Garrett Lisi (New Yorker 21 July issue)

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Sorry there was a fuss. Some people informed me that the authors of the email exchange, Ben Wallace-Wells and Bertram Kostant, had not given permission for their email to be posted online. So, at their request, I asked the PF administrators to remove the posts.

However, I have now emailed Ben and Bert and received their permission (encouragement in fact) to post part of the email exchange, which I have copied below. Thanks again to humanino and grosquet for the transcription.


Dear Ben Wallace-Wells,

The following is my response to your queries. In order to answer your question about the Lie group E(8), I found it necessary in the first paragraph to add some historical context. I hope it is not too burdensome to read.

Lie (pronounced Lee) group theory was developed by mathematicians towards the end of the 1800's. An important accomplishment at that time was also a classification of the simple Lie groups. It turned out there were 4 infinite families and 5 exceptional Lie groups, the largest (containing all the others) of which is E(8). There is an unfortunate double usage here of the word "simple". There is of course, the everyday usage (eu) meaning easy to understand and a technical use (tu) meaning not built up from other groups. For example, the title of Lisi's paper is "An Exceptionally Simple Theory of Everything". His use of Exceptionally Simple is a pun. The exceptional refers to the exceptional Lie groups and simple is (tu). Lie groups started entering physics in a serious way at the beginning of the twentieth century. Perhaps more prominent was Einstein's theory of special relativity, where the Lie group involved was the Lorentz group. This is a (eu) group and occupies only a very tiny sliver of something as sophisticated as E(8). Also Bohr's theory of atomic spectra uses the rotation group SO(3) and again is an (eu) and a very tiny sliver of E(8). For the most part, Lie groups were more or less put on the "back burner" by both mathematicians and physicists until the middle of the twentieth century, At that time, it became a serious object of study by mathematicians. I should make it perfectly clear that I am a research mathematician and not a physicist. My speciality is Lie groups and any use of physics terminology here is only what is common knowledge. On occasion I have been motivated by physics - for example, the marvelous development of quantum mechanics by physicists in the 1920's. I believe that there were some stirrings about Lie groups by physicists in the middle of the twentieth century. I have the following prescient story to tell. I was a visiting member of Princeton's Institute for Advanced Study in 1955. It was a Good Friday in April and Einstein was looking for the Institute bus to take him back home to 112 Mercer Street. Being Good Friday, the driver was on holiday amd I offered to drive him home. We had a wonderful conversation and at one point he asked me what I was working on. I told him Lie groups. He then remarked, wagging his finger, that that will be very important. Actually, I was quite surprised that he knew who Lie was. About a week later Einstein was dead. In the middle of the twentieth century, physicists developed what is called quantum field theory (Feynman, Schwinger, etc.) Also at that time, the powerful accelerators were producing a zoo of new particles. To deal with this menagerie of particles and to carry forward Einstein's program of finding a unified field theory (unifying all 4 forces of nature), physicists came up with what is called the Standard Model (Weinberg, etc.) This involved what is called a gauge group. In fact, in the Standard Model, the gauge group is a (eu) simple Lie group. A more refined development was the grand unified theory (GUT) of Glashow and Georgi. Here the gauge group (SU(5)) was more interesting. The GUT theory happily confers a desired fractional electric charge on such exotic particles as quarks. These theories also unified three of the four forces of Nature.

The latter part of the twentieth century also saw the development, by physicists, of string theory. String theory has had vast consequence for mathematics (excluding Lie groups). However, as far as I know, there have been no experimental verifications of the physics involved. (For his work in this area, the mathematical physicist Ed Witten was awarded the most prestigious prize in mathematics.)

A word about E(8). In my opinion, and shared by others, E(8) is the most magnificent "object" in all of mathematics. It is like a diamond with thousands of facets. Each facet offering a different view of its unbelievable intricate internal structure. It is easy to arrive at the feeling that a final understanding of the universe must somehow involve E(8), or otherwise put, (tongue in cheek) Nature would be foolish not to utilize E(8). There was a good deal of publicity about E(8) in the last few years when a team of about 25 mathematicians, using the power of present computers and a very complicated program, succeeded in determining all of the vast number of (to use a technical term) characters associated with it. Incidentally, one of the main leaders of the team was an ex-student of mine, David Vogan. It was Vogan who told me about Lisi's paper. Another person involved here is John Baez. Baez, (a relative of the singer Joan Baez) is a professor of mathematics at the Riverside campus of the University of California. Baez performs a great service to the Math-Physics community by publishing a very engaging weekly report on doings of mathematicians and physicists - explaining latest results in physics to mathematicians and latest results of mathematics to physicists. His week 253 report deals with Lisi's paper. In effect Lisi is saying that E(8) is the ultimate gauge group. Lisi's theory makes some astounding claims. Among them is that E(8) "sees" all the elementary particles in the Universe. In addition, Lisi claims that his theory unifies all 4 forces in Nature (the last being gravity) and thereby achieves Einstein's dream of a unified field theory. String theorists, by and large, heartily dismiss Lisi's theory. But among some prominent nonstring theorists (e.g., Lee Smolin), the paper has been acclaimed. Incidentally, string theorists utilize E(8), but not as a gauge group. According to Baez's week 253 report, one of Lisi's motivations in going to E(8) was that the Glashow-Georgi GUT theory "sees" only one generation of fermions. Apparently there are 3 "generations" of such particles. The Lie group E(8) has a triality construction and I believe that Lisi thought that this may be used to give all 3 generations. Since I had something to do with this triality construction, I became interested in Lisi's paper. I remind you, I am not a physicist and cannot comment one way or another on the physics involved. However, mathematically, I was able to show, using beautiful results of such finite group theorists as John Thompson, Robert Griess, Alex Ryba and an important input from Jean-Pierre Serre, together with some old results of mine, that E(8) not only "sees" GUT in a natural way, but in fact is itself (viewed through one of the facets) a composite of two copies of GUT. This is the subject matter of what I have been lecturing about. One such lecture was at UC Riverside, which was filmed and put on line by John Baez.

Having seem the film, Lisi sent me an enthusiastic E-mail, saying my results were brand new to him and speculating on what the meaning of the second GUT might be. I am too happy to forward Lisi's E-mail letter to you, if you wish to see it. At any rate, if there is any physical validity to E(8) as a gauge group, the ball is in the court of physicists to interpret what this doubling up of GUT might mean.

I am happy to cooperate with you on your New Yorker article. However, I think it is best to do this by E-mail and not via phone conversations. I wish to avoid all the misquotations attendant to the New Yorker publication having to do with the solution of the Poincare conjecture.

Bertram Kostant Professor Emeritus of Mathematics at MIT


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This is the main reason I think what you and Carl have been working on, with the CKM and MNS matrices, is interesting, and deserves more attention.
Wow, thanks for the compliment. I should read Physics Forums more often!

This seems like a good time to post the new results. On this particular subject, Marni's been providing the key insights and I've been doing the manual labor / number crunching. While CKM solution was found using a Java program, it's explainable without going through that.

The CKM and MNS are 3x3 matrices. They begin with experimental measurements, which are of the absolute values of the unitary matrix elements. To make such a matrix unitary, one multiplies the 9 components by complex phases in such a way that the rows and columns become orthonormal. Once you find such a unitary matrix, you can multiply a row or column by a complex phase and it will remain unitary. So there are a lot of choices to represent the same set of experimental numbers. Given a particular choice, or "parameterization", one needs 4 real variables to describe the possible sets of experimental numbers. A parameterization is supposed to associate each set of numbers with a specific unitary matrix.

The usual parameterizations are the K&M, Euler angles, and the Wolfenstein. They are given in the Wikipedia article on the CKM matrix. A week ago I found a pretty parameterization that chooses the phases such that a 3x3 unitary matrix splits into a real and imaginary part:

The real part is 1-circulant (each row shifts one to the right), while the imaginary part is 2-circulant (each row shifts two to the right). Like the other parameterizations, this is (more or less) a unique way of describing a unitary matrix. I'm attracted to the form partly because I'm a fan of density matrices which also eliminate arbitrary complex phases in an elegant way. The best explanation is an example.

The MNS experimental data is approximately given by the "tribimaximal" values:

&\nu_1&\nu_2&\nu_3\\ \hline

Rewriting this in real 1-circulant plus imaginary 2-circulant form we have a result dating to around June:
\pm i\left(\begin{array}{ccc}

The CKM matrix experimental data (from ) is:

&d&s&b\\ \hline

In real + imaginary form this is approximately given by:
\pm i\left(\begin{array}{ccc}
which dates to October 8:
Note that the two matrices each have only 3 real degrees of freedom for a total of 6.

This is basically the latest stage of a project Marni Sheppeard started back in June. She had noticed that the CKM matrix was approximately the sum of a 1-circ and a 2-circ. From there, it was natural to work out the MNS matrix, since it had a simple form that was easy to deal with. And she's pointed out that all this is related to the discrete Fourier transform.

In the last week or so, a flurry of emails and blog comments has given a few more results. I should type them up and make them into a post, but things are busy.

Ignoring an overall factor of sqrt(1/3), the discrete Fourier transform (DFT) of a vector of 3 elements (a,b,c) is defined as the three results:
A = a + b + c,
B = a + wb + w*c,
C = a + w*b + wc,
where [tex]w = \exp(2i\pi/3)[/tex]. This can be accomplished by multiplying the vector on the right by a matrix with the nine values:

To take the DFT of a 3x3 matrix, you multiply on the right by the above matrix, and on the left by the inverse transform. In the case of these matrices, this amounts to taking a DFT over generations for both the electrons and the neutrinos, or for both quark charges.

The eigenvectors of an arbitrary 1-circulant matrix are always (1,1,1), (1,w,w*), and (1,w*,w). So it turns out that the discrete Fourier transform of a 3x3 1-circulant matrix is a 3x3 diagonal matrix. And similarly, the DFT of a 2-circulant matrix is an "anti-diagonal" matrix.

The usual DFT is linear, and so is the matrix version. This means that the real 1-circulant + imaginary 2-circulant forms we've written the CKM and MNS matrices in amount to a choice of phase such that the DFT of the unitary matrix is as simple as possible.

The DFT converts the CKM and MNS matrices, when this new parameterization is chosen, into matrices that are very reduced: the degrees of freedom of the unitary matrices are written so that as many values are zero as possible. They're not as reduced as a diagonal matrix but it's close. You could say that this is the least possible non commutative generalization of diagonal.

The implication is that the MNS matrix is peculiarly simple in this parameterization because the DFT has something to do with the generation structure of the leptons. And from that, one supposes that there may be a pattern in the CKM matrix as well. But no one has found that pattern so far. Nevertheless, the CKM numbers are suggestive in that they follow the usual generation pattern in that one gets a sort of scale between the three values of about 9. Since the Koide mass formulas are also related to the DFT, we suspect that the 6 CKM entries are also so related.

Where is this going? Like the Koide formula the neutrinos, I think that the lepton version of this is sufficiently elegant that it should start showing up on arXiv. Right now I'm busy writing up a paper extending Koide's mass formulas to the hadrons, which I think is more important.


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I see I forgot to mention the relationship with the Koide mass formulas.

The Koide mass formulas for the charged leptons (electron, muon, and tau) are:
[tex]\sqrt{m_{en}} = \mu(\sqrt{1/2} + \cos(2/9 + 2n\pi/3))[/tex]
where [tex]\mu[/tex] is a constant of about 25.05 square root MeV.

The discrete Fourier transform of the MNS matrix has 6 terms. Three are on the diagonal. They are:
[tex]d_n = \sqrt{2} + e^{2i n\pi/3},[/tex]
which is somewhat similar to the charged lepton mass formula, but simpler. The other three terms, are similar:
[tex]d'_n = \pm (\sqrt{2} - e^{2i n\pi/3}),[/tex]

Hey, what's up with the LaTeX?


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Garrett gave a talk at TED, which I found delightful :
A beautiful new theory of everything
I did too. The computer animations visualizing the plain old Standard Model were just amazing! I'd suggest you start a thread just on Garrett's TED talk. More people would see it that way, than might visit this older thread on the piece in the New Yorker.

Here is Humanino's link again. Just click on it and wait a few seconds. The video will start.

I just finished watching it for the third time. A really great talk.
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