Back into print?

If you would like to suggest and/or vote for some out of print math book to come back to print, visit here:

outofprintmath

[via Terence Tao's blog, invited post by Timothy Chow].

Requardt’s papers

Back in 2005, when I started to read more and more papers on quantum gravity, I have found one paper that really interested (and still interests) me a lot, and was the subject of one of my first posts back at my older blog.

- The Continuum Limit of Discrete Geometries, by Manfred Requardt [http://arxiv.org/abs/math-ph/0507017]

It is a well written paper, with lots of interesting mathematics (Gromov’s geometric group theory, random graphs, etc). As far as I know, it has not been much discussed in other blogs.

Recently, Requardt posted another interesting one, which I am presently reading:

- About the Minimal Resolution of Space-Time Grains in Experimental Quantum Gravity, by Manfred Requardt [http://arxiv.org/abs/0807.3619]

Abstract:

We critically analyse and compare various recent thought experiments, performed by Amelino-Camelia, Ng et al., Baez et al., Adler et al., and ourselves, concerning the (thought)experimental accessibility of the Planck scale by space-time measurements. We show that a closer inspection of the working of the measuring devices, by taking their microscopic quantum many-body nature in due account, leads to deeper insights concerning the extreme limits of the precision of space-time measurements. Among other things, we show how certain constraints like e.g. the Schwarzschild constraint can be circumvented and that quantum fluctuations being present in the measuring devices can be reduced by designing more intelligent measuring instruments. Consequences for various phenomenological quantum gravity models are discussed.

I like his writting style and the various points that he covers in his papers with sensible criticism.

Bertram Kostant on E(8)

I include here an email from Prof. Bertram Kostant to Ben Wallace-Wells on his view about the E(8) group and some considerations on Lisi’s theory. According to Lisi, this part of the email was permitted to become publically available.

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

Garrett Lisi and Jacques Distler: debates revived — Part II

The text below was extracted from the original post by Distler over at the n-Category Café:

——–
[Begin of Distler's comment excerpt]

This is my summary of Lisi’s programme (at least, as best I have been able to understand it).

1. Choose an embedding of Spin(3,1)×SU(3)×SU(2)×U(1) in (noncompact) E 8. The generators of the Lie algebra, e 8, then transform as some representation of this subgroup.

* In particular, the action of the center of SL(2,ℂ)≃Spin(3,1) gives a ℤ 2 grading on e 8. The generators of e 8 which transform in spinorial representations of Spin(3,1) are “odd”; the generators which transform in tensorial representations are “even.”

* There is a similar ℤ 2 grading on the fields of any QFT: fermions are “odd” and bosons are “even.” The spin-statistics theorem requires that these be the same grading.

* While Lisi say that he doesn’t want to envoke a ℤ 2 grading, one is clearly physically required, and mathematically provided by the aforementioned embedding of Spin(3,1). He might as well say that he doesn’t want to speak in prose.

2. Use this ℤ 2 grading to build a Schreiber superconnection. The bosonic fields transform as 1-forms with values in various tensor representations of Spin(3,1); the fermionic fields transform as 0-forms in spinor representations of Spin(3,1).

* Lisi says that he isn’t using a Schreiber superconnection. Instead he’s doing ‘standard’ BRST. I can’t make head or tails of his usage of the term “BRST.” In the end, to each generator of e 8, he associates either a bosonic or a fermionic field. Spin-statistics dictates that he do this in a fashion compatible with the ℤ 2 grading. Which is to say that his fields comprise a Schreiber superconnection. Protestations to the contrary he, again, seems to be speaking in prose.

3. Use this Schreiber superconnection to build an action.

4. Quantize that action.

5. Try to extract some quasi-realistic physics from it.

Unfortunately, the construction falls down at step 1.

* Lisi wants there to be 192 odd generators, with respect to some embedding of Spin(3,1). This, of course, is impossible.

* Moreover, in his paper, Lisi embeds Spin(3,1)×SU(3)×SU(2)×U(1) via a D 4×D 4 subgroup of E 8. I classified all such embedding. They all lead (via the above prescription) to a non-chiral fermion spectrum. The closest one can come to the Standard Model spectrum of fermions is to get 1 generation and 1 anti-generation.

* This, in fact, is completely general. Any embedding SL(2,ℂ)×SU(3)×SU(2)×U(1)↪E 8 yields a nonchiral spectrum of fermions, with — at best — a generation and an anti-generation of Standard Model particles.

None of these statements is particularly hard to prove. In fact, once you know that there’s no ℤ 2 grading of e 8 with more than 128 odd generators, you know that it’s impossible to accommodate 3 generations. The best you could get is 2, but even that proves not to be possible.

That said, there is something kinda cool about the elements of the construction:

1. An embedding of Spin(d−1,1) in G gives a ℤ 2 grading on 𝔤.

2. Using the corresponding Schreiber superconnection, one naturally gets a theory with fermions, corresponding to the odd generators of 𝔤, transforming as spinors Spin(d−1,1).

It would be mildly interesting to see what sort of actions one could build with this construction.

[End of Distler's comment excerpt]
—————–

Although that site is perfectly adequate for rigorous discussions on the matter using mathematical language, I leave here a welcome space for comments on the above intrepretation by Distler in layman terms.

I have previously posted over at n-Cat café the following:

———–
Do I understand correctly that the “other stuff” that sits in the odd part is considered important for Distler (it is the “anti-generation” which for him is one of the points that would make the whole approach doomed to be incorrect), whereas for Lisi the “other stuff” – whatever it is – can be worked out, eventually avoiding a possible invalidation? Is this the point of tension?

——-

I have also posted the following remark (slightly edited):

——-
If I understand it correctly, it is agreeded on both parts that Lisi’s model as a whole results in a non-chiral spectrum (net number of generations = 0). Furthermore, Distler appears to have shown that there are no decompositions of E8 allowing the inclusion of the 3 SM generations. (Does Lisi agree with the latter?)

So, I was wondering – is it really all there is to be concerning the use of E8 (or any other group, for what is worth)?

I mean, on speculative grounds, is it possible that simply using the group “as it is” is not the whole story, but actually one could gain more room for analysis or insight by seeing the group from a different “perspective”?

What I have in mind here comes from something I was reading superficially about, groups of polynomial growth and the work of Gromov. Does E8 have any relation to such groups? If so, would it be possible to prove whether the “net # gen = 0” feature shown by Distler for the E8 is preserved (or not) when considering related groups of polynomial growth (if that is possible at all), in which the group is “seen from infinity”?
—-

I keep these posts here for personal record.

[Part I of the present post here.]

Lowest cost, highest benefit book ever

This is the book with the lowest cost ($7.95), highest benefit ever, in the physical sciences (in my opinion):

You submerge into Dirac’s mind and learn about his general Hamiltonian formalism with constraints, which is a great start for a subsequent, more modern treatment, given in (the much more expansive) Henneaux and Teitelboim‘s book.

There are, of course, other low cost, high benefit books, specially coming from Dover publications (several come to my mind). But Dirac’s book is presently my favorite in that respect.

A very concise, brilliant and rich little book that it’s easy to carry everywhere and keep your mind busy with important concepts and how to work towards new approaches and developments.

Research and Age

I reproduce here a brief comment that I wrote over at Backreaction, concerning the age factor affecting creativity and productivity.

—-

There are many internal and external factors, as well as historical factors (formative factors) that come into play for doing high-quality, relevant research. Age is only one of these factors.

As you get old, positive and negative points come into play, but their relative weights vary from person to person. The positive ones are maturity, amount of accumulated knowledge and more experience (specially in dealing with failure as part of the process of getting wiser). Also, the way one sees life and how research is part of his/her life is an important factor that has a meaning that follows oneself year after year without much change, when suddenly gets another distinct flavor at some point. This change can be used positively, but if one does not take appropriate internal action, it can be destructive.

Apart from the latter, the negative side of age may appear as some feeling of “getting slower” to learn or solve a problem as compared to the way one used to learn/solve things faster in the past. But getting slower also has a positive side, since one may be getting slower, but richer in the process. That is, the chances of missing important points may be higher when you are “faster” and lower when you are “slower”.

Capacity of deep thinking, continuous investiment in self-development, ability to get interest in different fields and working the brain towards increasing one’s particular gifts for creative processes, as well as passion for research, and energy and time for working on it are the points that really matter. Age may cause a disturbance on these factors, in a sense that at some point you can no longer control it. But even at this stage, you can be happy and keep something fruitfull to oneself until death comes. See the oldest blogger lady that Bee mentions at another post.

Garrett Lisi and Jacques Distler: debates revived

Debates on Lisi’s theory are back being discussed over at n-Category Café from this post on chronologically, so that you can check the exchange progress.

However, it is difficult to tell ahead the end of this story. It appears that some sort of agreement is slowly and painfully being reached, but it is clear that there is still a long way to go.

[Previous posts on Lisi's theory can be found here.]

[Edit 19-Jul-08: An interesting summary by Distler is found in a previous entry from the above mentioned exchange, dated May 22, 2008, and pointed out by Urs Schreiber. It is interesting not only for being a summary of what he interprets from Lisi's theory and the points that he indicate as being problematic, but also because he actually finds what he calls "something kinda cool about the elements of the construction" and that it "would be mildly interesting to see what sort of actions one could build with this construction".]

Maragogi

In the past week my family and I were on vacation at Maragogi, located at the northern coast of the state of Alagoas, Brazil, also known as the Coral Coast. We had a very nice stay at the Salinas do Maragogi Resort. The most interesting part of the trip was the diving in the natural pools formed at about 6 km from the coast, in which one can swim with the fishes at the very green waters of the Coral Cost, also known as the “Galés”.

I include some pictures here (click to enlarge):

The beach, the sky. The climate changed very quickly from sunny to rainy and back to sunny this time of the year.

The “coqueiros”.

The very green Galés.

My kid specially liked these guys.

For more pictures, try here with Google.

New Accepted Paper

I have a new paper accepted for publication in Physical Review B in the area of micromagnetism. Interested readers can access the paper via the arxiv number [0807.1978], already available.

arXiv:0807.1978
From: Christine Córdula Dantas
Date: Sat, 12 Jul 2008 13:55:38 GMT (1211kb)

Title: Micromagnetic simulations of small arrays of submicron ferromagnetic
particles
Authors: Christine C. Dantas and Luiz A. de Andrade
Categories: cond-mat.mtrl-sci
Comments: 24 pages, 8 figures, accepted for publication in Physical Review B
License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
\\
We report the results of a set of simulations of small arrays of submicron
ferromagnetic particles. The actions of dipolar and exchange interactions were
qualitatively investigated by analysing the ferromagnetic resonance spectra at
9.37 GHz resulting from the magnetization response of con- nected and
unconnected particles in the array as a function of the applied dc magnetic
field. We find that the magnetization precession movement (at resonance)
observed in individual particles in the array presents a distinctive behaviour
(an amplitude mismatch) in comparison to isolated, one-particle reference
simulations, a result that we attribute to the action of interparticle dipolar
couplings. Exchange interactions appear to have an important role in
modifying the spectra of connected particles, even through a small contact
surface.
\\