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June 19, 2009

Synchronicity

I was visiting the University of Michigan earlier this Spring, where Gordy Kane told me a story. He’d recently given a public lecture, and was somewhat taken aback when, during the question period afterwards, he was asked about the status of Garrett Lisi’s Theory of Everything. The questioner was convinced that Lisi was a key player in the unification 'biz, and was surprised that his theory had not received more attention (which is to say, had not been mentioned at all) in Gordy’s talk.

The very same day, I received an email from a mathematician working in Representation Theory. He was disgruntled that his student was being asked about Lisi’s work in the course of job interviews. He knew I had some blog posts about Lisi’s “theory”, but had I written these up anywhere? I responded that I didn’t see how it could possibly be worthwhile to publish a “refutation” of an unpublished work. And, in any case, what I had done was of such a mind-numbingly trivial nature that no respectable journal (in either Math or Physics) would consent to publish it. But Skip insisted that it would be helpful to someone like his student to be able to cite a paper in which this stuff had been debunked.

By coincidence, an exchange of comments, with the man himself, at the at the n-Category Café, convinced me that my blog posts have been less than efficacious.

So I decided to take Skip up on his suggestion and try to distil the arguments of the aforementioned blog posts and strengthen them into a theorem that some (not necessarily self-respecting) Math journal might publish.

The story can be summarized as follows. Consider any embedding of Spin(3,1) 0=SL(2,)Spin(3,1)_0= SL(2,\mathbb{C}) in a noncompact form of E 8E_8. Let GG be the centralizer of SL(2,)SL(2,\mathbb{C}) in E 8E_8, and HH be the maximal compact subgroup of GG. We decompose the 248 into representations of SL(2,)HSL(2,\mathbb{C})\cdot H. Irreducible representations of SL(2,C)SL(2,C) are labeled by a pair of positive integers, (m,n)(m,n). m+n=evenm+n=\text{even} lift to representations of SO(3,1) 0SO(3,1)_0. m+n=oddm+n=\text{odd} are “spinorial representations. In this notation, (m,n)=(n,m)¯(m,n) = \overline{(n,m)}. Lisi’s “idea” is that the 248 generators of E 8E_8 should correspond to either fermi fields (transforming in the given representation of HH), or to bosonic fields, which are 1-forms (again, transforming in the given representation of SL(2,)HSL(2,\mathbb{C})\cdot H). By the Spin-Statistics Theorem, m+n=oddm+n=\text{odd} are fermions and m+n=evenm+n=\text{even} are bosons. Any other assignment leads to a loss of unitarity. If we put an upper bound on the values of (m,n)(m,n) which appear in the decomposition of the 248, then the list of inequivalent embeddings is remarkably short. Imposing m+n<5m+n \lt 5, so that only spin-1/2 fermions appear, we can write 248=(2,1)R(1,2)R¯"bosons" 248 = (2,1) \otimes R \oplus (1,2)\otimes \overline{R} \oplus \text{"bosons"} The spacetime gauge group is some subgroup of HH containing (SU(3)×SU(2)×U(1))/ 6(SU(3)\times SU(2)\times U(1))/\mathbb{Z}_6. The spacetime theory would be chiral if RR formed a complex representation of the gauge group. Alas, it is easy to prove1 that RR is a pseudoreal (quaternionic) representation of HH (and hence, a real or pseudoreal representation of any subgroup of HH). Thus Lisi’s “theory” is always nonchiral, and cannot have anything to do with that thing we modestly call “the real world.” The proof consists of enumerating all the possible inequivalent embeddings of SL(2,)SL(2,\mathbb{C}), subject to our constraint on m+nm+n. The result can be summarized in the table below:
Embeddings of SL(2,)E 8SL(2,\mathbb{C})\subset E_8
Form of E 8E_8Centralizer (GG)Maximal compact subgroup (HH)Representation (RR)
E 8(24)E_{8(-24)} Spin(1,11)Spin(1,11) Spin(11)Spin(11) 3232
E 8(8)E_{8(8)} Spin(5,7)Spin(5,7) Spin(5)×Spin(7)Spin(5)\times Spin(7) (4,8)(4,8)
E 8(24)E_{8(-24)} Spin(9,3)Spin(9,3) Spin(9)×Spin(3)Spin(9)\times Spin(3) (16,2)(16,2)
E 8E_{8\mathbb{C}} Spin(12,)Spin(12,\mathbb{C}) Spin(12)Spin(12) 6464
E 8E_{8\mathbb{C}} Spin(13,)Spin(13,\mathbb{C}) Spin(13)Spin(13) 6464
E 8E_{8\mathbb{C}} E 7E_{7\mathbb{C}} E 7E_7 5656

In each case, RR is a pseudoreal representation of HH.

In related news, the meltdown of the world’s financial system has prompted Lee Smolin to move from Loop Quantum Gravity into Mathematical Finance. I wish him the best of luck.


1 The corresponding “Euclidean” statement, about embeddings of Spin(4)Spin(4) in compact E 8E_8 is completely trivial: G=H=Spin(12)G=H=Spin(12), whose irreducible spinor representations are pseudoreal.

Posted by distler at June 19, 2009 12:03 AM

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13 Comments & 1 Trackback

Re: Synchronicity

Thanks for the effort Jacques, but it seems that theorems and facts are of little consequence against a PR machine and willful ignorance.

Posted by: H-I-G-G-S on June 19, 2009 11:15 AM | Permalink | Reply to this

Re: Synchronicity

This is a fun argument but you are only using a very tiny portion of the physics constraints and risk a lot of possible loopholes by doing so.

For example, couldn’t one argue that one can take your last example, with E7(C), and further break it into an E6(C) with an E6(R) subgroup, a realistic GUT group that has complex representations?

Do you agree that the very idea that the Lorentz group may be embedded into a Yang-Mills group can’t possible give correct gravity dynamics in the bulk? After all, the diffeomorphism group is just not isomorphic to a bulk Yang-Mills group, is it?

This whole research of embedding gravity into bulk gauge groups is wrong, and your “constructive” approach, minimizing the group theory to analyze the situation, could on the contrary be encouraging for the people who try to study it. You have really helped them, in this sense - the only good luck is that they’re not efficient enough to realize that your centralizers have subgroups with complex reps. ;-)

Best wishes
Lubos

Posted by: Lubos Motl on June 24, 2009 2:39 AM | Permalink | Reply to this

Re: Synchronicity

Of course, HH has subgroups which admit complex representations. Indeed, in our paper, we exhibit embeddings of all your favourite GUT groups (SU(5)SU(5), Spin(10)Spin(10), Pati-Salam, …).

The point, however, is that if RR is a self-conjugate (real or pseudoreal) representation of HH, then it is necessarily a self-conjugate representation of any subgroup of HH.

There is no loophole there. You can’t get a chiral theory out of a nonchiral one.

Do you agree that the very idea that the Lorentz group may be embedded into a Yang-Mills group can’t possible give correct gravity dynamics in the bulk…

The bosonic sector of Lisi’s “theory” is even more of a train wreck than the fermionic sector I have focused on.

The reason I decided to focus on the fermions is that they “almost” make sense. Therefore, one can cleanly formulate what it would mean for the fermionic sector to “work” (and thence prove a no-go theorem).

The bosonic sector is such a train wreck, it’s not even clear what “work” would mean. So there’s no chance of proving a no-go theorem.

There is, to summarize, method to my madness. One gets a much stronger statement (in this case, a no-go theorem killing any possible variation of Lisi’s idea) by concentrating, not on the weakest points of his theory, but on the “strongest”.

I would hope that, upon reflection, you would see the wisdom of this course of action.

Posted by: Jacques Distler on June 24, 2009 9:19 AM | Permalink | PGP Sig | Reply to this

Re: Synchronicity

Out of curiosity, what prevents Lisi from simply finetuning his way out of experimentally verifiable terrain?

If he has a generation and its mirror, you would expect all those terms to acquire generically large gauge invariant masses. Otoh, the rest of his action is completely, horribly finetuned in the first place, so why not follow the rabbit all the way down the proverbial hole and simply tune every single parameter?

Whats the point I guess…

Posted by: Haelfix on June 24, 2009 3:08 PM | Permalink | Reply to this

Re: Synchronicity

Out of curiosity, what prevents Lisi from simply finetuning his way out of experimentally verifiable terrain?

Either there are no fermions at all, at low energies, because the generation and anti-generation pair up to obtain large, gauge-invariant, masses or — if you fine-tune — both the generation and the anti-generation are light (massless, in the limit of unbroken SU(2)×U(1)SU(2)\times U(1)).

You could, perhaps, give the anti-generation large Yukawa-couplings to the Higgs (so that they would have top-ish masses). But that is already well-excluded by precision electroweak data.

In any case, it’s impossible to get more fermions (than a generation and an anti-generation), so the theory is already dead-in-the-water on those grounds, alone.

For disposing of this “theory”, it suffices to note that the maximum dimension of the representation, RR is 32 (in the case of a noncompact real form of E 8E_8) – smaller than the 45 dimensional representation (for 3 generations), much less the 90 dimensional representation (for 3 generations and 3 anti-generations).

As I said, our theorem is vast overkill.

Posted by: Jacques Distler on June 24, 2009 5:28 PM | Permalink | PGP Sig | Reply to this

Re: Synchronicity

Yea I had in mind a Yukawa that bumps the mirrors up maybe an order of magnitude heavier than the top mass, but of course one worries about what that does with the Higgs mass. Too heavy and you push it into exluded zones.

Posted by: Haelfix on June 24, 2009 9:44 PM | Permalink | Reply to this

Re: Synchronicity

“But that is already well-excluded by precision electroweak data.”

Is there a definite write-up of the status of mirror matter somewhere? I haven’t been able to find one.

Posted by: Mark Srednicki on July 1, 2009 12:13 PM | Permalink | Reply to this

Mirror muck

I am unaware of one.

I would have thought such things would be definitively ruled out by their contribution to the ρ-parameter (the contribution being quadratic in the mass of the fermion). The top contribution is already significant, and the contribution of 15 more top-like fermions is surely completely ruled out.

Posted by: Jacques Distler on July 2, 2009 1:25 AM | Permalink | PGP Sig | Reply to this

Re: Synchronicity

“The spacetime theory would be chiral if R formed a complex representation of the gauge group. Alas, it is easy to prove … Lisi’s “theory” is always nonchiral …”

Regarding Lisi’s ad hoc derivation of a non-chiral SU(3)×SU(2)×U(1) from E(8), is it possible to manufacture a chiral SU(3)×SU(2)×U(1) (i.e. the SM) out of a non-chiral SU(3)×SU(2)×U(1)?

Specifically, can the Higgs sector be manufactured so that it gives not just mass to fermions, but also left-handedness? Woit argues in his representation theory paper that SU(2) chiral symmetry and mass may be closely associated:

‘An idea I’ve always found appealing is that this spontaneous gauge symmetry breaking is somehow related to the other mysterious aspect of electroweak gauge symmetry: its chiral nature. SU(2) gauge fields couple only to left-handed spinors, not right-handed ones. In the standard view of the symmetries of nature, this is very weird. The SU(2) gauge symmetry is supposed to be a purely internal symmetry, having nothing to do with space-time symmetries, but left and right-handed spinors are distinguished purely by their behavior under a space-time symmetry, Lorentz symmetry. So SU(2) gauge symmetry is not only spontaneously broken, but also somehow knows about the subtle spin geometry of space-time. Surely there’s a connection here… So, this is my candidate for the Holy Grail of Physics, together with a guess as to which direction to go looking for it.’

- http://www.math.columbia.edu/~woit/wordpress/?p=3

Posted by: Nigel on July 6, 2009 7:18 AM | Permalink | Reply to this

Re: Synchronicity

Sigh

SU(2)SU(2) has only self-conjugate representations. You could turn off the SU(2)SU(2), entirely, and the remaining SU(3)×U(1) YSU(3)\times U(1)_Y gauge theory would still be chiral.

But why did you think it valuable to dredge up that embarrassment from the distant past? You are truly a troll, Nigel.

Posted by: Jacques Distler on July 6, 2009 8:00 AM | Permalink | PGP Sig | Reply to this

Re: Synchronicity

Thanks for your patience, Professor.

Posted by: Nigel on July 6, 2009 8:27 AM | Permalink | Reply to this

Re: Synchronicity

Dear Jacques and Skip,

Thank you for sending your paper for my comments, and informing me that you are having trouble getting it past a referee. Your description of my private communication to you is misleading. Here is my position and my objection to your “ToE3” condition:


The Lie algebra of gravity and standard model gauge fields is a subalgebra of spin(3,11), which acts on one generation of fermions in its positive real 64 dimensional spinor representation space. When this spin(3,11) is in the spin(4,12) subalgebra of the quaternionic real form of E8, the Lie bracket of this spin(3,11) with a specific set of 64 generators in E8 is isomorphic to the action of spin(3,11) on one generation of fermions. Using this usual definition of the word “in,” the algebra of gravity and gauge fields acting on one generation of fermions is “in” E8. For you to make requirements on the properties of the other generators in E8, in the complement, as you do implicitly in ToE3, is to change the definition of the word “in.” To do this and use it as you have in your title, throughout the rest of your paper, and elsewhere, is blatantly dishonest.


I look forward to seeing this precise statement of my communication to you in the new version of your paper, if you are successful in getting it published. In the mean time, I will contribute it to your blog.

Sincerely,
Garrett

Posted by: Garrett on October 25, 2009 10:41 PM | Permalink | Reply to this

Re: Synchronicity

…and informing me that you are having trouble getting it past a referee.

We’re not.

The paper was accepted for publication in Communications in Mathematical Physics.

The referee suggested that the paper would be stronger if we responded to your privately-expressed but – apparently, widely circulated – “objection” to our work. And we obliged, by writing:

Our hypothesis (ToE3) says that the candidate “Theory of Everything” one obtains from subgroups SL(2,)SL(2,\mathbb{C}) and GG as in (ToE1) must be chiral in the sense of Definition 2.5.2

In private communication, Lisi has indicated that he objects to our condition (ToE3), because he no longer wishes to identify all 248 generators of Lie(E) as particles (either bosons or fermions). In his new — and unpublished — formulation, only a subset are to be identified as particles. In particular, V 2,1V_{2,1} is typically a reducible representation of GG and, in his new formulation, only a subrepresentation corresponds to particles (fermions). This is not the approach followed in [1], where all 248 generators are identified as particles and where, moreover, 20-odd of these are claimed to be new as-yet undiscovered particles — a prediction of his theory. As recently as April 2009, Lisi reiterated this prediction in an essay published in the Financial Times, [11].

Our paper assumes that the approach of [1] is to be followed, and that all 248 generators are to be identified as particles, hence (ToE3). In any case, even if one identifies only a subset of the generators as particles, all the fermions must come from the (1)(-1)-eigenspace, which is too small to accommodate 3 generations, as we now show.


2 Of course, there are many other features of the Standard Model that a candidate Theory of Everything must reproduce. We have chosen to focus on the requirement that the theory be chiral for two reasons. First, it is “physically robust”: Whatever intricacies a quantum field theory may possess at high energies, if it is non-chiral, there is no known mechanism by which it could reduce to a chiral theory at low energies (and there are strong arguments [10] that no such mechanism exists). Second, chirality is easily translated into a mathematical criterion — our (ToE3). This allows us to study a purely representation-theoretic question and side-step the difficulties of making sense of Lisi’s proposal as a dynamical quantum field theory.

(We then go on to recite an argument well-familiar to readers of this blog.)

But thanks for taking the time to look at what (I hope) will be the final, published, version of the manuscript.

P.S.: I’m not sure this is a conversation you really wanted to hold in the public view of all and sundry. If, perchance, it’s the result of a little too much wine imbibed at the Zuckerman banquet, let me know, and I’ll happily delete both your comment and my followup.

Cheers,
Jacques

Posted by: Jacques Distler on October 25, 2009 11:36 PM | Permalink | PGP Sig | Reply to this
Read the post Crib Notes
Weblog: Musings
Excerpt: Oh no! NOT Lisi, again!
Tracked: June 27, 2010 9:18 PM

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