taking the time.

    Of course we have octonions and the like. But it happens the same than with E8: they appear after the fixing of the number of dimensions, and as a consequence of this fixing. In the case of octonions, it is even pretty obvious, a 11 dimensional space must be decomposed in 7+4 or 4+7, and the most symmetric 7 dimensional manifold is the 7-sphere, bringing in all the Hopf fibrating parafernalia.

    It could be better if there were some first-principles way to these structures (to E8, to Octonions, to Quaternions, etc) but it is not the way as we are getting them, they come as consequences. Even the division algebras (Evans etc) do appear after requering supersymmetry, so as a byproduct of supersymmetry, and not as a justification of it.

    The only exploit, to make some "postulate" out of all the [clifford-octionion-quaternion-division-whatever] algebraic panorama could be the counting of degrees of freedom used to stablish the brane-scan.

    I first read about Octonions in A.A.Albert "Studies in Modern Algebra" (MAA) in the context of Composition Algebras - reals, complex, quaternions and octonions. One sees the antisymmetric version of Peano's axioms for Ordinary Arithmetic. There is no necessity to think of division algebras as a byproduct of supersymmetry. Some say that as we 'double' the complex algebra to get quaternions, we "lose" commutativity - I prefer to think we gain anti-commutativity, and doubling again we gain anti-associativity. After that we lose Alternativity and there are no more Normed Division Algebras. Since Physics is beholden to ordinary arithmetic, it is a mystery to me that anybody can ignore Octonions. It is also a mystery why so many refer to octonions as 8 dimensional when Graves first wrote them as 1 + a + b + c + ab + bc + ca + a(bc) . It looks sort of like a 3 dimensional multivector (with 8 components)
    or maybe triangular - for vertices rather than vectors. Clifford Algebras are similarly elementary. One could take any number of associative but anticommuting generators with any signature and expand it combinatorially and get a Clifford Algebra - if there are an even number of generators it is algebraically isomorphic to a direct product of quaternions - split and division in general. There is a neat connection to Slater Determinants but I dont have a ref handy.
    It is interesting that one may fiddle with the signature and number of generators by hand. It seems handy for N-Body rather than N-dimensions, and does not give us the structure of the world. For a highly compressed discussion of the theory of clifford algebras see A.A.Albert: Math Reviews Vol 10 (1949) p 180. I have it taped to the inside cover of Hestenes SpaceTime Algebra.
    Baez's paper "The Octonions" discusses the deal with E8 and Octonions.
    Howard Georgi discusses Octonions and their automorphism group containing SU(3) - in Lie Algebras for Physicsts.
    Frankly, though, I worry about trying to stuff Octonions into the context of quantum theory, in the fashion of Quaternionic QM. IMO one should wonder why Hamilton rejected Octonions. After all, if quaternions are so geometrical, one merely asks what is missing and the common sense response is "matter". Of course, in 1843 nobody knew squat about atoms and particles, so Hamilton's judgement is vindicated on practical grounds, but I think he should have left a bit of room - after all, one does wonder where that 'm' comes from in F = ma.

    See for instance

    For similarities between TGD and Banks: The 'only' difference is in the supersymmetry, but the octonions are there? Another striking similarity if you look at the references. - almost only to himself :)
    T. Banks 2010. Holographic space-time, cosmological SUSY breaking, and particle phenomenology
    String theory models are our only rigorously established models of quantum gravity, but none of the known models apply to the real world. They do not incorporate cosmology, and they do not explain the breaking of supersymmetry (SUSY) that we observe (not to talk of the biology-my comm.). an attempt to generalize string theory in order to resolve these problems. Its basic premise is a strong form of the holographic principle, formulated by myself and W. Fischler:
    - Each causal diamond in a d dimensional Lorentzian space-time has a maximal area space-like d − 2 surface in a foliation - holographic screen in Planck units. Every pair of causal diamonds has a maximal area causal diamond in their intersection. A holographic space-time is defined by starting from a d−1 dimensional spatial lattice,which specifies the topology of a particular space-like slice. In a time symmetric spacetime we think of the diamonds as having past and future tips (this is a zero energy ontology - my comm.).in Big Bang space time the sequence of unitaries U(n) may be thought of as a conventional time dependent Hamiltonian system with a discrete
    time, while for a time symmetric space-time they are instead “approximate S-matrices”
    - Although we have used geometrical pictures to motivate our constructions, they are entirely phrased in quantum mechanical language. The Lorentzian space-time is an emergent property of these quantum systems, useful in the limit of large causal diamonds
    - The lattice specifies only the topology of a space-like slice in the non-compact dimensions This topology does not change with time.
    information is encoded in the Cartan-Penrose equation
    One can argue that the states with all the momentum carried by one “particle” should actually be thought of as black holes that fill the causal diamond. the Dense Black Hole Fluid model of holographic cosmology
    - Quantum particles execute random walks in proper time. If we take the step size to be Planck scale, the area covered will also scale
    - The vanishing of one pair of the y couplings is introduced in order to have a dark matter candidate.
    - There is, as yet, no general prescription for calculating the scattering matrix of this super-Poincare invariant theory. (In TGD is primes used-my comment)
    T. Banks, “Cosmological breaking of supersymmetry or little Lambda goes back to the future. II,” arXiv:hep-th/0007146.
    that the observed SUSY breaking in the low energy world must be attributed to the existence of a nonzero cosmological constant. - took on the value 1/8, rather than its classical value 1/4. We attribute this large renormalization to the effect of large virtual black holes via the UV/IR correspondence. This implies that the cosmological constant is an input to the theory, rather than a quantity to be calculated.- The implication of these ideas is that SUSY breaking vanishes in the flat space limit, which is consistent with the fact that we have not succeeded in finding a string vacuum with broken SUSY and asymptotically flat spacetime. - On the other hand, even if one succeeded in finding a SUSY breaking vacuum which precisely describes the real world (we should be so lucky!) one would still have the uncomfortable task of explaining why the universe does not resemble one of the beautifully SUSic vacua. - one might try to find a nonsupersymmetric version of this correspondence (with
    an AdS space with curvature much less than the Planck scale) by searching for conformal field theories with certain properties. - I would like to suggest that we have been thinking about this problem the wrong way around. The flat space computation counts the zero point energy of the degrees of freedom in spacetime. One aspect of the subject that I find rather confusing is the relation of the fundamental theory to the low energy effective Lagrangian. I believe it is correct that physics below the Planck scale is governed by a locally supersymmetric effective Lagrangian. Local SUSY connected to the arbitrary choice of holographic screen, and should therefore be a fundamental symmetry, not to be broken. Since we expect the scale of SUSY breaking to be much smaller than the Planck scale there should be an effective Lagrangian description of low energy physics which is locally supersymmetric, which means that SUSY breaking appears spontaneously. The SUSY breaking scale and cosmological constant should simply be set by tuning parameters in this Lagrangian.
    T. Banks and L. Motl, “Heterotic strings from matrices,” JHEP 9712, 004 (1997) [arXiv:hep-th/9703218]
    T. Banks and L. Motl, “A nonsupersymmetric matrix orbifold,” JHEP 0003, 027 (2000) [arXiv:hep-th/9910164 Yes, the calculation above does not really agree with the usual supergravity calculus but I think that we don't know whether the calculus is correct after SUSY breaking anyway.

    Sorry for this long thing, but I think Lubos would know.

    Quaternions definitely play an important role in physics due to the existence of the quaternion waltz (c=ab/a). This number waltz does not get noticed in complex quantum mechanics. It comes forward when a unitary transform acts on an observation. When the microscopic effects are analyzed, then it becomes clear that the waltz is the source of special relativity. Via the waltz the quaternions cause things that have a Minkowski metric and that are no longer quaternions. These items are better treated with Clifford algebras. For more details see:
    If you think, think twice
    I couldn't open the link.


    I tried the link myself before publishing the comment. <?xml:namespace prefix = o ns = "urn:schemas-microsoft-com:office:office" />

    Please go to and go to the section English. Then move down to Science and select "Example quantum logical proposition".
    "How the world works" is a tale that tries to explain the stuff to people that hate formulas.
    Sorry, for the inconvenience. Hans

    If you think, think twice
    Bonny Bonobo alias Brat
    I was a bit skeptical about the 'truthiness' part of this paper until I looked it up on the Webster's dictionary - 1. truthiness (noun) See 1 : "truth that comes from the gut, not books" (Stephen Colbert, Comedy Central's "The Colbert Report," October 2005) 2 : "the quality of preferring concepts or facts one wishes to be true, rather than concepts or facts known to be true" (American Dialect Society, January 2006) Quite a useful word!
    My article about researchers identifying a potential blue green algae cause & L-Serine treatment for Lou Gehrig's ALS, MND, Parkinsons & Alzheimers is at
    My native language is Dutch, not English. Sometimes I write in double Dutch.
    If you think, think twice
    Bonny Bonobo alias Brat
    Sorry Hans, did you mean to write truthfulness and not truthiness in your paper?
    My article about researchers identifying a potential blue green algae cause & L-Serine treatment for Lou Gehrig's ALS, MND, Parkinsons & Alzheimers is at
    I think that that is what I thought and not wrote.
    If you think, think twice
    Bonny Bonobo alias Brat
    Hans, I can't pretend to understand your paper. I am new to Quantum Mechanics, Quantum Field Theory, weak interactions, boson and fermoins and the Standard Model. Hopefully one day I will be able to make an intelligent comment in this area. I have worked for many years as a systems analyst/computer programmer and IT Project Manager and I have always been good at logic and maths, but I have to admit I am finding this field enormous (for something so small) and very challenging. However, I keep reading and reading and bit by bit things are starting to fall into place. The areas I find really interesting are the uncertainty principle, the observer effect and wave particle dualism. As an ex IT professional I am also really interested in the Higgs Boson detection component of QFT at CERN. How can a photon or a speeding electron exhibit both particle and wave characteristics? What is the physical basis of a “probability wave?” Why is the velocity of light a constant for all observers? Its like walking through a minefield just hoping for the best, with only an ontology map, wikipedia and the Penguin dictionary to help me, but that is what I am doing and it is even quite enjoyable. I should also thank the contributors here at SB for their wonderful attempts to explain these concepts to people like me. You never know, one day I might get there.
    My article about researchers identifying a potential blue green algae cause & L-Serine treatment for Lou Gehrig's ALS, MND, Parkinsons & Alzheimers is at
    Helen - I was doing C programming until recently. To your question about particle-wave, the best reference I have seen is Feynman's "QED: the strange theory of light and matter" - esp the latest edition with an intro by A. Zee. You might consider that in actual fact, particles are NOT points we push around in space in the classical manner, but rather are oscillators with a complex phase, and this has vast implications. The gist of it can be seen in Feynman's explanation of the Double Slit experiment. One adds the complex numbers, and this produces "wave interference" and then we take the complex norm to get the Probablity. For the 'heavy' version see Feynman's "Quantum Electro Dynamics" 1962. Unfortunately the former book does not get into the Hydrogen atom - i am curious whether one of the implications would be that electrons want to be in an orbit with a whole number of oscillations.

    as to the constant velocity of light - David Hestenes "SpaceTime Algebra" is useful. There are entire websites devoted to Geometric Algebra. It is a property of Minkowski spacetime and rotations of 4-vectors in spacetime.

    You put many questions in a row. I will try to answer a few. May be it is worthwhile to read the tale "How the world works"
    My conviction is that quantum logic is at the base of physics. Quantum logic is close to classical logic. These logics only differ in one axiom. However, that difference has deep consequences for the structure of quantum logic. Classical logic has the same structure as the set of Venn diagrams. This is a rather simple structure. The set of propositions in quantum logic has the same (lattice) structure as the set of closed subspaces of an infinite dimensional separatable Hilbert space. Thus, the Hilbert space is a great mathematical environment to test quantum logical propositions by implementing them in Hilbert space. When I do this I try to use the utmost freedom that number systems allow. In that way I can analyze very sophisticated dynamics in Hilbert space.
    In Hilbert space it is possible to define a canonical conjugate operator with each normal operator that acts as a continuous observable. This can easily be done by defining the inner product of their eigenfunctions with a continuous function that is the exponential of the scalar product of the eigenvalues multiplied with an imaginary number and with a constant that represents the granularity of the set of eigenvalues. This measure not only specifies the canonical conjugate it also specifies a Fourier transform that relates the two operators. The observable that delivers position information represents the particle view of the item for which the operator delivers this characteristic. The canonical conjugate of this operator is the momentum operator. It does not commute with the position operator. Thus, both operators cannot at the same time give conclusive eigenvalues as characteristics for the considered item. This is the reason for the Heisenberg uncertainty relation. Where the position operator characterizes the particle view of the item, the momentum operator rather characterizes the wave view of that same item. The Fourier transform converts one view in the other. That Fourier transforms are important actors in nature can be seen from optics. Lenses are in fact Fourier transformers. Wave mechanics is optics taken one dimension higher. In this sense the hole in the wall of the camera obscure corresponds with the holographic screen that surrounds a black hole.
    The Fourier tranform is the cause of the uncertainty. The granularity of the set of eigenvalues is a closely related aspect. It is due to the fact that the dimension of the Hilbert space is infinite but at the same time countable. Thus the cardinality of the set is equal to the cardinality of the natural numbers. The cardinality of the reals is one level higher. The constant that relates to the granularity of the set of eigenvalues is Planck's constant.
    If you think, think twice
    Bonny Bonobo alias Brat
    What I don't understand is why do so many people have a problem understanding that the universe is infinite in every direction? Do they think that you can travel to the end of the universe and find a wall with a sign that says 'the universe ends here'? If it did then what would be on the other side? Even if its a vacuum it is still more space. If the universe is not curved but has a flat topology, it could be both unbounded and infinite. According to wikipedia (my bible) quote "The curvature of the universe can be measured through multipole moments in the spectrum of the Cosmic Background Radiation. As to date, analysis of the radiation patterns recorded by the WMAP spacecraft hints that the universe has a flat topology. This would be consistent with an infinite physical universe. The Planck spacecraft launched in 2009 is expected to record the Cosmic Background Radiation with ten times higher precision, and will give more insight into the question whether the universe is infinite or not." My money is on it being infinite which means that the 'Big Bang' theory and everything that goes with it, is wrong.
    My article about researchers identifying a potential blue green algae cause & L-Serine treatment for Lou Gehrig's ALS, MND, Parkinsons & Alzheimers is at
    Hi Helen. This statement is wrong:

    "My money is on it being infinite which means that the 'Big Bang' theory and everything that goes with it, is wrong."

    There is no contradiction between the big bang theory and an infinite universe. An infinite universe can still expand. We even have a toy mathematical model for this, in the form of an exact solution to Einstein's equations, called the Friedman-Robertson-Walker (or sometime Friedman-Lemaitre-Robertson-Walker) universe. Let me also clarify a common misunderstanding about the "big bang", while I'm at it. I think the best way to state the big bang theory is this:

    The universe was once very hot and very dense, and has since expanded and cooled.

    The evidence for this statement is varied and overwhelming. Any talk of singularities, the beginning of time, and so on, which often goes along with discussions of the big bang, is more or less nonsense, and actually misses the point.

    If the items in universe can be represented in Hilbert space, then universe can be infinite and still be countable. It can be densily covered with items and can still hold an infinite set of empty places. All base vectors of Hilbert space end on a unit sphere, the surface of an infinite dimensional ball. This surface can be mapped on a unit sphere in a finite dimensional number field. In this way the Hilbert space can obtain numerical (spherical) coordinates, like we have on earth.
    If you think about infinity, you must also think about the cardinality of that infinity. The universe must at least have the cadinality of the natural numbers. The integers and the rational numbers have that same cardinality. The real's, the complex numbers, the quaternions, the octonions and all 2n-ons have a cardinality which is one step higher than that of the natural numbers. The Hilbert space does not allow that cardinality for its base vectors. However it does allow it for the number field that is used to define its inner product. This also hold for quantum logical propositions. The set of quantum logical propositions is countable. That restriction does not hold for the set of sets of propostions.
    If you think, think twice
    Bonny Bonobo alias Brat
    No, I don't agree that it has 'the surface of an infinite dimensional ball'. That is your way of not understanding infinity. A dog will never be able to read and some people will never be able to understand infinity.
    My article about researchers identifying a potential blue green algae cause & L-Serine treatment for Lou Gehrig's ALS, MND, Parkinsons & Alzheimers is at
    Bonobos of course, will always be able to understand infinity.

    Please read It is a tale. At least it gives an idea how things are related.

    If you think, think twice
    Bonny Bonobo alias Brat
    Thanks Hans, I have read your suggestion and I would like to offer an alternative Interpretation. The book of laws contains a number of axioms that define the structure of quantum logic as an orthomodular lattice. This could otherwise be known as the Bible. Hilbert's bush stands for an infinite dimensional separable Hilbert space, that is defined over the number field of the quaternions. The set of the closed subspaces of the Hilbert space has the same lattice structure as quantum logic. This could possibly be Moses' flaming bush. The green poles represent an orthonormal base consisting of eigenvectors of the normal operator Q. This operator represents an observable quantity, which indicates the location of the state in space. This could possibly be the sheep in the good shepherd parable, The red poles represent an orthonormal base consisting of eigenvectors of the normal operator P. This operator is the canonical conjugate of Q and represents an observable quantity, which indicates the momentum of the state. This could possibly be the goats in the good shepherd parable. The bundle of silver white poles and the herd of sheep represent a closed subspace of the Hilbert space that on its turn represents a particular quantum logical statement. This statement concerns a particle or a wave packet in our surroundings. Q describes the thing as a particle. P describes the thing as a wave packet. This could possibly be the talents in the prodigal sons parable. The shepherd represents a complicated unitary operator Ut. that pushes the subspace, which is represented by his herd, around in the Hilbert space. This unitary operator is a complicated quaternionic Fourier transform. The operator Ut may be seen as a trail of infinitesimal unitary operators. It is a function of the trail progression parameter t. This is possibly what man calls God. The smells correspond to fields. The fields transport information about the conserved quantities that characterize the movements of the state and its elements. Each type of preserved quantity has its own field type. The operator Ut reacts on these fields. The smell is possibly the observor's reality. The operator Ut transforms the observation operators Q and P into respectively Qt = Ut-1•Q•Ut and Pt = Ut-1•P•Ut This distorts the correct observation and ensures that the observer experiences a speed maximum and a curved space. This is possibly Einstein’s Theory of relativity and QFT. The eigenvalues of Q and P and the trail progression parameter t characterize the space-time in our live space. This could possibly represent the current search for the Higgs Boson. De eigenfunctions of Ut control the (harmonic) internal movements of the particles. This is possibly the Large Hadron Particle Accelerator and all of its many operational appendages..
    My article about researchers identifying a potential blue green algae cause & L-Serine treatment for Lou Gehrig's ALS, MND, Parkinsons & Alzheimers is at
    I give you another consideration. During my career I worked extensively with Fourier transforms. For example I wrote toghether with Wolfgang Wittenstein most of the standard for measuring the Optical Transfer Function of imaging equipment.

    Fourier transforms have a very important property. They transform something that is small, i.e. something that is concentrated in a small location, into something that is widespread. and visa versa. If you apply this to the universe then an individual becomes something that is overal present in the 'other' space. The universe becomes an individual in the 'other'space.
    The universe as an individual can be seen as the creator. The creator of itself. This creator controls everything that happens in universe.
    When we die, our soul is spread all over the place and becomes an individual in the 'other'space.

    So, now everything has its place and you can start thinking of the consequences.

    If you think, think twice
    Or “Is it possible that a person's observations, actions, or even thoughts, all of which involve quantum-entangled matter, are influencing and influenced by those of others somewhere else in the universe through a quantum communication mechanism?”

    Read more at Suite101: Uncertainty and Quantum Entanglement: Properties of Elementary Particles and the Nature of Reality at

    Holism and zero energy ontology :)

    The wider the presence in both "spaces" the more chance there is for information transfer. Thus interaction with individuals in energy-momentum space is possible when they are not too much concentrated in a very tiny "location". Similarly if you give up your individuality a bit, then there is a chance that you get info from other places and other instances of times.

    You must also learn to interpret that info.

    If you think, think twice
    Bonny Bonobo alias Brat
    Well I'm not very good at giving up my individuality and as I mentioned earlier I am way out of my depth here, but one day I hope to be able to make some intelligent comments (I was only joking about the goats and the sheep parable) but thanks for all your advice Hans.
    My article about researchers identifying a potential blue green algae cause & L-Serine treatment for Lou Gehrig's ALS, MND, Parkinsons & Alzheimers is at
    What's exciting to me is that Lisi takes an approach similar to Einstein's - he starts outside the math box and uses a visualization to propel his own thought experiments. Since the amount of brain space that's given over to visual processing is vastly larger than that devoted to math-like reasoning, it represents an "ergonomic" approach to advanced physics. I'm reminded of the work that Buckminster-Fuller did with various multi-planar solids in the 1940s and 1950s. At that time Fuller joked that with his system, quantum mechanics could be taught in first grade. For some of us, like the dweeb with his face on the dinner plate, hiding behind the complex math allows us to feel superior. Certainty is a condition of lesser minds. Feynman used a similar, visual, approach in creating his diagrams, which seem ridiculously obvious to us all these years later. And he's not turning in his grave (but the 2 pi comment was LOL hilarious), he's grooving.