Musings

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

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

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.

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 $\mathrm{SU}(2)\times U(1)$).

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, $R$ is 32 (in the case of a noncompact real form of $E_8$) – 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.

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.

Sigh

$\mathrm{SU}(2)$ has only self-conjugate representations. You could turn off the $\mathrm{SU}(2)$, entirely, and the remaining $\mathrm{SU}(3)\times U(1{)}_Y$ 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.

…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 $\mathrm{SL}(2,ℂ)$ and $G$ 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, $V_{2,1}$ is typically a reducible representation of $G$ 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 $(-1)$-eigenspace, which is too small to accommodate 3 generations, as we now show.

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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