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November 21, 2007

A Little Group Theory …

I really wasn’t going to post about Garrett Lisi’s paper. Preparing a post like this requires work and, in this case, the effort expended would be vastly incommensurate with any benefit to be gained.

So I gritted my teeth through a series of credulous posts in the Physics blogosphere and the ensuing media frenzy. (Yes, Virginia, science reporters do read blogs. And if you think something is worth posting about, there’s a good chance — especially if it has the phrase “Theory of Everything” in the title — they will conclude that it’s worth writing about, too.) But, finally, it was Sean Carroll’s post that pushed me over the edge. Unlike the others, Sean freely admitted that he hadn’t actually read Lisi’s paper, but decided it was OK to post about it anyway.

So here goes.

I’m not going to talk about spin-statistics, or the Coleman-Mandula Theorem, or any of the Physics issues that could render Garrett’s idea a non-starter. Instead, I will confine myself to a narrow question in group representation theory. This has the advantage that

  1. It’s readily decidable, on purely mathematical grounds.
  2. Since it involves the starting point of Garrett’s analysis, a negative answer would render all of the other questions moot.

We would like to find an embedding of

(1)G=SL(2,)×SU(3)×SU(2)×U(1) G = SL(2,\mathbb{C})\times SU(3)\times SU(2)\times U(1)

in a suitable noncompact real form of E 8E_8, such that one finds 3 copies of the representation

(2)R= 2[(3,2) 1/6+(3¯,1) 2/3+(3¯,1) 1/3+(1,2) 1/2+(1,1) 1] + 2¯[(3¯,2) 1/6+(3,1) 2/3+(3,1) 1/3+(1,2) 1/2+(1,1) 1] \begin{aligned} R = &\mathbf{2}\otimes [(3,2)_{1/6}+(\overline{3},1)_{-2/3}+(\overline{3},1)_{1/3}+(1,2)_{-1/2} +(1,1)_{1}]\\ +&\overline{\mathbf{2}}\otimes [(\overline{3},2)_{-1/6}+(3,1)_{2/3}+(3,1)_{-1/3}+(1,2)_{1/2} +(1,1)_{-1}] \end{aligned}

in the decomposition of the 248 of E 8E_8. Here SL(2,)=Spin(3,1) 0SL(2,\mathbb{C})= Spin(3,1)_0 is the connected part of the Lorentz Group, the “gauge group” in the MacDowell-Mansouri formulation of gravity.

Now, the first question you might ask is, which noncompact real form of E 8E_8 are we talking about? There are two.

  • E 8(8)Spin(16)E_{8(8)} \supset Spin(16) as a maximal compact subgroup1. In E 8(8)E_{8(8)}, the 248 decomposes as 248=120+128 248 = 120 +\mathbf{128}
  • E 8(24)SU(2)×E 7E_{8(-24)}\supset SU(2)\times E_7 as a maximal compact subgroup. In E 8(24)E_{8(-24)}, the 248 decomposes as 248=(3,1)+(1,133)+(2,56) 248 = (3,1) +(1,133) +\mathbf{(2,56)}
where, in both cases, I’ve indicated the noncompact generators in bold.

Garret never deigns to tell us which real form of E 8E_8 he is using. But he does say that the embedding of GG in E 8E_8 is supposed to proceed via the subgroup F 4×G 2E 8F_4\times G_2\subset E_8, and he devotes page after mind-numbing page to describing the details of that embedding. Does this provide a clue?

Let us note 3 facts

  • Despite the fact the F 4×G 2F_4\times G_2 has rank 6, its commutant inside of E 8E_8 is discrete. The 248 decomposes as
    (3)248=(1,14)+(52,1)+(26,7)248 = (1,14) + (52,1) + (26,7)
  • The group GG that we are trying to embed also has rank 6, so — if it’s possible — the embedding in F 4×G 2F_4\times G_2 is essentially unique.
  • There are two noncompact real forms of F 4F_4
    • F 4(20)Spin(9)F_{4(-20)}\supset Spin(9) as a maximal compact subgroup. 52 =36+16 26 =1+9+16 \begin{aligned} 52 &= 36 +\mathbf{16}\\ 26 &= 1+9+16 \end{aligned}
    • F 4(4)SU(2)×Sp(3)F_{4(4)}\supset SU(2)\times Sp(3) as a maximal compact subgroup. 52 =(3,1)+(1,21)+(2,14) 26 =(1,14)+(2,6) \begin{aligned} 52 &= (3,1)+(1,21) +\mathbf{(2,14)}\\ 26 &= (1,14)+ (2,6) \end{aligned}

It turns out that F 4(20)×G 2E 8(8)F_{4(-20)}\times G_2 \subset E_{8(8)} and F 4(4)×G 2E 8(24)F_{4(4)}\times G_2 \subset E_{8(-24)}. We can see how this works by decomposing under the common (compact) subgroup. For E 8(8)E_{8(8)}, we have Spin(16)Spin(9)×G 2Spin(16)\supset Spin(9)\times G_2 and 120 =(1,14)+(36,1)+(1,7)+(9,7) 128 =(16,1)+(16,7) \begin{aligned} 120 &= (1,14) + (36,1) +{\color{red} (1,7) +(9,7)}\\ \mathbf{128} &= \mathbf{(16,1)} +{\color{red} \mathbf{(16,7)}} \end{aligned} and for E 8(24)E_{8(-24)}, we have SU(2)×E 7SU(2)×Sp(3)×G 2SU(2)\times E_7\supset SU(2)\times Sp(3)\times G_2 and (3,1) =(3,1,1) (1,133) =(1,1,14)+(1,21,1)+(1,14,7) (2,56) =(2,14,1)+(2,6,7) \begin{aligned} (3,1) &= (3,1,1)\\ (1,133) &= (1,1,14) + (1,21,1) + {\color{red} (1,14,7)}\\ \mathbf{(2,56)} &= \mathbf{(2,14,1)} +{\color{red} \mathbf{(2,6,7)}} \end{aligned} where, in each case, I’ve indicated the additional generators (the ones not in F 4×G 2F_4\times G_2) in red.

So that was, actually, no help. The only thing to do is to try to embed GG in F 4(20)×G 2F_{4(-20)}\times G_2 and in F 4(4)×G 2F_{4(4)}\times G_2 and see what happens.

To make a long story short, GG is not embeddable (actually, it is embeddable; there are just no suitable embeddings) as a subgroup of either F 4(20)×G 2F_{4(-20)}\times G_2 or F 4(4)×G 2F_{4(4)}\times G_2. This is not surprising. As I said, since the ranks are equal, there’s no “wiggle-room” in choosing an embedding.

A pessimist would probably pack it in, at this point. But let’s try to give Garrett the benefit of the doubt and relax our assumptions a bit.

Rather than attempting to embed GG in F 4×G 2F_4\times G_2, let’s just find some embedding of GG in E 8E_8. Clearly, that’s possible to do in quite a number of ways. Demanding that the representation RR appear in the decomposition of the 248 is, however, highly restrictive.

For the split real form, E 8(8)E_{8(8)}, the best you can do is obtain 2 copies of RR. To see how that goes, embed the maximal compact subgroup

(4)G 0=SU(2) MM×SU(3)×SU(2)×U(1) G_0 = SU(2)_{\text{MM}}\times SU(3)\times SU(2)\times U(1)

of GG in an SU(2)×SU(5)SU(2)\times SU(5) subgroup of Spin(16)Spin(16), such that2 120 =(1,24)+(3,1)+(1,10+10¯)+2(1,5+5¯)+5(1,1)+2(2,5+5¯)+4(2,1) 128 =(3,1)+(1,1)+2(2,5¯+10)+2(2,5+10¯)+2(2,1) \begin{aligned} 120 &= (1, 24) + (3, 1) + (1, 10+\overline{10}) +2(1, 5+\overline{5}) + 5(1, 1) + 2(2, 5+\overline{5}) + 4(2, 1)\\ 128 &= (3, 1) + (1, 1) + 2(2, \overline{5}+10) +2(2, 5+\overline{10}) +2(2, 1) \end{aligned} In addition to the (24-dimensional) adjoint of SU(5)SU(5), and the (6-dimensional) adjoint of SL(2,)SL(2,\mathbb{C}), the 248 contains

(5)“fermions”: 2[(2,5¯+10)+(2¯,5+10¯)]+(2,5+5¯)+(2¯,5+5¯)+3[(2,1)+(2¯,1)] “bosons”: 2(1,5+5¯)+(1,10+10¯)+6(1,1) \begin{aligned} \text{“fermions”:} & 2[(\mathbf{2}, \overline{5}+10) + (\overline{\mathbf{2}}, 5+\overline{10})] + (\mathbf{2}, 5+\overline{5}) +(\overline{\mathbf{2}}, 5+\overline{5}) + 3[(\mathbf{2},1)+(\overline{\mathbf{2}}, 1)]\\ \text{“bosons”:} & 2(1, 5+\overline{5}) + (1, 10+\overline{10}) +6(1, 1) \end{aligned}

I’ve put scare quotes around “fermions” and “bosons”, for reasons that are obvious to anyone who has taken more than a passing glance at Garrett’s paper. No matter. We have failed to find an embedding of three copies of RR. The best we were able to do was embed 2 copies.

I leave it as an exercise3 for the reader to repeat the analysis for E 8(24)E_{8(-24)}.

Update (11/23/2007):

Those who think I have been too harsh in condemning the Physics blogosphere as an intellectual wasteland can probably point to Wikipedia as being measurably worse. If the article doesn’t make your head explode, try reading the Talk page.

Update (11/29/2007):

David Vogan, from MIT, wrote me to point out that I was too fast in saying that GG does not embed in F 4×G 2F_4\times G_2. It is possible to find such an embedding, but it necessarily leads to a completely nonchiral “fermion” representation (and hence contains no copies of RR). I simply didn’t bother considering such embeddings, when I was preparing this post. For the record, though F 4(20)Spin(8,1)Spin(3,1)×Spin(5)SL(2,)×SU(2)×U(1) F_{4(-20)} \supset Spin(8,1) \supset Spin(3,1)\times Spin(5)\supset SL(2,\mathbb{C})\times SU(2)\times U(1) and F 4(4)Spin(5,4)Spin(3,1)×Spin(2,3)SL(2,)×SU(2)×U(1) F_{4(4)} \supset Spin(5,4) \supset Spin(3,1)\times Spin(2,3)\supset SL(2,\mathbb{C})\times SU(2)\times U(1) In the latter case, one obtains 26=1+9+16 =(1,1) 0+(4,1) 0+(1,3) 0+(1,1) 2+(1,1) 2 +(2,2) 1+(2,2) 1+(2¯,2) 1+(2¯,2) 1 52=36+16 =(Adj,1) 0+(1,3) 0+(1,1) 0+(1,3) 2+(1,3) 2+(4,3) 0+(4,1) 2+(4,1) 2 +(2,2) 1+(2,2) 1+(2¯,2) 1+(2¯,2) 1 \begin{aligned} 26 = 1 + 9 + 16 &= (\mathbf{1},1)_0 + (\mathbf{4},1)_0 + (\mathbf{1},3)_0 + (\mathbf{1},1)_2 + (\mathbf{1},1)_{-2}\\ &\quad + (\mathbf{2},2)_1 + (\mathbf{2},2)_{-1} + (\overline{\mathbf{2}},2)_1 + (\overline{\mathbf{2}},2)_{-1}\\ 52 = 36 + 16 &= (\mathbf{Adj},1)_0 + (\mathbf{1},3)_0 + (\mathbf{1},1)_0 + (\mathbf{1},3)_2 + (\mathbf{1},3)_{-2} + (\mathbf{4},3)_0 + (\mathbf{4},1)_2 + (\mathbf{4},1)_{-2}\\ &\quad + (\mathbf{2},2)_1 + (\mathbf{2},2)_{-1} + (\overline{\mathbf{2}},2)_1 + (\overline{\mathbf{2}},2)_{-1} \end{aligned} In the former case, there are two distinct embeddings of SU(2)×U(1)Spin(5)SU(2)\times U(1)\subset Spin(5). For the one under which 4=2 1+2 14 = 2_1 +2_{-1}, one obtains the same result as above. For the one under which 4=2 0+1 1+1 14= 2_0 + 1_1 + 1_{-1}, one obtains 26 =2(1,1) 0+(4,1) 0+(1,2) 1+(1,2) 1 +(2,2) 0+(2,1) 1+(2,1) 1+(2¯,2) 0+(2¯,1) 1+(2¯,1) 1 52 =(Adj,1) 0+(1,3) 0+(1,1) 0+(4,1) 0+(1,1) 2+(1,1) 2+(1,2) 1+(1,2) 1+(4,2) 1+(4,2) 1 +(2,2) 0+(2,1) 1+(2,1) 1+(2¯,2) 0+(2¯,1) 1+(2¯,1) 1 \begin{aligned} 26 &= 2(\mathbf{1},1)_0 + (\mathbf{4},1)_0 + (\mathbf{1},2)_1 + (\mathbf{1},2)_{-1}\\ &\quad + (\mathbf{2},2)_0 + (\mathbf{2},1)_1 + (\mathbf{2},1)_{-1} + (\overline{\mathbf{2}},2)_0 + (\overline{\mathbf{2}},1)_1 + (\overline{\mathbf{2}},1)_{-1}\\ 52 &= (\mathbf{Adj},1)_0 + (\mathbf{1},3)_0 + (\mathbf{1},1)_0 + (\mathbf{4},1)_0 + (\mathbf{1},1)_2 + (\mathbf{1},1)_{-2} + (\mathbf{1},2)_1 + (\mathbf{1},2)_{-1} + (\mathbf{4},2)_1 + (\mathbf{4},2)_{-1}\\ &\quad + (\mathbf{2},2)_0 + (\mathbf{2},1)_1 + (\mathbf{2},1)_{-1} + (\overline{\mathbf{2}},2)_0 + (\overline{\mathbf{2}},1)_1 + (\overline{\mathbf{2}},1)_{-1} \end{aligned} Putting these, together with the embedding of SU(3)G 2SU(3)\subset G_2, 7 =1+3+3¯ 14 =8+3+3¯ \begin{aligned} 7 &= 1+3+\overline{3}\\ 14 &= 8+3+\overline{3} \end{aligned} into (3), one obtains a completely nonchiral representation of GG.

Update (12/10/2007):

For more, along these lines, see here.

Correction (12/11/2007):

Above, I asserted that I had found an embedding of GG with two generations. To do that, I had optimistically assumed that there is an embedding of SL(2,)SL(2,\mathbb{C}) in a suitable noncompact real form of A 4A_4, such that the 55 decomposes as 5=1+2+25=1+2+2. This is incorrect. It is easy to show that only 5=1+2+2¯5= 1+2+\overline{2} arises. Thus, instead of two generations, one obtains a generation and an anti-generation. That is, the spectrum of “fermions” is, again, completely non-chiral. I believe (but haven’t proven) that this is a completely general result: for any embedding of GG in either noncompact real form of E 8E_8, the spectrum of “fermions” is always nonchiral. Let’s have a contest, among you, dear readers, to see who can come up with a proof of this statement.

I apologize if I’d gotten anyone’s hopes up, with the above example. Not only can one never hope to get 3 generations out of this “Theory of Everything”; it appears that one can’t even get one generation.

Update (12/16/2007):

This post is still receiving a huge number of hits, but no one has taken me up on my challenge above. So let me give the easy part of the proof. Consider, instead of the Minkowskian case (associated to some noncompact real form of E 8E_8), the “Euclidean” case (associated to the compact real form). Instead of SL(2,)SL(2,\mathbb{C}), we’re embedding Spin(4)=SU(2) L×SU(2) RSpin(4)=SU(2)_L\times SU(2)_R. Consider the left-handed “fermions” (the (2,1)(2,1) representation of Spin(4)Spin(4)), which transform as electroweak doublets. If they lie in a generation, then they transform as 3 1/6+1 1/23_{1/6} + 1_{-1/2} under SU(3)×U(1) YSU(3)\times U(1)_Y. If they lie in an anti-generation, then they transform as 3¯ 1/6+1 1/2\overline{3}_{-1/6} + 1_{1/2}. But the 248 is real, ergo the number of generations and anti-generations must be equal, and the theory is non-chiral. QED.

That much was trivial. The gnarly bit is to work out what happens for embeddings of SL(2,)SL(2,\mathbb{C}) which are not related by “Wick rotation” to embeddings of Spin(4)Spin(4) in the compact real form.

Final Update (Christmas Edition)

Still no responses to my challenge. I suppose that the overlap between the set of people who know some group theory and those who are (still) interested in giving Lisi’s “Theory of Everything” a passing thought is empty.

But, since it’s Christmas, I guess it’s time to give the answer.

First, I will prove the assertion above, that there can be at most 2 generations in the decomposition of the 248. Then I will proceed to show that even that is impossible.

What we seek is an involution of the Lie algebra, e 8e_8. The “bosons” correspond to the subalgebra, on which the involution acts as +1+1; the “fermions” correspond to generators on which the involution acts as 1-1. Note that we are not replacing commutators by anti-commutators for the “fermions.” While that would make physical sense, it would correspond to an “e 8e_8 Lie superalgebra.” Victor Kač classified simple Lie superalgebras, and this isn’t one of them. Nope, the “fermions” will have commutators, just like the “bosons.”

We would like an involution which maximizes the number of “fermions.” Marcel Berger classified such involutions, and the maximum number of 1-1 eigenvalues is 128128. The “bosonic” subalgebra is a certain real form of d 8d_8, and the 128128 is the spinor representation.

We’re interested in embedding GG in the group generated by the “bosonic” subalgebra, which is Spin(8,8)Spin(8,8) in the case of E 8(8)E_{8(8)} or Spin(12,4)Spin(12,4), in the case of E 8(24)E_{8(-24)}. And we’d like to count the number of generations we can find among the “fermions.” With a maximum of 128 fermions, we can, at best find

(6)128=?2(2,+(1,1) 0)+2(2¯,¯+(1,1) 0) 128 \stackrel{?}{=} 2 \left(\mathbf{2},\mathfrak{R}+(1,1)_0\right) + 2 \left(\overline{\mathbf{2}},\overline{\mathfrak{R}}+(1,1)_0\right)

where =(3,2) 1/6+(3¯,1) 2/3+(3¯,1) 1/3+(1,2) 1/2+(1,1) 1 \mathfrak{R} = (3,2)_{1/6}+(\overline{3},1)_{-2/3}+(\overline{3},1)_{1/3}+(1,2)_{-1/2} +(1,1)_{1} That is, we can, at best, find two generations.

Lisi claimed to have found an involution which acted as +1+1 on 56 generators and as 1-1 on 192 generators. This, by Berger’s classification, is impossible.

In the first version of this post, I mistakenly asserted that I had found a realization of (6). This was wrong, and I had to sheepishly retract the statement. Instead, it — and Lisi’s embedding (after one corrects various mistakes in his paper) — is nonchiral

(7)128=(2,+(1,1) 0+¯+(1,1) 0)+(2¯,+(1,1) 0+¯+(1,1) 0)128 = \left(\mathbf{2},\mathfrak{R} + (1,1)_0 + \overline{\mathfrak{R}} + (1,1)_0\right) + \left(\overline{\mathbf{2}},\mathfrak{R} + (1,1)_0 + \overline{\mathfrak{R}} + (1,1)_0\right)

The reason why (6) cannot occur is very simple. Since we are talking about the spinor representation of Spin(164k,4k)Spin(16-4k,4k), we should have 2128120 \wedge^2 128 \supset 120 In particular, we should find the adjoint representation of GG in the decomposition of the antisymmetric square. This does not happen for (6); in particular, you won’t find the (1,8,1) 0(1,8,1)_0 in the decomposition of the antisymmetric square of (6). But it does happen for (7). So (6) can never occur. It doesn’t matter which noncompact real form of E 8E_8 you use, or how you attempt to embed GG.

Quod Erat Demonstratum. Merry Christmas, y’all!


1 E 8(8)E_{8(8)} is the split real form of E 8E_8, which also engendered a slew of blog posts.

2 Start with the decomposition of the fundamental representation 16=(1,5)+(1,5¯)+2(2,1)+2(1,1) 16 = (1,5) + (1,\overline{5}) + 2(2,1) + 2(1,1)

3 As before, once you specify how the 22 of SU(2)SU(2) and the 5656 of E 7E_7 decompose under G 0G_0, everything else is determined. And there just aren’t that many possibilities…

Posted by distler at November 21, 2007 11:50 PM

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91 Comments & 9 Trackbacks

Re: A Little Group Theory …

Thank you Jacques for taking the pains going through and put an end to this nonsense!

It is just incredible how the blogosphere distorts things, fueled by the dearest wish to outsmart string physicists and “kick string theory in the back” (Lisi). All this hype and media engineering would be worth a news story on its own.

Boy, those have just no idea of what it takes to do sensible science and create a “better” model inspite hundreds of physicists have tried for decades, and more often than not in such a naive way.

Posted by: Moveon on November 22, 2007 4:26 AM | Permalink | Reply to this

Re: A Little Group Theory …


“Garrett never deigns to tell us which real form of E8 he is using.”

Actually he does, on page 29: E8(24).

“We would like to find an embedding of G=SL(2,ℂ)×SU(3)×SU(2)×U(1)”

But he doesn’t use SL(2,ℂ), he uses SO(4,1). It’s an LQG thing.

Posted by: mitchell porter on November 22, 2007 8:27 AM | Permalink | Reply to this

Re: A Little Group Theory …

Actually he does, on page 29: E 8(24)E_{8(24)}.

I stand corrected. On the last page of the paper, in the penultimate paragraph, he says

This relationship may also shed light on how and why nature has chosen a non-compact form, E IX, of E8.

It’s a bit of a mystery how (and where) that figures into his analysis.

Suffice to say that I did look at both noncompact forms of E 8E_8, and chose to present the one which came closer to actually succeeding.

But he doesn’t use SL(2,ℂ), he uses SO(4,1). It’s an LQG thing.

No. In his paper, he exclusively talks about the “gravitational so(3,1)” group. SO(4,1) never appears. Needless to say, trying to embed the latter group in E 8E_8 would make the problem even worse.

Posted by: Jacques Distler on November 22, 2007 8:53 AM | Permalink | PGP Sig | Reply to this

Re: A Little Group Theory …

Yes, I should have said SO(3,1). Though he does talk about SO(4,1).

Posted by: mitchell porter on November 22, 2007 9:17 AM | Permalink | Reply to this

Re: A Little Group Theory …

Yes, I should have said SO(3,1).

Sigh. As I explained in my post, we’re not interested in SO(3,1)SO(3,1), we’re interested in its double-cover, Spin(3,1)Spin(3,1). And the connected component of Spin(3,1)Spin(3,1) is SL(2,)SL(2,\mathbb{C}).

Posted by: Jacques Distler on November 22, 2007 9:42 AM | Permalink | PGP Sig | Reply to this

Re: A Little Group Theory …

I have the impression that in loop quantum gravity they often don’t use the double cover. But it doesn’t affect your real point, which Garrett has conceded.

Posted by: mitchell porter on November 22, 2007 5:19 PM | Permalink | Reply to this

Re: A Little Group Theory …

Hello Jacques,
It wasn’t my intent to instigate this media frenzy. I apologize for the pressure it put on you to read and discuss the paper, but at the same time I am happy that you did. Your analysis is entirely correct, and focuses on exactly the issue that I agree needs a better understanding in this theory. What this theory does currently have is an embedding of G and one copy of R in E8. The other two copies of R are related to the first by a triality rotation. Under a strict interpretation of the embedding, as you present here, these other two copies of R do not have the same quantum numbers as the first – so by this interpretation they are not good copies. However, they do have the correct quantum numbers under the triality rotated G. This, admittedly, is unsatisfactory hand waving – and I have discussed this inadequacy clearly in the paper, going so far as to explicitly state it is currently the main problem with the theory. Addressing this inadequacy in the theory is what I will be working on over the coming months, and if I or someone else can’t figure out a good way to solve this problem, the theory won’t work. Nevertheless, I consider it nontrivial that G and one copy of R fit in E8 in such a way that the standard model and GR Lagrangian may be efficiently constructed. For this reason, and the prospect of relating the other two generations to R through E8 triality, I consider this to be a developing theory that is worth my time to work on, as a long shot.
Best,
Garrett

Posted by: Garrett on November 22, 2007 12:07 PM | Permalink | Reply to this

While you’re here …

Addressing this inadequacy in the theory is what I will be working on over the coming months, and if I or someone else can’t figure out a good way to solve this problem, the theory won’t work.

I can honestly say that I have no idea what you are hoping for.

Still, since you are here, perhaps you can explain something that has eluded me.

Do you claim that GG can be embedded as a subgroup of F 4(4)×G 2F_{4(4)}\times G_2? If so, can you please enlighten me and write down the decomposition of the (26,1)(26,1) and of the (1,7)(1,7) under GG?

Thanks.

Posted by: Jacques Distler on November 22, 2007 7:01 PM | Permalink | PGP Sig | Reply to this

Re: While you’re here …

Sure, you’re a very smart guy, and I’ll probably learn some things here. The GG is not embedded in F4×G2F4 \times G2 (thanks for the benefit of the doubt). The GG is embedded in a D4×D4D4 \times D4 subgroup of E8E8. If we write GG in terms of the Lie algebra breakdown, g=so(3,1)+su(2)+u(1)+su(3) g = so(3,1) + su(2) + u(1) + su(3) then this is in a so(7,1)+so(8)so(7,1) + so(8) of e8e8 via the Pati-Salam, left-right symmetric model, g=so(3,1)+su(2) L+su(2) R+su(4) g' = so(3,1) + su(2)_L + su(2)_R + su(4) The so(3,1)+su(2) L+su(2) Rso(3,1) + su(2)_L + su(2)_R is in so(7,1)so(7,1), the su(4)su(4) is in so(8)so(8), and a u(1)u(1) from the su(2) Rsu(2)_R and a u(1) BLu(1)_{B-L} from the su(4)su(4) combine to give the weak hypercharge u(1)u(1) in gg. The u(1) BLu(1)_{B-L} in the su(4)su(4) is not in F4F4 or G2G2. I wrote the paper by starting with G2G2 because it was pedagogically useful as the simplest example for getting the point across that I was including the fermions in the adjoint.

The so(7,1)+so(8)so(7,1) + so(8) acts on the 8 S+×8 S+8_{S+} \times 8_{S+} as the first generation of fermions. That part works great. The structure of E8E8 suggests that the second and third generations relate to the triality partners of the first, 8 S×8 S8_{S-} \times 8_{S-} and 8 V×8 V8_{V} \times 8_{V}, but I don’t understand this relationship yet. As you know, and as I described in the paper, these second and third triality partners cannot literally be the second and third generation particles as the theory is currently constructed – the relationship is merely suggestive, and I suspect something more interesting is going on. I will probably end up using a slightly different (non-triality) assignment of the fermions, and may even end up using a different group for gravity. Or I might not be able to get it to work. I’ve tried to be very clear, both in the paper and to the press, that this idea is still in development. Most physicists seem to understand this theory is work in progress, and treat it accordingly – but thank you for spending the time to elucidate this fact so that others will understand.

Posted by: Garrett on November 23, 2007 11:08 AM | Permalink | Reply to this

Re: While you’re here …

Garrett, I will take my opportunity:
Congratulations, very appealing theory - there is a certain amount of beauty (and importance) in it regardless if it’s 100% true. Your paper is very readable, and I’m sure that everyone looking here appreciates it very much that you do your science in such an open way. The same amount of gratitude to Jacques.

Posted by: msal on November 23, 2007 11:35 AM | Permalink | Reply to this

Re: While you’re here …

The G is embedded in a D4×D4D4\times D4 subgroup of E8. If we write G in terms of the Lie algebra breakdown, g=so(3,1)+su(2)+u(1)+su(3)g=so(3,1)+su(2)+u(1)+su(3) then this is in a so(7,1)+so(8)so(7,1)+so(8) of e8e8.

I’m sorry. Now you have me baffled again.

Above, Mitchel Porter pointed me to the passage in your paper where you state that GG is embedded in E 8(24)E_{8(-24)}. Now you say it’s embedded in Spin(7,1)×Spin(8)Spin(7,1)\times Spin(8). That’s not a subgroup of E 8(24)E_{8(-24)}. It is a subgroup of E 8(8)E_{8(8)}, which is the case I analyzed in detail in my post.

If you want people to have a fighting chance of understanding what you are talking about, it would be best to pick one story and stick to it.

Moreover, while the spinor representations of Spin(8)Spin(8) are real, and related by triality to the vector representation, the spinor representations of Spin(7,1)Spin(7,1) are complex, and unrelated by any symmetry to the vector representation (which is real).

Posted by: Jacques Distler on November 23, 2007 12:23 PM | Permalink | PGP Sig | Reply to this

Re: While you’re here …

Ah, I knew I’d learn something here. I’ve been building this model from the ground up, and only recently encountered the potential match to E8E8. It sounds like I made a mistake in thinking that so(7,1)+so(8)so(7,1) + so(8) is in the Lie algebra of EIXE IX, when in fact it’s in the split real form. Thanks.

Posted by: Garrett on November 23, 2007 1:59 PM | Permalink | Reply to this

Re: A Little Group Theory …

That was something that caught my eye and confused me when I looked at the paper the first time.

The triality symmetry you take acts like its simply a symmetry from a larger group as opposed to acting like its something internal. It doesn’t seem like the triality im used too.

Further, its not clear to me at all what generates different mass hierarchies between generations. Even imposing it by hand makes it seem like the spontaneous symmetry breaking terms are working from a much bigger group.

What am I missing?

Posted by: Haelfix on November 22, 2007 5:30 PM | Permalink | Reply to this

Re: A Little Group Theory …

The bottom line is that there isn’t a single “E8 theory” here (I’m just talking classically, since the quantum theory hasn’t been defined). There are three theories, related by triality, and they each only get one generation right.

Posted by: mitchell porter on November 22, 2007 6:50 PM | Permalink | Reply to this

Re: A Little Group Theory …

Gee, Mitchell, you’re on to it. And I don’t think Garrett was claiming to have understood the generation structure.

Posted by: Kea on November 22, 2007 7:12 PM | Permalink | Reply to this

Re: A Little Group Theory …

> I don’t think Garrett was claiming to have understood the generation structure

I am not sure at this point what new insight Garrett *is* claiming to have found.
What do you call a knife without blade where the handle is missing?

Posted by: wolfgang on November 22, 2007 8:53 PM | Permalink | Reply to this

Re: A Little Group Theory …

What do you call a knife without blade where the handle is missing?

An idea of a knife contains the idea of a cutting edge.

Posted by: Kea on November 22, 2007 9:18 PM | Permalink | Reply to this

Re: A Little Group Theory …

An idea of a knife

Yeah, it’s that element of the infamous “wishful thinking” in hep physics:

with no idea being easily proved right, we are left with estimating which ideas are promising. Agreeing on that is much harder than agreeing on what is right, which can already be pretty hard.

Anyway, one idea I definitely like is that of blogs: I found the above quite informative, and mainly because of it containing both an entry and the comments by Garrett.

Posted by: Urs Schreiber on November 23, 2007 6:24 AM | Permalink | Reply to this

Generations

Anyway, one idea I definitely like is that of blogs: I found the above quite informative, and mainly because of it containing both an entry and the comments by Garrett.

It would be more informative if Garrett would answer my question.

For Kea’s benefit, I will add one more remark, perhaps elucidating why this “triality” idea is a non-starter.

Let HH be the representation H=(1,1,2) 1/2H=(1,1,2)_{-1/2} of GG, and H¯\overline{H} be its complex conjugate. Let R=R c+R¯ cR=R_c+\overline{R}_c, where R cR_c is the first line of (2) and its complex conjugate, R¯ c\overline{R}_c, is the second line. There are GG-invariant trilinear forms λ l,λ d: 2R cH λ u: 2R cH¯ \begin{aligned} \lambda_l,\lambda_d:& \wedge^2R_c\otimes H \to \mathbb{C}\\ \lambda_u:& \wedge^2R_c\otimes \overline{H} \to \mathbb{C} \end{aligned}

Since there are 3 generation of RR, each of the λ\lambda’s is a 3×33\times 3 matrix in “generation-space”.

These matrices not simultaneously diagonalizable. At best, you can simultaneously diagonalize two of the three (say, λ l\lambda_l and λ u\lambda_u). The remaining one is ineluctably non-diagonal, and its off-diagonal entries have been measured to be nonzero in the real world.

Posted by: Jacques Distler on November 23, 2007 9:46 AM | Permalink | PGP Sig | Reply to this

Re: Generations

Jacques, thank you for your interest and informative post on this topic. I suppose that many people are looking into this, and it is a great chance to educate, your post brings its best.

I am also waiting for response from Garrett, but regardless if the embedding you ask for is possible, it looks to me that Garrett just found some embedding in E8 where one generation of fermions agrees with everything beautifully, two others only after triality rotation.

What is your opinion on what Garrett proposed in his paper(p.13): that physical fermions may be linear combinations of triality partners (maybe, for some reason, in experiments we only get quantum numbers’ sets we know, others still being there but hidden…?).

Also, as someone already remarked here, it might be that “triality” transformation comes from some larger group.
Thanks for your great post (and exercises).

Posted by: msal on November 23, 2007 11:19 AM | Permalink | Reply to this

Re: Generations

Thanks, Jacques. I realise you probably don’t care, but I’m actually not interested in a triality arising from ordinary representation theory, as has been made clear at great length on my blog.

Posted by: Kea on November 23, 2007 9:09 PM | Permalink | Reply to this

Re: A Little Group Theory …

Jacques Distler wrote:

Those who think I have been too harsh in condemning the Physics blogosphere as an intellectual wasteland can probably point to Wikipedia as being measurably worse.

Wow. That article manages to quote both Motl and Woit, without exploding in a matter-antimatter annihilation. The fact that one can’t learn any physics from it is almost inconsequential by comparison.

Posted by: Blake Stacey on November 23, 2007 10:03 PM | Permalink | Reply to this

Re: A Little Group Theory …

What is the upside to adopting such a contentious and dismissive tone?

If you’re wrong, you’re the poster boy for the leaden establishment physics community.

If you’re right…you still look like an asshole.

Apparently, between you and Motl, it takes sharp elbows to get anywhere in physics. Which I suppose explains why physics has gone nowhere in 20 years.

Posted by: Molon Labe on November 24, 2007 12:14 AM | Permalink | Reply to this

Haggard old crone

I really didn’t want to post on this subject. (It took over a week before I finally broke down and wrote this post.)

You wanna know why?

One of the reasons (admittedly, not the only reason) was that I knew that, if I did write this post, I would have to field comments from people who, unironically, use phrases like “the leaden establishment physics community.”

Thanks for reminding me why I should steer clear of subjects like this in the future.

Posted by: Jacques Distler on November 24, 2007 1:40 AM | Permalink | PGP Sig | Reply to this

Re: Haggard old crone

Thanks for reminding me why I should steer clear of subjects like this in the future.

Dear Jacques,

I actully learnt some E8 representation theory from your post, and also saw why Garrett’s paper doesn’t even work field-contentwise.

When a long paper appears which claims to solve “everything”, and when people like Smolin who are part of the academia endorse it, then the question of “to-read or not-to-read” begins to become a serious one.

Assuming I am not the exceptional (no pun intended) case, this is one of those situations where your scientific journalism saved a lot of people at least some amount of time. So I am not sure that an unqualified “steering clear” of things like this is the right thing to do in the future.

Maybe you should pick and choose?

Warm regards from cold Brussels,
Chethan.

Posted by: chethan on November 26, 2007 5:35 AM | Permalink | Reply to this

Labour-saving devices

Assuming I am not the exceptional (no pun intended) case, this is one of those situations where your scientific journalism saved a lot of people at least some amount of time.

Thanks for the kind words, Chethan.

Obviously, there’s a great collective saving of labour in sparing the rest of y’all the need to go through the tedious group-theoretical analysis yourselves.

However, I’d much rather spend my time discussing topics where the result is actually interesting — where your reaction to reading my post is

Boy, I’d like to look into that more closely!

rather than

Whew! Glad I don’t have to read that paper.

On the other hand, I’ve gotten quite a number of emails, thanking me for this post, for essentially the reasons you have. You’d be surprised at some of the people who, apparently, read this blog.

Under the circumstances, I think I will take consolation in that knowledge.

Posted by: Jacques Distler on November 27, 2007 2:11 AM | Permalink | PGP Sig | Reply to this

Re: Labour-saving devices

You have my thanks as well (and my gratitude also goes to Chethan for making my point before I could find the words).

Having lurked around and learned at least a little from your discussions of topics “where the result is actually interesting”, I hope you continue doing that for a long time to come!

Posted by: Blake Stacey on November 28, 2007 11:38 AM | Permalink | Reply to this

Re: A Little Group Theory …

If you want to take a clue as to the generation structure from Koide’s lepton mass formulas, it is hardly a surprise that the three generations do not match each other in quantum numbers. The three generations show up as a real singlet (the tau generation) and a complex pair (the electron and muon).

The symmetry breaking between the electron and muon families is by a rotation by the mysterious angle delta = 0.22222204717. Gerald Rosen uses 2/9.

The formulas (ignoring an overall scale) are:
sqrt(m) = 1 + sqrt(2)cos(delta + phi)
where phi is 0 for the tau (singlet) and +/- 2 pi/3 for the electron and muon.

Posted by: Carl Brannen on November 24, 2007 12:15 AM | Permalink | Reply to this

I have a theory…

Carl (and the rest of the Anne Elks among you): this is not a forum for airing your pet theories of physics.

If I have to start deleting comments, I will.

Posted by: Jacques Distler on November 24, 2007 12:30 AM | Permalink | PGP Sig | Reply to this

Re: I have a theory…

Carl (and the rest of the Anne Elks among you): this is not a forum for airing your pet theories of physics. If I have to start deleting comments, I will.

As far as me changing the subject to a pet theory, I am the “C. Brannen” that is mentioned on Lisi’s blog with regard to Koide’s mass formula, and my comments are exactly in line with what you are discussing.

And oh, please do delete my comments. I publish things on the web to get priority, not convince arrogant morons of their errors or to get web traffic. You might consider a similar attitude.

Posted by: Carl Brannen on November 24, 2007 4:33 PM | Permalink | Reply to this

Re: A Little Group Theory …

For the SO(3,1) the (3,1) is both spacetime and vectors so I would think the vector part of the Triality should be an emergent spacetime (there’s already an emergent gravity why not an emergent spacetime)? For the E6 GUTs of superstrings, is the vector part of the Triality the bosons? Spacetime seems better than bosons even and still leaves only spinors for the fermions.

The reason D4xD4 seems to be needed for the color/electroweak bosons is that Lisi uses for bosons not just the adjoint for the D4 Triality but also uses its Hodge dual. That’s certainly unusual. In an A-D-E series view of E8, the Hodge dual would be up at the E7 and E8 level. That’s above the E6 GUT level. E7 and E8 are up at the level where Smolin started thinking about the Triality in big Jordan Algebras and how to apply that to string theory (yes Virginia, Smolin has worked on string theory).

Posted by: John G on November 24, 2007 10:08 AM | Permalink | Reply to this

Stringing buzzwords into syntactically-correct sentences

For the SO(3,1) the (3,1) is both spacetime and vectors so I would think the vector part of the Triality should be an emergent spacetime (there’s already an emergent gravity why not an emergent spacetime)? For the E6 GUTs of superstrings, is the vector part of the Triality the bosons? Spacetime seems better than bosons even and still leaves only spinors for the fermions.

The reason D4xD4 seems to be needed for the color/electroweak bosons is that Lisi uses for bosons not just the adjoint for the D4 Triality but also uses its Hodge dual. That’s certainly unusual.

I have no idea what you are talking about.

But, as I explained above, there is no triality for Spin(7,1)Spin(7,1) and, even if there were, it would not help.

(yes Virginia, Smolin has worked on string theory).

Don’t even get me started…

I’m annoyed enough at the apparent intellectual standards of the physics blogosphere.

Posted by: Jacques Distler on November 24, 2007 11:35 AM | Permalink | PGP Sig | Reply to this

Re: Stringing buzzwords into syntactically-correct sentences

Yes I have no reason to doubt you as far as using Triality for the 3 generations but if one uses just the spinors for fermions that leaves the vectors for spacetime. To make the spacetime complex you actually use D5 as in D5/D4xU(1). The spinors will also be complex at E6 as in E6/D5xU(1). As for signature problems, I’m just an unauthorized messenger, but perhaps this from Tony Smith helps:

“Since the more realistic Minkowski Physical SpaceTime with -+++ Signature has Quaternionic Structure, a useful Clifford path is this Clifford Path that is Quaternionic from Cl(3,5) through Cl(2,4) to Cl(1,3) and from Cl(2,6) through Cl(2,5) and Cl(2,4) to Cl(1,3):

http://www.valdostamuseum.org/hamsmith/ClifQPath.gif

D5 is taken to be SL(2,O) = Spin(1,9), and spinors are R16 + R16 of Cle(1,9) = Cl(1,8), where Cle(p,q) = Cl(p,q-1) and Cle(p,0) = Cl(0,p-1)”

All I’m saying is that perhaps Lisi needs to stay unconventional in some areas (like having gravity in the GUT) and become more conventional in other areas (use only spinors for fermions as in E6 GUTs for strings)

Other areas are perhaps more open to debate, like moving bosons into one D4 instead of D4xD4. This would get everything down into an E6 GUT and leave E7 and E8 open to ideas like this from Horowitz and Susskind (perhaps you prefer them to Smolin, this was the paper Smolin was referencing anyways):

http://xxx.lanl.gov/abs/hep-th/0012037

Posted by: John G on November 26, 2007 1:45 PM | Permalink | Reply to this

Re: Stringing buzzwords into syntactically-correct sentences

All I’m saying is that perhaps Lisi needs to stay unconventional in some areas (like having gravity in the GUT) and become more conventional in other areas (use only spinors for fermions as in E6 GUTs for strings)

If you want to include the MacDowell-Mansouri Spin(3,1) 0Spin(3,1)_0, along with the Standard Model gauge group, in E 8E_8, then there is not enough “room” to also include 3 generations of quarks and leptons in the 248. That was what Lisi was aiming for. And I think we are all agreed that it doesn’t work.

If you have some alternative proposal, I invite you to write it up (not here, though).

Posted by: Jacques Distler on November 26, 2007 2:17 PM | Permalink | PGP Sig | Reply to this

Re: Stringing buzzwords into syntactically-correct sentences

Yes I agree you only get one generation. The other two would have to be some kind of “composite” particles with their own math outside of the E8 GUT. Tony Smith may have written some of this up when the archive was at Los Alamos. Since Smith and his former advisor, David Finkelstein, are both thanked in Lisi’s paper, maybe Finkelstein and Georgia Tech might endorse some newer stuff from Smith. Finkelstein seems just as enthusiastic about Lisi as Smolin is.

Those Georgia Tech people seem good at doing things behind the scenes with Clifford Algebra. I personally have more trouble thinking in Clifford Algebra terms. My previous use of the term Hodge dual could have been appropriate if I was talking two D4-like things in Cl(8) but for two D4s in E8, my use of the term was nonsense (I think).

Posted by: John G on November 27, 2007 9:53 AM | Permalink | Reply to this

some wiki herald1X ((delete-item))


“In the language of flowers, the thistle (Germ. DISTEL) (like the burr) is an ancient Celtic symbol of nobility of character as well as of birth, for the Wounding Or Provocation Of A Thistle Yields Punishment. For this reason the thistle is the symbol of the Order of the Thistle, a high chivalric order of Scotland.”

I think instead of engaging in a pleasant conversation with you i rather turn to my sweet cacti.

-Anne Elk (orsomethin)

Posted by: muse on November 24, 2007 12:20 PM | Permalink | Reply to this

Re: A Little Group Theory …

A Layman’s Lament:

It would be clear to the world, if the world were inclined to read this blog, that science is hard! It would be apparent that the interpersonal complexities in the field of science outweigh the intellectual complexities 3 to 1 (or 2.998765 to 1 to be exact ;-)

As a layman, I was led here through 3 other linked sites that have collectively given me sufficient insight, I believe, to place Mr Lisi’s theory in the right context. To that end, the “blogosphere” works.

It suits media to write attractive articles, but that should not condemn every author of an idea to be treated with disrespect, or condemn the reading public as gullible and ignorant. I applaud Mr Distler’s factual critique and similar contributions of others for assisting me set the context for the Mr Lisi’s theory. But I think that for the lack of a pinch of decorum and interpesonal restraint obvious throughout the sites on this issue, the bar has sadly been lifted for the level of bravery needed to have an idea in science. Sigh…..

Posted by: Kin Man on November 24, 2007 6:56 PM | Permalink | Reply to this

Re: A Little Group Theory …

……the bar has sadly been lifted for the level of bravery needed to have an idea in science. Sigh;…..

lol tell that to Galileo

Posted by: damian on November 24, 2007 9:53 PM | Permalink | Reply to this

Re: A Little Group Theory …

Brevity is the soul of wit, and the Truth sets us free:
1) Lisi’s paper does not present a theory of everything.
2) Lisi’s paper makes no predictions.
3) Lisi’s paper offers no novel, sensible calculations.
4) Lisi’s paper is replete with errors, as noted by Distler et al, and it adds fermions and bosons.
5) Smolin called Lisi’s paper “fabulous” and hyped it to the media.
6) Without reading and understanding the paper, Woit gave it a lot of attention on his powerful blog, further fueling the media storm.
7) Bee, who is employed by Smolin, presented the paper as gospel, and only backtracked and reason and logic were brought to the table by others.
8) Lisi’s paper was knowingly mistitled.
9) Lisi must refrain from “ironic” lying titles in the future, so as not to confuse Fox News: http://www.foxnews.com/story/0,2933,311952,00.html :
Laid-Back Surfer Dude May Be Next Einstein — “For his part, Lisi self-mockingly calls his finding “An Exceptionally Simple Theory of Everything.”
In the name of Truth and Science, the following must be done:
1) Lisi should retract his error-riddled, mistitled paper from arxiv.org, whereupon he can feel free to correct the errors, retitle it, and resubmit it, if it is still worth submitting.
2) Smolin should step forth and either apologize to the scientific community for calling the paper “Fabulous,” or he should elaborate on what he meant by the word “Fabulous.”
3) Peter should use his powerful and influential blog to aid in serving truth and science and setting the record straight.

Posted by: Can We Turn The Tide? on November 25, 2007 1:52 PM | Permalink | Reply to this

“Should”

Bee, who is employed by Smolin, presented the paper as gospel, and only backtracked and reason and logic were brought to the table by others.

I am going to take strong exception to your characterization of Bee as Smolin’s lackey. I was very disappointed by her post, but I do not believe it was written at someone else’s behest.

As to what various people “should” do, I would not hold my breath, if I were you.

Posted by: Jacques Distler on November 25, 2007 2:34 PM | Permalink | PGP Sig | Reply to this

Re: Can We Turn The Tide?

if there are mathematical inconsistencies (and even ill defined objects) in Lisi’s paper, the tide will ebb away (almost) on its own (the math. kids will not take long to identify the logical cracks)

Do you believe in the Power of Math?!?

If you wish for the “other side” to lose face (orsomething) you will get served your dish cold after a time (and that is the best way to have it - as the saying goes(!))
I personally hope that we at least will have had some valuable new insights into a somewhat novel approach to a problem of common fascination. Lisi is a coworker, who’s efforts should be appreciated(!). The somewhat sexy title of his paper can’t be called other than apt. The media winds follow their own quirky laws and should not couple too strongly to the monks of the order of True Truth.

Is Lisi a son of Neptun (for all his surfing) who commands the storms? :)

What we should commonly fear is not the erosion of the Broader Ethical Code (of the Broad-sheets, i mean) but the code of ethics in the profession.

Posted by: muse on November 25, 2007 2:40 PM | Permalink | Reply to this

Re: A Little Group Theory …

Nice post. It would make a good final exam question for a first course on baby Lie theory, too. (A 4th year undergrad course, these days, I should point out; something worth pondering.)

And, as no else has mentioned it: there is no Spin_16 in E_8; the group that appears is
a quotient of Spin_16 by a Z/2Z in the center; its the quotient that isn’t SO_16. (This doesnt invalidate anything in the post; in physicist speak it just imposes some restrictions on the multiplets that can arise.)

Posted by: time_wasting on November 26, 2007 2:10 PM | Permalink | Reply to this

Re: A Little Group Theory …

Nice post. It would make a good final exam question for a first course on baby Lie theory, too. (A 4th year undergrad course, these days, I should point out; something worth pondering.)

I was going to point out that everything I had written above could easily be worked out by a bright undergraduate. But then I thought better of it.

I’m guessing that the noncompact real forms of the exceptional Lie groups are still a bit beyond the level of a contemporary undergraduate course. But we’re awfully close.

And, as no else has mentioned it: there is no Spin 16Spin_{16} in E 8E_8; the group that appears is a quotient of Spin 16Spin_{16} by a /2\mathbb{Z}/2\mathbb{Z}.

You are absolutely right. I was deliberately not being careful about such factors of 2\mathbb{Z}_2. In similar fashion, the maximal compact subgroup of E 8(24)E_{8(-24)} is not, as I wrote, SU(2)×E 7SU(2)\times E_7, but rather (SU(2)×E 7)/ 2(SU(2)\times E_7)/\mathbb{Z}_2.

This doesn’t really affect anything I said. So, for ease of presentation, I omitted these details.

Posted by: Jacques Distler on November 26, 2007 2:29 PM | Permalink | PGP Sig | Reply to this
Read the post Can We Turn The Tide? comments on "A Little Group Theory ..."
Weblog: Thorny Path of 7 News
Excerpt: Great, This is now on my Thorny Path.
Tracked: November 27, 2007 1:14 AM

Re: A Little Group Theory …

Can we turn the tide:

Bee, who is employed by Smolin, presented the paper as gospel, and only backtracked and reason and logic were brought to the table by others.

I am not employed by Smolin. He neither knew what I was writing, nor am I sure he read it.

Jaques:

I was very disappointed by her post, but I do not believe it was written at someone else’s behest.

Thanks. I don’t know much about the group structure of E8, and I prefer to stick with writing about what I know. I think my opinion comes across very clearly in what I’ve written.

Best,

B.

Posted by: Bee on November 27, 2007 12:17 PM | Permalink | Reply to this

Your post

I think my opinion comes across very clearly in what I’ve written.

I don’t think I said your post was unclear.

In brief, you praised the group-theoretical aspects of Lisi’s work

I really like this part. He unifies the SM with gravity while causing only a minimum amount of extra clutter.

but went on to complain that his construction for reproducing the Standard Model is ad-hoc, and not particularly beautiful

I find this construction neither simple nor particularly beautiful. …The most unattractive feature are the extra assumptions he needs to write down an action that gives the correct equations of motion.

This all sounds very “reasonable” and “balanced.” It is also utterly false.

As we’ve seen, the group-theory is wrong. And the action he constructs is, most certainly not the Standard Model. (Hint — to pick one of many problems: where’s the Kobayashi-Maskawa matrix in Lisi’s action?)

You say you’re not familiar with E 8E_8? Fine … but surely you are familiar with the Standard Model. If you’d actually taken the time to look at his action, I don’t know how you could have come up with the characterization of it that you did

[W]ith the chosen action he is able to reproduce the adequate equations of motion. … Given that he has to choose the action by hand to reproduce the SM, one can debate how natural this actually is.

Posted by: Jacques Distler on November 27, 2007 1:58 PM | Permalink | PGP Sig | Reply to this

Re: A Little Group Theory …

Hi Jacques: Excuse me, but what ‘I find neither simple nor beautiful’ or consider the least attractive feature is my opinion. You can disagree on my liking or disliking, but “utterly false” is not quite appropriate. I admit that I did not check the details of the group decomposition, and I want to sincerely apologize if I missed a point that you consider essential, but I was not writing a referee report (and I would never have agreed on writing one for good reasons). Given that my main objection is that Garrett has to pick the action by hand to make things work at least somehow, why would I spend days on the details of the ‘somehow’ if I find the whole procedure inappropriate to begin with if one wants to call it a TOE? It is not hard to approximate your behavior and predict that you will cheerfully make fun of this view of mine, because I consider other things important than you do. Maybe you should take a deep breath and remind yourself that you should be glad not everybody thinks the way you do. - B.

Posted by: Bee on November 27, 2007 5:27 PM | Permalink | Reply to this

Reading…

Hi Jacques: Excuse me, but what ‘I find neither simple nor beautiful’ or consider the least attractive feature is my opinion. You can disagree on my liking or disliking, but “utterly false” is not quite appropriate.

Please, Bee. What’s “utterly false” is the statement that Lisi writes down an action that “reproduce[s] the SM”. That you manage to pass æsthetic judgement on something that Lisi doesn’t actually do in his paper is another matter.

Posted by: Jacques Distler on November 27, 2007 7:37 PM | Permalink | PGP Sig | Reply to this

Re: A Little Group Theory …

Now now Jacques, you’re just being mean–how could he get the right CKM matrix without having three generations to begin with :-) ?

More seriously, while you have done a very nice job of showing that Lisi’s work doesn’t even rise to the level of impressive numerology (which it might have done had the group theory worked out), you might want to expand a little on all the physics reasons it doesn’t make any sense. As many people pointed out, in this work the E8 plays no dynamical role since the action breaks it at order 1–and there can be no regime where the E8 does make sense with any usual notion of “make sense” for the reasons of, um, not being able to combine fermions and bosons, (never mind C-M, though, I have to say, saying that dS is the loophole to C-M is classic–hey, the metric around the earth isn’t minkowski space either, in fact the deviation is a lot bigger than from the dS expansion of the universe–what a great loophole!). But perhaps we are being too harsh, like all great ideas in physics, this idea for a TOE-putting things “together” then pulling them apart to get the equations of motion-goes back to Feynman and his famous “U = 0” theory, where U = (F - ma)^2 + dot dot dot…

One can go on and on. Lubos did a bit of the physics analysis. As usual, he was a terrible jerk and a*hole, but also as usual, his analysis was correct. Unfortunately in the physics blogosphere, it is more important to be courteous and polite than to be right. This went a long way to fueling the media storm. Though I think the hype and fuss is not all Lisi’s fault; the LQG community and Smolin in particular are largely to blame for giving it cred to the press.

There is one positive thing that will come out of this debacle. Physicists outside the field perhaps couldn’t judge the relative standards of the LQG/alternative physics and string communities before. Now they get to see what kind of work Smolin thinks is “fabulous”, what Rovelli wonders why he missed, and what Woit thinks is “serious work”. They can then draw their own conclusions….


Posted by: funfunfun on November 27, 2007 6:31 PM | Permalink | Reply to this

Re: A Little Group Theory …

The thing that really p*sses me off about this is that it is not that long ago that a certain unnamed physicist based not a million miles from Toronto was giving smug patronising lectures to the rest of the community, and worse to the general public, on `ethics in science’, the importance of not overstating results, and how the LaLaLoop community was a group of smart, skeptical, original, independent minds who were careful to report results accurately.

And then: `Fabulous - the most compelling unification model in many, many years’

I don’t blame Lisi. He seems like a nice enough guy, even though the paper is wrong. But theoretical physics is a really important subject, and there is a responsibility to the public to be accurate. If very senior full professors say this to journalists, they will listen - why shouldn’t they? They don’t know E_8, they rely on experts to evaluate the work accurately.

Theoretical physics has become a public laughing stock because of this and it is humiliating for the subject.

Posted by: piscator on November 29, 2007 3:22 PM | Permalink | Reply to this

Re: A Little Group Theory …

Bee said that Garrett “… unifies the SM with gravity while causing only a minimum amount of extra clutter …”.

Jacques replied “… the group-theory is wrong. And the action he constructs is, most certainly not the Standard Model. …
where’s the Kobayashi-Maskawa matrix in Lisi’s action? …”.

Jacques also said in an earlier comment:
“… If you want to include the MacDowell-Mansouri Spin(3,1) 0, along with the Standard Model gauge group, in E 8, then there is not enough “room” to also include 3 generations of quarks and leptons in the 248. That was what Lisi was aiming for. And I think we are all agreed that it doesn’t work. …”.

Garrett said in an earlier comment that he “made a mistake” using EIX = E8(-24) when he should have been using EVIII = E8(8).

When you modify Garrett’s work in terms of E8(8), you can see that it not only works in general terms, but also allows things that you (Jacques) and Bee have requested, such as calculation of coupling constants, Kobayashi_Maskawa parameters, etc.,
although, as you (Jacques) correctly pointed out, the second and third generations of fermions are not directly in the 248 by triality.

I have put up a pdf file about that on dotMac at
web.mac.com/t0ny5m17h/Site/E8GLTSCl8Cl16.pdf

Would you endorse it for posting on hep-th where Garrett’s 0711.0770 is posted ?

Tony Smith

Posted by: Tony Smith on November 27, 2007 10:12 PM | Permalink | Reply to this

Re: A Little Group Theory …

funfunfun,

Whoever you are, even if you think it’s acceptable behavior to attack people from behind the cover of anonymity, you should at least make some effort to not put things in quotes that I didn’t write. As far as I can tell the description “serious work” you attribute to me is not something I wrote.

As for what physicists outside the field think of all this, if they read blogs I suspect they would agree with your characterization of Lubos’s personality, but are also noticing that he’s not the only one…

Posted by: Peter Woit on November 27, 2007 11:01 PM | Permalink | Reply to this

Re: A Little Group Theory …

I can’t speak for Bee, but personally I just took it on faith that Lisi’s model unifies all fields of the Standard Model and gravity as he explicitly claims in the abstract of his paper. Why? Because of the reactions and comments of Smolin and Rovelli. Surely they wouldn’t have said what they did if such a basic thing was wrong. Don’t they care about their reputations?

Like everyone else I appreciate Prof. Distler’s clarification of the situation. Actually, Aaron Bergman had already pointed out somewhere in the blog discussions that all was not well with the group theoretical stuff, but it is nice to have it explicitly laid bare here.

Posted by: amused on November 28, 2007 12:51 AM | Permalink | Reply to this

Re: A Little Group Theory …

Nothing that was not in Lisi’s paper has been “laid bare” here.
Also adding fermions to bosons is perfectly correct due to special representation features of E8.
I don’t know what will be the final judgment on E8 theory, but apparently critics have made up their minds long before applying reason.

Posted by: observer on November 29, 2007 8:55 PM | Permalink | Reply to this

Magical thinking

[A]dding fermions to bosons is perfectly correct due to special representation features of E8.

Care to explain what “special representation features of E8” those might be?

Are there other groups that share those “special representation features”? Or is E 8E_8 the only group sprinkled with the magic pixie dust?

Posted by: Jacques Distler on November 29, 2007 9:06 PM | Permalink | PGP Sig | Reply to this

Re: Magical thinking

Sure. I’ll just cite Lisi here:

The second trick is that we’re also including all the fermions in this superconnection, as Lie algebra valued Grassmann numbers. Now, at first look, this second trick shouldn’t work. When we calculate the dynamics of this connection by taking its curvature, the interactions between fields will come from their Lie bracket. But we know gravity and the gauge fields interact with the fermions in fundamental representations. The fermions, such as this Dirac spinor column of spin up and spin down left and right chiral fields, live in a fundamental representation space, and these certainly don’t appear to be Lie algebra elements. So how can this possibly work? Well, it turns out that for all five exceptional Lie groups, there are Lie brackets that act like the fundamental action. The structure of these algebras is such that some Lie algebra elements ARE fundamental representation space elements. This fact makes it possible to include the fermions in the connection as Lie algebra valued fields.

(this is an excerpt from a seminar given by Lisi to physicists, Smolin among them)


So, in fact fermions are fundamental representation space elements but also like Grassman numbers and interaction of these through Lie bracket might define Standard model interactions. Though, I must admit this looks magic.

Posted by: observer on December 1, 2007 8:05 AM | Permalink | Reply to this

Re: Magical thinking

If you think there something wrong with that, i’ll listen carefully.
I’m trying to include links to the transcript from the seminar:
transcript, PDF slides, audio

Posted by: observer on December 1, 2007 8:25 AM | Permalink | Reply to this

Re: Magical thinking

If you’re going to quote long verbatim passages from Lisi, the least you could do is choose ones relevant to the point at hand.

In the passage you quoted, Lisi explains why he thinks he can find copies of the representation RR inside the 248-dimensional adjoint representation of E 8E_8.

That was the subject of this post.

The fact that some of the generators of the 248 are to be interpreted as boson, while some others as fermions (rather than bosons transforming in the 22 of SL(2,)SL(2,\mathbb{C})), is an entirely different matter.

It’s this latter issue that people are talking about when they complain that Lisi is “adding fermions to bosons.”

Wanna try again?

And this time, try putting the argument in your own words, rather than uncomprehendingly quoting passages of Lisi’s that you think might be relevant.

Posted by: Jacques Distler on December 1, 2007 10:02 AM | Permalink | PGP Sig | Reply to this

Re: Magical hoping, critical thinking

From Lubos Motl’s post I got the impression that he made exactly this accusation - that you can’t add fermions to bosons because they’re of a different kind (different representation space), and Lisi explanation given sort of responds to that.

I think it’s clear that Standard Model cannot be fit exactly into E8(8) (thanks to your post too, but you can find it in Lisi’s paper), but fits nicely modulo triality. If triality rotation cannot be explained physically (another quantum number?, other fermion families unification?) then E8 no longer stands. Because of the appealing beauty of the theory, I hope it will work, and certainly hope it will be treated by science properly. Are you 100% sure that the case is over? (if i make some basic mistakes - sorry - my (limited) expertise is only mathematics, not physics, and not group theory, in such case: sorry for wasting your time)

Posted by: observer on December 1, 2007 11:32 AM | Permalink | Reply to this

Magical thinking

… but fits nicely modulo triality.

No it doesn’t, as I have already explained.

First, because there is no triality for Spin(7,1)Spin(7,1), and second because — even if there were, and even if Lisi found a way of using it to build a theory — invoking triality would not be compatible with the nontrivial (“Kobayashi-Maskawa”) generation-mixing that one finds in the Standard Model.

Are you 100% sure that the case is over?

If you’re asking a sociological question, I think that, thanks to the enthusiastic endorsements of Smolin, Rovelli and others, this “Theory of Everything” will live on for quite some time.

If you are asking a scientific question, …

Posted by: Jacques Distler on December 1, 2007 3:36 PM | Permalink | PGP Sig | Reply to this

Re: Magical thinking

? am I completely lost here?
Garrett responded to you that he embedded G in a D4×D4 subgroup of E8(8) and the mistake he made was only(?) thinking that so(7,1)+so(8) fits into EIX, while it is E8(8), thus I concluded that his paper still stands after the name of the group is changed (see link below) (and Garrett should, and I think he will, thank you in the next version of the paper).

Speaking of second reason - why would I need it if the first one holds? So are you saying that Garrett is completely wrong with his fit? Then, I know that fixing triality problem is difficult, but I understood that this is not so clear and other physicists issuing comments weren’t seeing this as non starter at all.

Here is what Garrett has to say about the mistake you’ve found: Lisi’s post at AdvancedPhysics

Posted by: observer on December 1, 2007 6:18 PM | Permalink | Reply to this

Re: Magical thinking

? am I completely lost here?

Apparently.

Spin(8)Spin(8) has three inequivalent real representations, the vector representation, 8 v8_v, and the two spinor representations, 8 s8_s and 8 c8_c. And it has a triality symmetry (an outer automorphism) which permutes these three representations.

For Spin(7,1)Spin(7,1), the vector representation is real, as before. But the two spinor representations are complex (and are complex conjugates of each other). There is no triality symmetry permuting the three representations.

(There is, as for any Spin(2n1,1)Spin(2n-1,1), a parity symmetry which exchanges the two spinor representations, but that’s all.)

Both groups go by the name “D4.” But they are different real forms and have very different properties.

Speaking of second reason - why would I need it if the first one holds?

Because you want to reproduce the Standard Model.

[O]ther physicists issuing comments weren’t seeing this as non starter at all.

I will leave you to draw your own conclusions from that observation.

Posted by: Jacques Distler on December 1, 2007 6:49 PM | Permalink | PGP Sig | Reply to this

Re: Magical thinking

Ok, I got it - the three fermions families in Lisi’s representations are like e ik2π/3,k=0,1,2e^{i\cdot k \cdot 2\pi / 3}, k = 0,1,2 but without e i2π/3e^{i \cdot 2\pi/3} element in the group to rotate around them, only conjugation (hence any new quantum number does not emerge from triality)

Lisi hopes that this still can give some insight into SM “somehow”, you think it has already failed by what you’ve shown. As for me, I can only wait (and learn). Thank you a lot.

Posted by: observer on December 1, 2007 7:32 PM | Permalink | Reply to this
Read the post contempt in physics
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Excerpt: A cautionary tale for our times.
Tracked: November 28, 2007 10:25 AM
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Tracked: November 28, 2007 9:32 PM

Re: A Little Group Theory …

Jacques, it’s a bit inconsistent to slam Bee for just focusing on the physics side of things while letting this comment from “funfunfun” go unremarked:

“Lubos did a bit of the physics analysis. As usual, he was a terrible jerk and a*hole, but also as usual, his analysis was correct. “

Lubos didn’t pick up on the fact that Lisi had failed to embed the whole standard model + gravity into E8 either…and unlike Bee he also failed to notice that the ad hoc action Lisi cooks up is not E8 gauge invariant (otherwise he (Lubos) wouldn’t have been going on about Coleman-Mandula). The E8 noninvariance means that, as it stands, Lisi’s model can no way be regarded as a “unified theory”, something Bee rightly emphasized and Lubos failed to mention.

I look forward to you slamming Lubos at least as hard as Bee ;-)

Posted by: amused on November 30, 2007 5:31 AM | Permalink | Reply to this

Food-Fight

I look forward to you slamming Lubos at least as hard as Bee ;-)

If I had read Luboš’s blog post (which I have just now done), I probably would have concluded that he hadn’t really read the paper either, before criticizing it.

But, if I were to write such a thing, Luboš would, justifiably, feel obligated to show up to defend himself. And then his hordes of detractors would show up and, in the ensuing food-fight, we could kiss this once-useful comment-thread goodbye.

Thanks alot, fella!

(Now, you’ll excuse me while I go and delete some comments from Luboš-detractors, who’ve shown up early for the food-fight.)

Posted by: Jacques Distler on November 30, 2007 8:06 AM | Permalink | PGP Sig | Reply to this

Re: A Little Group Theory …

As a side issue, what is the status of Coleman-Mandula for gravity? In vol III, Weinberg says that (as far as he knows) there is no C-M theorem for nonabelian gauge theory in a Coulomb phase, because you can’t give the gauge bosons an IR-regulator mass. This presumably would apply to gravitons as well.

Seems like a technical loophole, but does anyone know if it’s ever been closed?

Posted by: Mark Srednicki on November 30, 2007 3:08 PM | Permalink | Reply to this

Coleman Mandula

Do we really need an IR-regulator mass? Or is that just a way to ensure the existence of an S-matrix (which is not a given, in the presence of massless particles).

My impression is really the latter.

But, even if I’m right, you’ve only pushed off the question to: “Is there really an S-matrix for gravity?”

Posted by: Jacques Distler on November 30, 2007 9:21 PM | Permalink | PGP Sig | Reply to this

Re: A Little Group Theory …

Well, as far as I know, there’s never an S-matrix with massless particles in d=4. Only infrared-safe things like jet-jet cross sections exist. According to your colleague, in QED you can give the photon a fictitious mass, invoke Coleman-Mandula, then construct infra-red safe thingies, then take the photon mass to zero. You’re left with infrared-safe thingies that can’t have nontrivial symmetries mixed with Lorentz symmetry (because you ruled that out at the intermediate stage when you did have an S-matrix with massive photons). This intermediate step appears to fail in gravity (and Coulomb-phase nonabelian gauge theory). It seems to me very likely that this is a mere technicality, but it would be nice if the loophole were closed (or, better, slipped through in an interesting way!).

On the other hand, in d>6, there are no infrared divergences, the S-matrix exists, and presumably Coleman-Mandula holds.

Posted by: Mark Srednicki on November 30, 2007 10:43 PM | Permalink | Reply to this

Infrared divergences

On the other hand, in d>6d\gt 6, there are no infrared divergences, the S-matrix exists, and presumably Coleman-Mandula holds.

Yeah, sure, I was tacitly assuming that we were in sufficiently high dimensions that an S-matrix could plausibly exist. For d<7d\lt 7, tree-level graviton exchange makes a divergent contribution to the total elastic cross-section, and hence there’s no S-matrix. This figured in an old post here on this very blog.

But you really want to talk about 4 dimensions. As you say, one would need some sort of “soft-graviton resummation,” of the sort one does with soft photons.

I don’t know of any such construction, either. Where are the experts when we need them?

Posted by: Jacques Distler on November 30, 2007 11:14 PM | Permalink | PGP Sig | Reply to this

Re: Infrared divergences

Hi Jacques
You asked:
“Where are the experts when we need them?”
I think the answer to that question is: “just down the corridor”. See the paper:

Infrared photons and gravitons.
by Steven Weinberg
Phys.Rev.140:B516-B524,1965.

where he notes in the abstract:
“It is shown that the infrared divergences arising in the quantum theory of gravitation can be removed by the familiar methods used in quantum electrodynamics. An additional divergence appears when infrared photons or gravitons are emitted from noninfrared external lines of zero mass, but it is proved that for infrared gravitons this divergence cancels in the sum of all such diagrams. (The cancellation does not occur in massless electrodynamics.)”

Thanks for taking the time to wade through the group theory for this post; you’ve saved many people a lot of time.

Best regards to both you and Mark.

Bruce

Posted by: bruce on December 1, 2007 1:48 PM | Permalink | Reply to this

Re: Infrared divergences

Thanks, Bruce! That’s exactly the reference Mark and I were seeking.

And, yes, I probably should have just asked Steve. But, then I wouldn’t have gotten to hear from you.

So this works out well …

Posted by: Jacques Distler on December 1, 2007 3:45 PM | Permalink | Reply to this

Re: Infrared divergences

I know the Weinberg paper (a true classic); it’s not on point here. Weinberg shows how to compute IR-safe cross sections in gravity (as long as the matter fields are massive). But the distinction between abelian theories and nonabelian theories (including gravity) is that, in abelian theories, you can give the gauge fields a mass and construct an S-matrix as an intermediate stage in constructing the IR-safe cross section. This doesn’t work for nonabelian theories (which was pointed out by Weinberg in his book). With the intermediate-stage cross section, you can invoke Coleman-Mandula. Without it, you can’t. That’s the issue (that I was raising). I thought perhaps someone had closed this loophole in the intervening years.

Not that any of this is relevant for Lisi’s proposal, which appears to me to be content free.

Posted by: Mark Srednicki on December 1, 2007 8:31 PM | Permalink | Reply to this

Re: A Little Group Theory …

A seperate issue and one of the physics issues that is relevant to the paper afaics is how to make the limit between where an Smatrix exists (lambda = zero) and where it doesn’t well defined.

One of the experts can chime in maybe, but isn’t there a certain amount of issues even in the well understood AdS case, where we send lambda –> zero. How exactly does the Smatrix emerge in that limit, it seems to me to be quite discontinous.

Posted by: Haelfix on December 1, 2007 1:34 AM | Permalink | Reply to this

Re: A Little Group Theory …

I am grateful for this posting. I didn’t walk away from my physics education with the skills to make the group calculation that Jacques does here. Is this a skill you would learn in a Algebra class?

Take care,
mike v

Posted by: Mike V on December 1, 2007 12:46 PM | Permalink | Reply to this

Re: A Little Group Theory …

Where can I learn this SL(2,ℂ), Spin(16) -stuff? Could you give a URL to a good group theory book?

Thanks!

Posted by: Henri Heinonen on December 5, 2007 4:04 AM | Permalink | Reply to this
Read the post A Little More Group Theory
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Excerpt: Lisi again???
Tracked: December 10, 2007 2:44 AM

Re: A Little Group Theory …

A naive question: if the only massless particles in your theory are Goldstone bosons, there are no infrared divergences and the S-matrix exists, is this correct?

Thanks!

Posted by: anon on December 12, 2007 12:07 AM | Permalink | Reply to this

Goldstone bosons

Goldstone bosons do, indeed, decouple at zero momentum, so if the only massless particles were Goldstone bosons, there would be no problem (in d3d\geq 3).

Unfortunately, we’re talking gravity here. For d6d\leq 6, there’s no S-matrix. But it is possible to formulate “infrared-safe” quantities, as in QED. Unlike QED, it’s not 100% clear that Coleman-Mandula applies to those infrared-safe quantities.

Posted by: Jacques Distler on December 12, 2007 12:29 AM | Permalink | PGP Sig | Reply to this

Re: Goldstone bosons

Thanks for your reply!

On a different note, I’m confused about the statement that the S-matrix, for theories with gravity, exists in d>6. Don’t infrared divergences arise in diagrams with soft graviton emission from external particle lines? It seems like the d=4 analysis I learned from Weinberg vol. 1 ch. 13 (the additonal particle propagator diverges as the graviton momentum goes to zero, but interaction vertex does not vanish in this limit) holds for any number of dimensions. Am I missing something?

Posted by: anon on December 12, 2007 12:49 PM | Permalink | Reply to this

Soft gravitons

The infrared divergences, discussed in Section 13.2 of Weinberg, associated to soft virtual gravitons, go away for d7d\geq 7. So, then, does the necessity to “cancel” those divergences, by the procedure of Section 13.3.

More prosaically (the first place you see the problem): the elastic scattering amplitude (and hence the differential cross section) still diverges in the strictly forward direction. But that divergence is integrable for d7d\geq 7.

Posted by: Jacques Distler on December 12, 2007 1:11 PM | Permalink | PGP Sig | Reply to this

Re: Soft gravitons

I see, that helps a lot! Thanks for taking the time for these silly questions.

Posted by: anon on December 12, 2007 2:57 PM | Permalink | Reply to this

Re: A Little Group Theory …

Thanks, Jacques, and a happy silly season.

Posted by: Kea on December 26, 2007 12:49 PM | Permalink | Reply to this

Re: A Little Group Theory …

It was pointed out somewhere that looking for physical fermions and an embedding of D4+D4 might be irrelevent cause Lisi’s use of fermions doesn’t require physical ones and his D4+D4 is after symmetry breaking. I think you have to be up at E8/D8 to have physical fermions and use of the E8 symmetry. This isn’t Lisi’s view however, he I think might be looking at complexified E8 (I’m not personally familiar with complexified E8) which I guess would produce something other than E8/D8?

Posted by: John G on December 28, 2007 5:38 PM | Permalink | Reply to this

Re: A Little Group Theory …

Is complex E8 where the polytope is in 4 complex dimensions instead of 8 real ones?

Posted by: John G on December 28, 2007 5:44 PM | Permalink | Reply to this
Read the post Une théorie de Tout...en lisant Science et Vie (épisode 3)
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Excerpt: Le plus diffusé des magazines de vulgarisation scientifique, Science et Vie, consacre un dossier de couverture à la théorie de Garrett Lisi (voir les épisodes précédents ici et là. J'aurais préféré, par fierté confraternelle, apprendre qu...
Tracked: January 15, 2008 8:15 AM
Read the post Superconnections for Dummies
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Excerpt: A brief review of Quillen superconnections, and an introduction to the "Schreiber" superconnection.
Tracked: May 11, 2008 10:02 PM
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Excerpt: By popular demand, another post on this stuff.
Tracked: September 15, 2008 10:48 AM
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Excerpt: Like Freddy Krueger ...
Tracked: June 19, 2009 12:06 AM

Re: A Little Group Theory …

I don’t understand two things:

1) If usage of E8 group by Garret is so wrong, why string theory is using it too?
2) Why not to use some larger group, for example Monster group? It’s evident, real world is not just 24 dimensional..

Posted by: Zephir on March 7, 2010 2:42 AM | Permalink | Reply to this

Re: A Little Group Theory …

I don’t understand two things

Really? Only two things?

Why not to use some larger group, for example Monster group?

The Monster Group is a finite group.

Posted by: Jacques Distler on March 14, 2010 11:19 AM | Permalink | PGP Sig | Reply to this

Re: A Little Group Theory …

As I suspected, it didn’t take you long to see through Zephir’s ‘Moonshine’.

Thanks for a fascinating read or three!

Posted by: Muggins on May 20, 2010 10:04 AM | Permalink | Reply to this

Re: A Little Group Theory …

Hello, Physicists. I went to school with Garrett, and we used to call him the “Anaconda,” or, “El Snake (pronounced snah-kay),” because he used to cut in front of people with his longboard on waves. I know, Garrett, that was a long time ago. Is that what he’s doing to you guys now?
James

Posted by: James on April 13, 2010 8:30 PM | Permalink | Reply to this
Read the post Crib Notes
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Excerpt: Oh no! NOT Lisi, again!
Tracked: June 27, 2010 9:19 PM

Re: A Little Group Theory …

Garrett has posted a new paper, about a month ago..”An Explicit Embedding of Gravity and the Standard Model in E8”.

My understanding is that this new paper is about how the algebra of the gauge fields acting on fermions can be described by real matrices which relate to E8. I think the punchline is that he predicts a range of particles for the Higgs boson.

Not sure. Did anyone have any thoughts about the new paper ? Did he fix the bugs.

My understanding of the criticisms leveled here against the original paper were that they are pretty fundamental.

However, I am not sure if the new paper addresses the issues raised?

Posted by: turnerbroadcasting on September 8, 2010 12:47 PM | Permalink | Reply to this

Re: A Little Group Theory …

I really wasn’t going to post about Garrett Lisi’s paper. Preparing a post like this requires work and, in this case, the effort expended would be vastly incommensurate with any benefit to be gained.

How do you feel about all the effort now almost 4 years later? :p

I’m not a mathematician, nor particularly versed in physics, but I do love astronomy and astrophysics topics.

I just saw an episode of “Through the Wormhole” that talks about Garrett’s “Theory”. After googling his name and E8 I came across this blog entry.

I’ve read through most of the replies and all, and while Garrett may be wrong and you may be right, I find you sir, as a person lacking in social decency, in your responses and your initial post. Arrogant, mean spirited, and just generally rude.

If you were a poster child for mathematics and physics recruitment, I think I and most everyone I know would run the other way.

I think I’d much rather know Garrett on a personal level (not that I do, or ever will) even if his theory turns out to be bull$#!+.

Posted by: Jack on June 26, 2011 4:36 PM | Permalink | Reply to this

Re: A Little Group Theory …

How do you feel about all the effort now almost 4 years later? :p

Completely and utterly wasted.

Multiple blog posts, and a duly-refereed paper, published in the most prestigious journal in mathematical physics, had not the slightest impact on the Lisi publicity juggernaut (as you, yourself, have so amply demonstrated).

I just saw an episode of “Through the Wormhole” that talks about Garrett’s “Theory”….

A more depressing example of the state of science-journalism could hardly be imagined.

Posted by: Jacques Distler on June 28, 2011 6:13 AM | Permalink | PGP Sig | Reply to this

Re: A Little Group Theory …

Ouch. I remember reading this article when it was first published. I too noticed the Lisi “publicity juggernaut”, and was a little sceptical of the lack of substance behind many of the claims. This post helped to reinforce those doubts.

I see Lisi tried to hand-wave his way around your critique of his paper, and still to this day appears to have no useful or falsifiable predictions from his theory.

As an engineering scientist, I need a pretty solid foundation in physics but I’m not a theoretical physicist, and could not have carried out the analysis performed here myself (although I did learn something from it!). It isn’t a big deal, but I thought you should know someone out there appreciates the time and effort you put into this, Jacques!

Posted by: Spence on July 17, 2011 12:06 PM | Permalink | Reply to this

Re: A Little Group Theory …

The wikipedia page on Lisi’s theory has been protected from editing for the multiple attempts to revert it to a previews version. If people here are interested they can go give their two cents in the talk page, here:
http://en.wikipedia.org/wiki/Talk:An_Exceptionally_Simple_Theory_of_Everything#What_is_the_purpose_of_hiding_the_truth.3F

(don’t be supporters or detractors, just state things calmly and give your opinion supported with facts to help the dispute, it seems like the page is finally coming to a correct formulation and it would be good to have support by impartial physicists)

Thanks!

Posted by: Dan on July 2, 2011 4:00 PM | Permalink | Reply to this

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