Particle physics is not my area of expertise, so there is no way I can make technical considerations on Garrett Lisi‘s recent paper, An Exceptionally Simple Theory of Everything.
I am glad that this paper is under scrutiny (at least this is what one would infer from blogs, if they can serve as thermometer for what is going on, with the caveat that the blogosphere tend to augment things). One can learn more about the discussions (with the participation of Garrett himself on them) over at Sabine’s blog, Woit’s blog and Physics Forums.
Garrett acknowledges that his theory is a boolean entity; there are only two possibilites: it can be right or wrong. If that is so, and if it can be developed enough to offer predictions, be falisfied, etc, then it is good science to me.
A few days after appearing in the arxiv under hep-th (High Energy – Theory), his paper was reclassified, as it seems, by the moderators, to the General Physics classification. It makes me wonder whether the new arxiv numbering system, which no longer carries the subject field label on it, was implemented to ease such reclassifications. In any case, it is not clear what is the criteria used by the moderators to move the papers around in the repository, not to say other issues like acceptance, endorsement system, etc.
Garrett Lisi is an independent scientist who received a grant from the FQXi Foundation.
I hope that his new theory is scrutinized to exhaustion, like any theory should be. Specially if it claims to be a first step towards a ‘theory of everything’.
Updates:
1- Garrett’s paper has been reclassified back to hep-th.
2- Garrett was invited to talk at the International Loop Quantum Gravity Seminar (ILQGS), yesterday. You can find his slides and audio at their site. I recommend listening to his talk.
Additional update:
For some reason, the trackback of this post (automatically sent by wordpress) have not yet appeared at Lisi’s trackback list at the arxiv.
And further update:
A critical assessment on the mathematics (group representation theory) side of Lisi’s paper has been recently given by Distler at his blog. A brief comment by Garrett can also be found there.
And further further update:
More on Lisi’s paper over at Dynamics of Cats: here and here.
(Last?) update:
A personal account by Lisi on the whole story can be found here.
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Disclaimer: Here I add a compilation of some technical exchanges that occurred over at Cosmic Variance. I have made several edits which leave out parts that I have judged non-technical. The interested reader is urged to go to the original posts if they would like to read the complete posts for citing specific passages or for their own personal interpretations. I post them here just for my own interests. Although I had great care in making the present editing, eventual errors in the compilation/edition process may have occured, so one more reason to check the originals yourself.
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Lee Smolin on Dec 11th, 2007 at 10:15 am
I did look at Dsitler’s blog and there is, to my understanding, only one issue was raised in his discussion of Lisi’s paper that was not already raised by Lisi himself in his paper and talks: this is that Lisi muffed the nomenclature for non-compact forms of E8.
The question is whether the open issues are solvable issues or not. Distler thinks not, Lisi thinks perhaps yes. (…) Some of the open issues are straightforward to address, given that there is a literature on this kind of unification, beginning with Peldan in 1992. So in my recent paper I describe how to make a fully gauge invariant action for proposals of Peldan and Lisi’s type, and I also suggest an alternative approach for the fermions which mght resolve some of the other open issues. The gauge invariant action is, btw, the starting point for quantization using LQG and spin foam methods.
(…) the CM theorem (…) concerns global symmetries of the S matrix. In a gravitational theory, which Lisi’s is, global symmetries are symmetries only of solutions or of asymptotic conditions and are not the same as the local gauge symmetries. Indeed, even though my local gauge symmetry is some semi-simple G, I display a solution whose global symmetry is a subgroup of G, namely SO(4)+H where H is the largest compact subgroup of G/SO(4). (The same would work with Lorentzian signature.) Even if G=E8 this is in accord with the CM theorem.
(…)
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Lee Smolin on Dec 14th, 2007 at 9:37 am
(…)
1) There is no issue for the euclidean, compact case.
3) (…) While I haven’t checked in detail, (…) there is no mention in Jacques’s post of the Pati-Salam chirally symmetric theory, as that is what Lisi’s paper shows is embedded in E8. While I haven’t worked out the details, the Pati-Salam is a vector theory where parity is only broken spontaneously. Fermions in the Pati-Salam model are in parity symmetric reps, because of the overall parity invariance of the theory. In Pati-Salam parity is broken spontaneously, leaving chiral fermions at low energy. (…)
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Jacques Distler on Dec 14th, 2007 at 10:39 am
A hint to (…) the Pati-Salam model. The left-handed fermions in that model transform in a complex representation of SU(4)xSU(2)xSU(2), specifically, the (4,2,1)+(4bar,1,2).
Lisi also has the electroweak SU(2) embedded in an SU(2)xSU(2). There, however, the similarity with Pati-Salam ends.
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Jacques Distler on Dec 14th, 2007 at 10:33 pm
(…) I added an appendix, where the phrase [“Pati-Salam”] is used liberally.
(…) Lisi does not even get one generation of quarks and leptons, let alone three(…)
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H-I-G-G-S on Dec 15th, 2007 at 4:23 pm
A small clarification. Let us define a parity invariant theory to be one in which there is a Z_2 symmetry which includes inversion of the spatial coordinates (in three spacetime dimensions). Let us also define a chirally invariant theory as one in which, after writing all fermions as left-handed Weyl fermions, the fermions are in a real representation of the gauge group. Lee uses these two terms interchangably, which is unfortunate, and can lead to confusion. The Pati-Salam model with SU(4)xSU(2)xSU(2) gauge symmetry is parity invariant. To achieve this invariance one must extend the “usual” parity symmetry by a Z_2 which interchanges the two SU(2) factors in the gauge group. However, as correctly pointed out by Jacques, the PS theory is not chirally invariant because the fermions are not in a real representation. Since Jacques showed that the embedding used by Lisi gives a real fermion representation, he is correct in saying that the Lisi embedding does not contain the PS model. Lee is incorrect in saying the opposite (…) One can (and should) break the parity symmetry in PS spontaneously, but this does not suddenly generate chiral fermions from non-chiral fermions. The fermions were chiral to begin with.
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Lee Smolin on Dec 15th, 2007 at 7:11 pm
(…)
What is usually called the Pati-Salam model was defined in two papers, one of which is accessible through the KEK archive and is Jogesh C. Pati, Abdus Salam, LEPTON NUMBER AS THE FOURTH COLOR.Phys.Rev.D10:275-289,1974. (I should note this paper has 2700+ citations, so this is not an obscure work.) Another clear paper is H.S. Mani, Jogesh C. Pati, Abdus Salam, ‘NATURALNESS’ OF ATOMIC PARITY CONSERVATION WITHIN LEFT-RIGHT SYMMETRIC UNIFIED THEORIES. Phys.Rev.D17:2510,1978. In these papers it is clearly explained that their theory is parity invariant and parity is broken only spontaneously. They are explicit that for every left handed current in their model there is a parity related right handed current with equal coupling constants. Hence before spontaneous symmetry breaking the fermion rep for Pati-Salam must be parity invariant. More specifically, in the first paper above, just after eq 3, they state that the fermions are in the representation (4-bar,2,1)+(4-bar,1,2) of SU(4) x SU(2)_L x SU(2)_R, which is parity invariant. (Note that parity exchanges the left and right handed SU(2)’s.)
In Distler’s post he asserts to the contrary that in the Pati-Salam model the fermions are in the representation he calls R_ps= (4,2,1)+(4-bar,1,2) of SU(4) x SU(2)_L x SU(2)_R, which is not parity invariant. (Parity changes this to (4,1,2)+(4-bar,2,1) which is not equivalent to R_ps.) This disagrees with what is stated in the above paper, by the omission of a single bar. This small change has a major impact on the discussion because it turns a parity symmetric theory into a parity non-symmetric theory which is not the Pati-Salam model.
It seems to me this invalidates Distler’s discussion of Pati Salam and by extension suggests that his second post on Lisi is incorrect. While Lisi’s scheme is not quite the same as the above, because the electroweak gauge symmetry is unified first with local lorentz, then with G2, the moral is the same because Lisi breaks E8 to a version of electro-weak unification which is parity symmetric. This is ok for the same reason Pati-Salam is ok, because parity can be broken spontaneously.
(…)
Ps to HIGGS.
One does not need a special extension of parity to switch SU(2)_L with
SU(2)_R, because they couple to left and right handed currents in the usual sense, so parity switches them. And, in case there is any confusion, the above discussion is not affected by any terminological confusion as to the meaning of chiral.
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Jacques Distler on Dec 15th, 2007 at 7:39 pm
(…)
Start by writing everything in terms of left-handed 2-component Weyl fermions (whose Hermitian conjugates are right-handed Weyl fermions).
The quantum number of a generation of fermions are as I stated in my post (and as can be confirmed by a myriad of contemporary sources). To check, it suffices to look at what happens when you break the Pati-Salam group down to the Standard Model gauge group to see that what I wrote down give the correct Standard Model content, and that what you wrote down does not.
In fact, the representation you wrote down is anomalous, so could not possibly be correct.
H-I-G-G-S stated the distinction between Parity and Chirality correctly. The names of the two SU(2) groups in Pati-Salam are just names.
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H-I-G-G-S on Dec 15th, 2007 at 7:46 pm
(…)
In the Pati-Salam paper what they say is that \psi_L is in the (2,1 \bar 4) and \psi_R in the (1,2 \bar 4). It is common
in the study of grand unification to write all fermions as left-handed fermions. \psi_R^*, after multiplication by a suitable matrix, is left-handed. Let us call it \psi’_L. We then have
\psi_L: (2,1, \bar 4)
\psi’_L: (1,2, 4)
Note that complex conjugation does not change the SU(2) rep because the 2 is pseudo-real. The representation in which all fermions are left-handed is the one that Jacques was correctly using and it agrees with Pati-Salam. The relevance of the reality of the representation when written this way is that a real representation allows a gauge invariant mass term whereas a complex one does not. (…)
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Lee Smolin on Dec 15th, 2007 at 7:49 pm
Jacques,
You are claiming 1) that the original Pati-Salam paper I refer to is incorrect about what their own theory is and 2) even though their big point is that the fundamental dynamics could parity invariant that they should have based the model on a parity non-invariant representation (4,2,1)+(4-bar,1,2) instead of the parity invariant one they specify?
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H-I-G-G-S on Dec 15th, 2007 at 7:57 pm
Lee,
(…) Jacques agrees with PS. He is simply using a basis in which all fermions are left-handed. PS are not. But they have exactly the same degrees of freedom and are
describing the same theory. (…)
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Lee Smolin on Dec 15th, 2007 at 8:36 pm
Dear HIGGS
I see, if it is then just a terminological mixup that is of course fine for this issue. (…)
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Lee Smolin on Dec 17th, 2007 at 11:47 am
Dear HIGGS,
(…)
“3) He has nothing new to say about dynamics so I expect the theory to be non-renormalizable, to the degree the theory is even defined.
5) The E8 symmetry is not a symmetry at all since there is no limit proposed in which it is restored.”
These two points are addressed in my paper, by the giving of an action which is fully E8 gauge invariant and by the proposal of a spin foam quantization based on that E8 invariant action. While we do not yet know if that leads to an ultraviolet finite quantum theory, it is known that related forms of spin foam actions are uv finite, for reasons that may apply here. In particular, spin foam models for just general relativity are ultraviolet finite and well defined. So there is a solid basis to proceed to investigate a quantum theory of an E8 unified theory. (Note that my paper applies to a general class of theories that include Lisi’s E8 proposal.)
(…)
“4) He starts with all couplings equal but does no RGE analysis to see whether they take reasonable values at low-energies. This seems unlikely since the low-energy couplings do not unify without superpartners or some other new structure at low energies.”
I agree it would be good to a RG analysis but I also hope that the first paper proposing a new theory does not already have to present a full renormalization group analysis. (…) is it not true that in any spontaneously broken gauge theory, broken at a scale M, the couplings must unify above the scale M? Is it not also true that there must be many ways to fill in the desert so as to explain a unification of couplings at a scale M, the MSSM gives just one way to do this? The MSSM is sufficient, but not necessary to complete grand unified theories to give a unification of gauge couplings.
“1) It does not contain the standard model fermions in a chiral rep. In fact it contains fermions in real reps, so they will presumably have large masses and one will not get the chiral structure of the SM.”
This is claimed (…) Let me ask two questions, related to this issue in the whole class of models I study including E8. (…) In these models a semi-simple gauge group G with exact G invariant dynamics is spontaneously broken to a subgroup
H= local-lorentz x Y,
where Y is a yang-mills gauge group. The definition of chirality you use involves treating the left and right handed parts of fermion fields differently, so it cannot be applied directly to reps of G. Now do you know the answer to the following question: what property does a representation of G have to have so that it gives rise, after spontaneous symmetry breaking, to spacetime spinors which are chiral in Y? (…)
Second, (…) Distler’s post (…) seems to involve two distinct claims. First that the particular gauge groups Distler, in an earlier post, claims are subgroups of a parrticular non-compact form of E8, are not. Second that the construction Lisi gives of fermions arising from certain pieces of the adjoint of E8 do not give rise to chiral reps (in the senses that you use it) after the spontaneous symmetry breaking. (…) In particular, if Distler is making the first claim, why is it at all necessary to bring up the issue of chiral fermion reps?
(…)
“2) It mixes bosons and fermions with (…) BRST but no definite proposal about what the mathematical structure is that lies behind this. For example, are the physical states defined by BRST cohomology classes?”
Lisi is definite about the mathematical structure he is referring to, he is using a certain definition of a “BRST extended connection”, but he is not using it the way BRST is often used to construct gauge invariant amplitudes for yan-mills theory. In any case, as I indicated in my paper, in case Lisi’s approach fails there is another way fermions could arise in such a theory. This does not in particular, limit them to coming from generators in the adjoint of E8. This is one reason why I am interested in the more general question I raised above.
(…)
I do not have a big stake in how the issue turns out with Lisi’s fermions because I have a different proposal for how fermions can arise in the kind of gravity-gauge theory unifications his proposal fits into. This proposal does not limit them to arising from certain generators in the adjoint, as Lisi’s does. (…) I am happy either way, I just want to understand what is true.
(…)
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Jacques Distler on Dec 17th, 2007 at 1:16 pm
I find the mixing of these two issues confusing. In particular, if Distler is making the first claim, why is it at all necessary to bring up the issue of chiral fermion reps?
(…) Even though the particular real form of D_4xD_4, chosen by Lisi, does not embed in any real form of E_8, there are other real forms of D_4xD_4 which do. So a thorough analysis would study those other real forms as well, to see what one can obtain with them.
That is what I did (I fact, I looked at what you get for all five cases where G can be embedded as a subgroup, though I only reported the result for the two “Pati-Salam”-like cases.
[W]hat property does a representation of G have to have so that it gives rise, after spontaneous symmetry breaking, to spacetime spinors which are chiral in Y?
In a nutshell, if you look at the piece of the adjoint representation that transforms as a “2″ (as opposed to a “2bar”) of SL(2,C), then that should be a complex representation of SU(3)xSU(2)xU(1). Ideally, it should contain 3 copies of the (3,2)_{1/6}+(3bar,1)_{1/3}+(3bar,1)_{-2/3} +(1,2)_{-1/2}+(1,1)_1.
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Lee Smolin on Dec 17th, 2007 at 1:29 pm
Jacques,
Thanks re the first. Re the second, yes, I knew that, the question is, is there some general property of a rep of G that guarantees that or for which this is never the case? Can it ever be true for a real rep?
Let me add a third question, is the issue of embedding of Lie algebras-leaving the fermion question aside-clear for the euclidean case? That is does SO(4) + SU(3)+ SU(2) + SU(2) + U(1) embedd in the real form of the lie algebra of E8, using the decomposition suggested by Lisi? Is there any sense in which the lie algebra SL(2,C) + SU(3)+ SU(2) + SU(2) + U(1) fits in a complexification of the lie algebra E8?
Finally, what is the best reference to understand why there are only these two non-compact forms of E8 and where the results you quote come from? At the level of the lie algebra it would seem there might be more freedom. Is it easy to say why there is not?
(…)
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Jacques Distler on Dec 17th, 2007 at 1:41 pm
is there some general property of a rep of G that guarantees that or for which this is never the case? Can it ever be true for a real rep?
See here for an argument that applies quite generally to the “Euclidean” (compact real form) case of any group (not just E_8). It is almost, but not quite a proof for the Minkowskian (noncompact real form) case. There are some potential loopholes there. (At least, they seem like potential loopholes to me; someone who knew more about the representation theory involved could probably complete the proof pretty quickly.)
Finally, what is the best reference to understand why there are only these two non-compact forms of E8 and where the results you quote come from?
See the article by Marcel Berger that I linked to in my post. To define a noncompact real form, you need an involution of the Lie algebra that acts as +1 on the compact generators, and as -1 on the noncompact generators. Such an involution also defines a symmetric space structure, and Berger classified those.
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Jacques Distler on Dec 17th, 2007 at 1:51 pm
Actually, to find the noncompact real forms, you need to classify Riemannian symmetric spaces. That classification long predates Berger’s paper.
See the textbook by Helgason.
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Lee Smolin on Dec 17th, 2007 at 4:12 pm
Dear Jacques,
(…) Following your post, the Pati-Salam fermions are in a rep given by
R_ps = (2,r) +(2-bar, r-bar)
where these refer to their transformation properties under the sum of the spacetime lorentz algebra so(3,1) and H, where H is the Pati-Salam algebra
H=su(4)+ su(2)+su(2)
And r = (4,2,1)+(4-bar,1,2)
Now we established before that r is not equivalent to its complex conjugate, so R_ps is in the standard terms chiral. It is also the case that parity takes R_ps to itself. Now, R_ps is also pseudo-real, ie it is equivalent to its complex conjugate, as it is the sum of itself and its complex conjugate. So suppose there were a bigger lie algebra G that contained so(,1)+H as a subalgebra. Could not R_ps arise from the decomposition of a pseudo-real representation of G?
(…)
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Jacques Distler on Dec 17th, 2007 at 4:29 pm
Dear Lee,
Perhaps it would be best if we retreated, for a moment, to the “euclidean” case, where we are embedding Spin(4), instead of SL(2,C). Then
R = (2,1, r) + (1,2, rbar)
which is still complex, if r is a complex representation of H.
In any case, Lisi’s game (interpreted most broadly) is to embed things in the adjoint representation of some real form of some Lie group. That is always a real representation.
(…)
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Lee Smolin on Dec 17th, 2007 at 4:43 pm
Jacques,
But in the lorentzian case where complex conjugation exchanges the left and right factors,
R_ps = (2,1, r) + (1,2, rbar)
of SU(2)_L+SU(2)_R+H
is equivalent to its complex conjugate.
(…)
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Jacques Distler on Dec 17th, 2007 at 5:24 pm
Lee,
That’s the point! A priori, in the Lorentzian case, R is a real (or pseudo real) representation. So you might think that you could get a chiral spectrum (a net number of copies of R). But, for any embedding of SL(2,C) in the noncompact real form, which is related by Wick rotation to an embedding of Spin(4) in the compact real form, this never happens. I have just explained to you why it never happens.
More broadly, there are other embeddings of SL(2,C) in the noncompact real form, not related by Wick rotation to embeddings of Spin(4) in the compact real form, which also necessarily yield a nonchiral spectrum. Specifically, I have looked at embeddings that proceed via SL(4,R) and via SU(2,2).
What I have not done is show that these are the only remaining possibilities. So there’s still a challenge outstanding, to any readers of my blog, to close that gap.
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Jacques Distler on Dec 18th, 2007 at 2:15 am
I wrote
What I have not done is show that these are the only remaining possibilities. So there’s still a challenge outstanding, to any readers of my blog, to close that gap.
But let me be very clear that this “potential loophole” is irrelevant to your paper, which concerned the Euclidean case (embedding Spin(4) in the compact real form of some Lie group), and to Lisi’s paper, which concerned an embedding via D_4xD_4.
Neither of these can ever lead to a chiral spectrum (let alone to 3 Standard Model generations).
(…)
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Lee Smolin on Dec 18th, 2007 at 10:17 am
Dear Jacques,
I am not sure I understand your last point. Let me review. We have been discussing the situation encountered unifying the Pati-Slam gauge symmetry with the local lorentz transformations. This leads us to consider
R_ps = (2,r) +(2-bar, r-bar),
with r = (4,2,1)+(4-bar,1,2)
as described in my 149 above.
We have established 1) that r is chiral, in the sense that it is in a complex rep of the YM gauge symmetry that acts on left handed spinors, 2) that R_ps is parity invariant and 3) that R_ps is equivalent to its complex conjugate.
Thus, if there were a unification of gravity and Yang-Mills in terms of the connection of a larger G which has as a subalgebra local lorentz+H_ps, along the lines of Peldan, myself and others, R_ps could arise from the decomposition of a rep of G which is also equivalent to its complex conjugate.
Do I understand correctly that you agree with this but are arguing that in the particular case of if G=E8, R_ps cannot arise from decomposition of the adjoint of E8 ?
Note: the above is all assuming lorentzian signature. It is true that in my paper I worked with Euclidean signature, but only for simplicity.
(…)
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Jacques Distler on Dec 18th, 2007 at 10:45 am
(…)
If the embedding of SL(2,C) in the noncompact real form of G is related by Wick rotation to an embedding of Spin(4) in the compact real form of G, then R is nonchiral.
This has nothing to do with the specific choice G = E_8.
——
(…)
For concision, I will phrase things in terms of embeddings of Lie algebras, instead of groups.
Let g_C be the complex Lie algebra, which has (at least) two real forms: a compact real form, g_e, appropriate to the Euclidean case, and a noncompact real form, g_l, appropriate to the Lorentzian case. Let h be the lie algebra of the SM or Pati-Salam, and h_C the corresponding complex Lie algebra (of which h is the compact real form).
We are interested in embeddings of
so(4) x h ⊂ g_e
and
so(3,1) x h ⊂ g_l
I say “these embeddings are related by Wick rotation” if they stem from the same embedding of complex lie algebras
d_2 x h_C ⊂ g_C
by choosing different real forms.
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Lee Smolin on Dec 19th, 2007 at 12:39 pm
(…)
What we established (…) was that the fermions are in a rep r which has all the following properties:
1) It is chiral in the standard sense (see 99 and 151)
2) When taken in full as a rep R_ps, as defined in previous posts, of lorentz+H where H=su(4)+su(2)+su(2) it is parity invariant.
3) R_ps is also pseudo-real.
It was important to establish these points, because they bear on Lisi’s proposal and more generally for model building of this type. Pati-Salam is not a trivial point to be filled in, it is the key to whether Lisi succeeds or fails. This key point was not commented on by Distler or anyone here until I brought it up.
I also asked what properties a Lie algebra G has to have so that it has a subalgebra of lorentz + H, with a representation that on breaking to the subalgebra gives R_ps. My question, central to understanding if anything like Lisi’s proposal can work, was based on the fact that the situation was rather different than the case of the usual standard model where 2 and 3 don’t hold.
Distler did in 165 offer a general statement, “If the embedding of SL(2,C) in the non-compact real form of G is related by Wick rotation to an embedding of Spin(4) in the compact real form of G, then R is non-chiral.” So far as I can tell this is false because G=SO(3,1)+H is itself a counterexample to it, given the above. Were there further discussion this would be the place to start.
(…)
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Jacques Distler on Dec 19th, 2007 at 1:17 pm
(…)
1. Pati-Salam is not a vector-like theory. It is inherently chiral.
2. (…) You can never turn a vector-like theory into a chiral theory, “leaving chiral fermions at low energies” by spontaneously breaking parity.
In #102, you said
In Distler’s post he asserts to the contrary that in the Pati-Salam model the fermions are in the representation he calls R_ps= (4,2,1)+(4-bar,1,2) of SU(4) x SU(2)_L x SU(2)_R, which is not parity invariant.
…
It seems to me this invalidates Distler’s discussion of Pati Salam and by extension suggests that his second post on Lisi is incorrect.
which (…) was (…) wrong.
In the sense you are using it (viewing the complete fermion representation as (2,r)+(2bar,rbar) of SL(2,C)xH), property 3 holds for the Standard model and, indeed, for every unitary quantum field theory (…).
What is true of Pati-Salam, which is not true of the SM, is that there is a definition of “parity” such that property 2 holds.
But that is completely irrelevant (as far as I can tell) to the rest of your argument.
(…)
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Brother Love on Dec 19th, 2007 at 4:39 pm
Perhaps I can help resolve confusion, just in case anyone is, by digressing. The PS model has chiral fermions (so P \Psi \neq \Psi) in the fundamental rep and has a left-right symmetry (under P the Langrangian looks the same). The SU(2)_R group factor may be broken to a U(1) (isospin) via a Higgs in the fundamental representation of SU(2)_R acquiring a VEV. (Maybe when we are seeing this referred to as a vector model somewhere above, it is referring to this Higgs being in the fundamental representation of the SU(2).) Consequently, the L-R symmetry is now broken. However, the fermions were always chiral.
(…)
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