New blog

Announcing my new (tentative) blog, Toy Universes.

Blog is shutdown.

All contents will be available as far as there is a host out there in the internet “cloud” making it accessible. I will not delete this blog, so you may be able to access the archives for some time, I guess.

Comments will be shutdown, however.

Thanks and good-bye!

News on Garrett Lisi’s E8 theory

Lisi posted this yesterday over at Physics Forums; I reproduce here:

Hello PF folk.

If you believe the Dirac equation in curved spacetime, and you believe Spin(10) grand unification, then a Spin(3,11) GraviGUT, acting on one generation of fermions as a 64 spinor, seems… inevitable.

Also, it’s pretty.

And it’s up to you whether or not to take seriously or not the observation that this whole structure fits in E8. Personally, I take it seriously. Slides are up for a talk I gave at Yale:

http://www.liegroups.org/zuckerman/slides.html

Best,
Garrett

I am not certain whether it addresses Distler’s previous objections (as I am not certain whether the issue was even settled at that time– see here and here, which goes as far as I could follow. More (older) personal opinions can be found here, here and here in reverse chronological order).

Edit: I forgot to add. I do find the theory beautiful and interesting. I hope it can be properly tested.

Edit: Here are further links that are relevant to this post.

There is no “Theory of Everything” inside E8 by Jacques Distler and Skip Garibaldi.

Here is Distler’s blog entry on his paper.

There was a discussion of Distler and Garibaldi’s paper at Physics Forums some time ago.

There was also a discussion at n-Category Café some time ago.

Connexions

Just learned about this:

Connexions is:

a place to view and share educational material made of small knowledge chunks called modules that can be organized as courses, books, reports, etc. Anyone may view or contribute:

authors create and collaborate
instructors rapidly build and share custom collections
learners find and explore content

This sounds a formidable initiative.

The Potential Nobel

Fellow blogger Clifford from Asymptotia suggests:

How about alternative prizes for this week’s categories? Prizes to work (or authors of the work) that while extremely promising, certainly has not met the promise yet (for which the jury is still out since the work is not done).

Very easy.

The Potential Nobel (physics, literature, peace?) goes to… me.

When I was 17 years-old, I thought I had so much to realize. Today, much older and wiser, I certainly have not yet realized even a small fraction of my earlier envisions. Nor I think I will improve much on that. I have not yet done the work of my life. I do not even know exactly what it is. My opus is pure potential. It will always be. Life is too short. And I am not as clever as I once thought.

I deserve the Potential Nobel prize.

My non-FQXi Essay

Yes, this is true. I have just written my non-FQXi Essay, in the sense that I have not submitted it to the presently running edition, now featuring the theme “What’s Ultimately Possible in Physics?”.

You may be wondering why I have not submitted it. After consideration, I have found some reasons, but let me tell you only the short one: I concluded that it is undignified to compete for a prize on speculation.

But I have written my short essay anyway. It took me only a couple of hours, and I must point out that it is not a scientific work, nor a philosophical work. It is a speculative work. But this fact does not mean that it is not a serious speculative work.

So here it is, in case you are interested:

Title: When Response Nullifies
Author: Christine Córdula Dantas
3 pages, 72 Kb, pdf format

Update 05 Oct 2009: A few typos, corrections, stylistic improvements and additions were made. Please replace previous version with the current one. Other corrections are welcomed. Thanks!

[English][Portuguese] Announcement / Aviso

[English] I have decided to run a separate blog on mathematics for Brazilian students, the Matemática Replay!. This blog [Theorema Egregium] continues with the usual material. Thanks.

—

[Português] Eu decidi manter um blog separado sobre matemática para estudantes Brasileiros, o Matemática Replay!. Este blog [Theorema Egregium] continua com o material usual. Obrigada.

“New Look” and Twitter

I hope you like the “new look” of Theorema Egregium; notice also that I am “twittering” now– see the side bar. Just experimentally.

Open letter

I make here publically available my letter to Sabine (Backreaction) concerning ou recent exchange of comments over at her blog. I will not post her letter because it was personally addressed to me. However, since my letter was general enough, and perhaps elucidative enough, here it is.

—-

Dear Sabine,

Thanks for your email. I think that it is quite possible that I have not expressed myself the best possible way. My comments were an attempt at a criticism on the topics based on those that you have mentioned in your post, which I had assumed to be representative of the conference. All I want to say is that I do not care what people want to work on, but I am tired to see professional scientists working on non-scientific issues (viz., those which the scientific method is not applied) as scientific. I consider this very non-ehtical and a dis-service to the public.

Theory must provide a means to experimental verification (in principle, at least), if not, you have an unproven hypothesis. Some people at the frontier of physics are not taking care of this very important concept and elevate their unproven hypotheses to principles of truth, from which they base all their subsequent work. We cannot rely on our subjective judgements to consider some theory acceptable or not: this is why the scientific method exists as a pillar for science.

I think FQXi is perfect as a funding agency for non-mainstream, alternative approaches, which nevertheless are perfectly scientific. Also, philosophical themes (which is a completely different class of discipline, with its own epistemological rules). However, it appears that this question is not clear enough.

I am glad to learn that your work was well received. I hope that you have understood that my criticism was not aimed at your work on phenomenology, which is evidently scientific enough.

Best regards,
Christine
—-

EDIT: Sabine writes that “most of them [the projects] eventually won’t lead anywhere – that being the nature of the business”. The problem is not that some projects lead nowhere, but that any non-scientific project leads nowhere by construction. If one’s work is based on an unproven hypothesis which is elevated to a principle of truth, from which all subsequent work is based, then it is highly probable that it will indeed lead nowhere. Or the conclusions will be most probably false.

She also insists that I point to specific projects that I consider non-scientific. As I already emphasized, my comments were based on the themes that she highlighted on her post, not on the program, which was not made avaliable on the FQXi site at the time of the postings. In any case, I leave the excercise to the reader to apply the scientific method and find out the answer by him/herself.

—
Related posts:

Smolin Against the Timeless Multiverse

The Universe

Universes Everywhere

What is science for you? Up to 50 words

What is science – for me

Science is no longer scientific

Dear Traveller of the Future,

If for some reason this set of electronic ramblings reach you, I salute you.

But my salute is embeded in deep sorrow. I have just realized that I am a living testimony of the end of science — a human activity arduously conceived during centuries in order to objectively probe nature into her deepest substrate: a triumph of the human mind.

For some reason, many people engaged into such a dignified activity slowly gave up on the arduous road, and chose the easiest paths, which unfortunately often lead to the swampy terrain of incertitude and ignorance.

Although there are still many advancements, specially technical ones, there is Death waiting at the frontier.

Science is no longer scientific!

Best wishes to you, Traveller of the Future, to whom the idea of what was gained and what was lost may never be perceived, but I still hope that the starlight inspire your soul, somehow.

Christine

PS- This anecdote was inspired by a discussion developed here.

Smolin against the timeless multiverse

There is a new article by Lee Smolin at Physicsworld.com, “The unique universe“, where he exposes his metaphysical position on the multiverse and the notion of time as fundamental, not emergent.

My thoughts are close to Smolin on those issues, see my previous post on the multiverse here:

The Universe

and, in a funny side, my cartoon here:

Universes Everywhere

Concerning the question whether time is fundamental or not, my philosophical position is that it is fundamental, although there is a facet of it which can be made artificially emergent. See my essay on concurrent time here .

Sabine and Stefan in Brazil

It was a great pleasure and honor to meet on January 06 my blogger colleagues Sabine and Stefan (Backreaction), here in my city (São José dos Campos) while in their recent holliday trip to Brazil.

Here is a picture that my 9 years-old son took of us (from left to right: Sabine, Stefan, me and Fabiano, my hunsband):

We are standing at the border of the so-called “Banhado“, a large green preservation area on a low plateau, a turistic point of the city. (More pictures of the Banhado can be found here).

Thanks, Sabine and Stefan, for taking your time and energy to visit us! It was a pity, however, that your passage here was too brief to allow us further conversation in person…

[Sabine also reports on our brief meeting over at Backreaction].

Dyson’s birds and frogs

[Via Peter Woit]

On the Nature of Time — essay competition

I have submitted an essay to the FQXi competition. If you are interested in reading it, click here.

Title: On the Nature of Time – Or Why Does Nature Abhor Deadlocks?

Essay Abstract

This essay aims at introducing a novel point of view on the nature of time, inspired by a synthesis of three seemingly unrelated concepts: Bergson’s notion of duration, Dijkstra’s notion of concurrency, and Mach’s notion of inertia.

Edit (June 9th 2009): Apparently, the essays on the nature of time are no longer available at the FQXi site. I have made a very few small corrections and modifications in my essay and a new version is available here (pdf file).

The most extraordinaire Brazilian soprano

I invite you to watch this video — be patient until the grand finale. What you will see is Niza de Castro Tank when she was 50+ years old, in the 80′s, singing an aria of the opera “Il Guarany” by the Brazilian composer Carlos Gomes. I had the opportunity to meet her in some occasions and in a few master classes. She is still singing. A truthful phenomenon. She is not very well-known outside Brazil. Enjoy.

Edit: if you would like to read some of the comments (some available in English) over at the youtube site, click here.

Physical limits of inference – Theories of almost everything?

There is a review at Nature’s News and Views section by P.-M. Binder about a recent article by David H. Wolpert from NASA Ames Research Center, entitled “Physical limits of inference“. Binder writes:

A provocative contribution to the logic of science extends the theorems of Kurt Gödel and Alan Turing, and bears on thinking about prediction, the standard model of particles, and quantum gravity.

From the abstract of the paper, one reads

We show that physical devices that perform observation, prediction, or recollection share an underlying mathematical structure. We call devices with that structure “inference devices”. We present a set of existence and impossibility results concerning inference devices. These results hold independent of the precise physical laws governing our universe. In a limited sense, the impossibility results establish that Laplace was wrong to claim that even in a classical, non-chaotic universe the future can be unerringly predicted, given sufficient knowledge of the present. Alternatively, these impossibility results can be viewed as a non-quantum-mechanical “uncertainty principle”.

[Yeah, Laplace was wrong even classically, according to my SF novel... ]

and

(…) We informally discuss the philosophical implications of these results, e.g., for whether the universe “is” a computer.

I find it very surprising that this was published in Physica D: Nonlinear Phenomena, and not in a philosophical journal. I have no criticisms against this work in particular (I did not read the paper in full), it is just that it does not seem, from a first impression, a physics paper per se, as much as interesting as it may seem.

Another (somewhat funny, I must admit, but it may be a reflection of my present pessimistic/sarcastic mood) excerpt from Binder’s review is this:

The other limitation is our inability to bring quantum mechanics and gravity into a single theory, although several viable alternative theories are being studied [9]. Quantum electrodynamics, a refinement of quantum mechanics, is defined by just two parameters (the charge and mass of the electron), whereas quantum gravity would require infinitely many parameters, and hence infinite experiments to determine those parameters, making it so far a meaningless theory.

BTW, Ref. [9] above is Wilczek’s book, The Lightness of Being.

To be young again…

Children are great scientists, the best. Carl Sagan noticed that very well. But when they grow up into adults, most of them loose their once genuine curiosity to really understand how the world works. They stop doing the right questions (usually on the mark) and the instinctive use of the scientific method through testing their ideas and hypothesis against what they observe. All the naturalness of the understanding process that encompasses a genuine scientific activity is somewhat lost.

What happens during the growing process of children into adults is quite complex, but should be taken into serious consideration, because something very essential and important is lost. Is it just innocence? If so, then we should be more “innocent”. We should be more uncorrupted and pure. Science should be an open activity for all, and the level of exploring it only dependent on one’s technical knowledge, honesty, wisdom. The merit of one’s work should be evaluated against those qualities. PhD titles should be only marks that certifies that you have those qualities and that you have made contributions to the field. But someone without a PhD should not be dismissed from the scientific endeavor.

Of course, what I see in reality is many PhDs that are a complete distortion of what a scientist should be. And conversely, I know people without a PhD that would make honest scientists.

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.

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.

Universes everywhere

A universe could form inside this room and we’d never know.

Sean Carroll

Hmmm… Why does it sound quite familiar to me?

Wheeler (1911-2008); some days after, and what is yet to come

After a few days when I first learned about the death of the great physicist John Archibald Wheeler, and after reading all the comments on this beautiful tribute to him, a strange feeling is left on me.

I have never met him, but how many times and hours have I happily devoted myself to his tour de force Gravitation (with Misner and Thorne)? How many times have I submerged in intellectually stimulating thoughts that he so intelligently raised? How much have I learned from him? I cannot gauge that.

After these few days when I learned about his death, I can only think of the impact… of the absence of a great scientist… the work he left… the really one of the last from a generation of great physicists… what is yet to come?… and think about myself and what is left from me, as an ever-learning scientist and person.

Rest in peace.

[Update:] An obituary written by Peebles and Unruh has been released yesterday by Nature (you need a subscription). There one can learn the following:

John Wheeler was an exuberant person. He happened to be in the office of one of us, Jim Peebles, when a student, Dan, came in with his new dissertation.

John: What’s it about?
Dan: A search for correlations of position angles of galaxies.
John: Gödel would be interested.
Dan: Who’s Gödel?
John: To say Gödel is the greatest logician since Aristotle would be to slight Gödel.

John picked up the phone, got Gödel on the line, and handed the phone to Dan. We don’t know what Gödel made of this call (he haddiscovered a rotating-universe model), but John had got to do what he loved best: inspire a young person.

I confess…

When I saw it, I sweared in utter bafflement and astonishment…

MacBook Air

Best Christmas baroque performance

No, I’m not religious at all. However, my favorite composer is Johann Sebastian Bach, specially his sacred works. In my opinion, if I had to choose a music to represent mankind, it would be his St. Matthew Passion.

There is no other musician that can be compared with Bach. (No, don’t try to convince me otherwise!).

For Christmas, my favorite performance is this: Christmas Oratorio (Weihnachtsoratorium), conducted by Nikolaus Harnoncourt, with soloists Peter Schreier (tenor, perfect), Robert Holl (bass), Concentus Musicus (tenor and bass choral) and the wonderful boy’s choral Tolzer Knabenchor.

In that performance, recorded around 1981, I call your attention for the 3 excellent boy sopranos soloists. One of them I have identified as being Allan Bergius, the one with the most “angel-like” voice, who sings the difficult and beautiful terzett “Ach, wenn wird die Zeit erscheinen?”, along with Peter Schreier and the boy alto soloist Stefan Rampf, who also has a splendid perfomance overall (being the only alto soloist in whole oratorio), given the difficulty of the cantatas and the need for a maturity that is impossible to ask for such young boys. Another boy soprano soloist (a curly-haired boy) has an incredible soprano-dramatic voice, very rich in the middle, low range, with potent high notes and great musicality, but I could not identify his name. To him, is given the lovely “echo-aria”, “Flößt, mein Heiland”, as well as the dramatic and demanding aria “Nur ein Wink von seinen Händen”. The third boy soloist is the blond, long-haired one, who I also could not identify his name– he has a very soft and small voice, but exceptional musicality, who is given the sweet duett “Herr, dein Mitleid, dein Erbarmen”, along with the bass soloist Robert Holl.

The expressiveness of Nikolaus Harnoncourt also makes this DVD a very special one.

I think I have watched this performance more than a hundred times, and I can sing all soprano parts (even other parts) from my heart. I recommend it to anyone, anywhere, anytime. But, specially, this time.

Overview of the choral.

The boy soprano with an angel voice, identified as the very young Allan Bergius. Wonderful perfomance of difficult parts.

Excellent boy soprano, dramatic and versatile voice.

Soft voiced soprano with great expressiveness and musicality.

Beautiful alto voice; complete control of all alto pieces. Identified as Stefan Rampf.

A wonderful Oratorio passage: tenor Peter Schreier with his very young (but very professional) singer colleagues in a demanding part.

Competitive Cycle

Over at the Cosmic Variance discussion on Lisi’s TOE, the question of competition versus collaboration in science, and the existence of arrogance in many scientists in doing their professional activities has been raised.

Sure, one can be very successful making a science career being very competitive and aggressive. In fact, the system favors these characteristics. Why? Because science is made by people, and those who are more competitive and aggressive naturally tend to be more successful in climbing into key positions in the system. So the present situation in science is a somewhat closed cycle in this respect.

So yes, it is possible to do science under that scheme and be very successful. However, is it an optimum system in the long term?

I don’t think so.

Competitive and aggressive scientists probe nature under very specific motivations — that is, “how to be successful” in their careers. This may lead him/her to important discoveries and there are certainly many examples of them. But there are many fundamental and deep questions in nature that I really doubt that can be adequately advanced if you do not have a truthfully inquisitive, contemplative, genuine curious mind about nature’s deepest secrets. People with such an approach to science have learned the meaning of the words “humbleness” and “collaboration”, can also be successful, and there are certainly some examples of them, but I really think they are a minority, at least in our current epoch. The system definitely does not favor such people in science.

On purely philosophical grounds, I consider it very strange that we, supposedly intelligent people, let the situation develop into that closed cycle. The cycle can be easily explained, as I did above, but it is really a shame that we let ourselves into this. Is it not obvious that we need a healthy collaborative environment, with plenty of genuine modesty and humbleness, towards the understanding of the vastness of nature?

Aggressiveness and competitiveness can be the means to accomplish that goal, and you can follow them if they are attractive to you and to your aims in life, and indeed the achievements that may result from such a path can certainly make many satisfied. Yet, a vast sea of unknown is doomed to be hidden by construction to them, because the understanding of nature and our place in the universe are not things to be conquered (like territory and hunt were things to be conquered by our ancestors in order to survive), but things of the human intellect, that must be nurtured, grown and shared to all, with pain, persistence, reason, humility, curiosity. With our minds filled with wonder.

Update: Some technical discussions between Distler and Smolin finally appeared in the comment’s section over at CV after I wrote this post. If I find time, I’ll try to make a compilation of them in my blog.

Update: I have added my compilation in the post about Lisi’s paper.

Garrett and Smolin, to boldly go…

I should warn the reader that all that follows is a personal interpretation from a curious person that is attempting to learn new subjects. Your corrections are welcome.

—–

As I have previously mentioned, Garrett Lisi’s work sounds extremely interesting, and any reasonable person expects that it finds its way to a serious critical analysis before far reaching conclusions and speculations take over. There are many issues that need further work. This is what Garrett explicitly have said several times.

Lisi’s paper got attention: a relatively heavy media hype (considering perhaps the title of the paper and the fact that Lisi is not in the academia, and enjoys a different lifestyle) and inumerous discussions all over the blogosphere. Now that things appear to have calmed down (although we can still hear some fresh echoes of “Lisi’s wave”), a new paper by Smolin, citing Lisi’s work, appeared in the scene.

From a first reading of Smolin’s paper, the most interesting aspect of it (as I see it) appears to be the introduction of an extension of the Plebanski action that accepts any large group structure (containing the local Lorentz group), showing that, with a symmetry breaking mechanism, the resulting dynamics is that of a unified Einstein-Yang-Mills, plus corrections. Smolin suggests his framework could be used for building up the dynamical sector of Lisi’s proposal (by applying the group E8 in the proposed extended action), a part of Lisi’s work that has not been worked out satisfactorily. According to Smolin, it would give “a fully gauge invariant action, which has solutions which spontaneously break the symmetry and give, when expanded around, the bosonic part of Lisi’s action, plus corrections”.

Smolin also suggests techniques from LQG, more specifically, the notion of “disordered locality”, as an alternative for the description of fermions and scalars to that used by Lisi (BRST extension of the connection). That part of the paper is not sufficiently clear to me, but while I try to figure it out, I would gladly accept a clarification.

Although Smolin recognises that there are open issues in Lisi’s work, specially whether the three generations of quarks and leptons are realized in the proposed structure of the E8 group, he boldly advances Lisi’s idea into aspects that as far as I was able to follow have not emerged from the whole body of discussions elsewhere. But I guess these ideas have been exchanged privately for some time (as inferred from the acknowledgement section of both Smolin and Lisi’s paper, as well as the mentioning by Lisi in his paper that the “quantum E8 theory follows the methods of quantum field theory and loop quantum gravity — though the details await future work”).

In any case, the main idea of Smolin’s paper seems to be independent of Lisi’s proposal. That is, suppose the E8 group ends up to be the wrong route. And instead one finds another group that realises the unification. Would not Smolin’s framework be consistent enough in order to allow the study of the resulting dynamics of the “correct” group, whatever it is?

So I think we have two separate issues here. One is the validity of Lisi’s proposal first of all. The other is how far Smolin’s work is dependent on it. As I mentioned, it does not appear to be completely dependent on it. He uses Lisi’s proposal as an example of application. But I may be wrong.

In any case, it is of fundamental importance to go through all the necessary details and assumptions systematically before reaching any conclusions.

On that regard, Jacques Distler has posted twice about Lisi’s work on his blog (here and here) where he argues the impossibility of getting all three fundamental particle generations through Lisi’s embedding into the E8 group. What calls the attention is Distler’s purely mathematical argumentation involving group theory representation only.

So it seems that we have reached a very first bottleneck on Lisi’s idea, in which there appears to be concrete technical (mathematical) issues that are absolutely necessary to be cleared up before continuing. Since it is a purely mathematical issue, it is indeed a point of tension that should be released before anything else. Otherwise the whole edifice of unification through E8 symmetry seems to face an early collapse.

This is unambiguous — as far as mathematics is: is Distler right or wrong?

Is Smolin’s results dependent on Lisi’s work?

If you would like to discuss specifics of the first question, please do it here. To the second question, you are invited to go here. These posts are for technical comments only. For general or more qualitative comments, you may use the present post.

As a final remark, I wonder whether those papers are being considered for refereed publication. I think that well moderated blogs can be a remarkable arena where scientific works can be widely discussed. But at the same time they do have a limited scope. Also, the peer review system has its problems; this is well known and acknowledged. It is not clear whether the authors will submit their papers to a journal. So what are we left with?

Not every member of the scientific community, whatever the field, is connected to the blogosphere. And even if they were connected, would it be acknowledged that blogs are the place where a consensus on the validity of a scientific work is to be met? Where should they be met anyway? We live the revolution of information and communication that the internet has brought, so there is a visible change in the scientific arena of discussions. Things are not as before.

In any case, Nature is the final judge. If you have a theory that can be cleary tested, then that is what is needed at the end of the day.

Except, of course, if you have got wrong mathematics to begin with.

Update: Garrett Lisi invites a technical discussion on his paper in this thread over at PF.

Physics needs independent thinkers

Particle physics is not my area of expertise, so there is no way I can make technical considerations on Garrett Lisi‘s recent paper, An Exceptionally Simple Theory of Everything.

I am glad that this paper is under scrutiny (at least this is what one would infer from blogs, if they can serve as thermometer for what is going on, with the caveat that the blogosphere tend to augment things). One can learn more about the discussions (with the participation of Garrett himself on them) over at Sabine’s blog, Woit’s blog and Physics Forums.

Garrett acknowledges that his theory is a boolean entity; there are only two possibilites: it can be right or wrong. If that is so, and if it can be developed enough to offer predictions, be falisfied, etc, then it is good science to me.

A few days after appearing in the arxiv under hep-th (High Energy – Theory), his paper was reclassified, as it seems, by the moderators, to the General Physics classification. It makes me wonder whether the new arxiv numbering system, which no longer carries the subject field label on it, was implemented to ease such reclassifications. In any case, it is not clear what is the criteria used by the moderators to move the papers around in the repository, not to say other issues like acceptance, endorsement system, etc.

Garrett Lisi is an independent scientist who received a grant from the FQXi Foundation.

I hope that his new theory is scrutinized to exhaustion, like any theory should be. Specially if it claims to be a first step towards a ‘theory of everything’.

Updates:

1- Garrett’s paper has been reclassified back to hep-th.

2- Garrett was invited to talk at the International Loop Quantum Gravity Seminar (ILQGS), yesterday. You can find his slides and audio at their site. I recommend listening to his talk.

Additional update:

For some reason, the trackback of this post (automatically sent by wordpress) have not yet appeared at Lisi’s trackback list at the arxiv.

And further update:

A critical assessment on the mathematics (group representation theory) side of Lisi’s paper has been recently given by Distler at his blog. A brief comment by Garrett can also be found there.

And further further update:

More on Lisi’s paper over at Dynamics of Cats: here and here.

(Last?) update:

A personal account by Lisi on the whole story can be found here.

=======================================

Disclaimer: Here I add a compilation of some technical exchanges that occurred over at Cosmic Variance. I have made several edits which leave out parts that I have judged non-technical. The interested reader is urged to go to the original posts if they would like to read the complete posts for citing specific passages or for their own personal interpretations. I post them here just for my own interests. Although I had great care in making the present editing, eventual errors in the compilation/edition process may have occured, so one more reason to check the originals yourself.

=====START OF COMMENTS EXCHANGE=====

Lee Smolin on Dec 11th, 2007 at 10:15 am

I did look at Dsitler’s blog and there is, to my understanding, only one issue was raised in his discussion of Lisi’s paper that was not already raised by Lisi himself in his paper and talks: this is that Lisi muffed the nomenclature for non-compact forms of E8.

The question is whether the open issues are solvable issues or not. Distler thinks not, Lisi thinks perhaps yes. (…) Some of the open issues are straightforward to address, given that there is a literature on this kind of unification, beginning with Peldan in 1992. So in my recent paper I describe how to make a fully gauge invariant action for proposals of Peldan and Lisi’s type, and I also suggest an alternative approach for the fermions which mght resolve some of the other open issues. The gauge invariant action is, btw, the starting point for quantization using LQG and spin foam methods.

(…) the CM theorem (…) concerns global symmetries of the S matrix. In a gravitational theory, which Lisi’s is, global symmetries are symmetries only of solutions or of asymptotic conditions and are not the same as the local gauge symmetries. Indeed, even though my local gauge symmetry is some semi-simple G, I display a solution whose global symmetry is a subgroup of G, namely SO(4)+H where H is the largest compact subgroup of G/SO(4). (The same would work with Lorentzian signature.) Even if G=E8 this is in accord with the CM theorem.

(…)

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Lee Smolin on Dec 14th, 2007 at 9:37 am

(…)

1) There is no issue for the euclidean, compact case.

3) (…) While I haven’t checked in detail, (…) there is no mention in Jacques’s post of the Pati-Salam chirally symmetric theory, as that is what Lisi’s paper shows is embedded in E8. While I haven’t worked out the details, the Pati-Salam is a vector theory where parity is only broken spontaneously. Fermions in the Pati-Salam model are in parity symmetric reps, because of the overall parity invariance of the theory. In Pati-Salam parity is broken spontaneously, leaving chiral fermions at low energy. (…)

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Jacques Distler on Dec 14th, 2007 at 10:39 am

A hint to (…) the Pati-Salam model. The left-handed fermions in that model transform in a complex representation of SU(4)xSU(2)xSU(2), specifically, the (4,2,1)+(4bar,1,2).

Lisi also has the electroweak SU(2) embedded in an SU(2)xSU(2). There, however, the similarity with Pati-Salam ends.

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Jacques Distler on Dec 14th, 2007 at 10:33 pm

(…) I added an appendix, where the phrase [“Pati-Salam”] is used liberally.

(…) Lisi does not even get one generation of quarks and leptons, let alone three(…)

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H-I-G-G-S on Dec 15th, 2007 at 4:23 pm

A small clarification. Let us define a parity invariant theory to be one in which there is a Z_2 symmetry which includes inversion of the spatial coordinates (in three spacetime dimensions). Let us also define a chirally invariant theory as one in which, after writing all fermions as left-handed Weyl fermions, the fermions are in a real representation of the gauge group. Lee uses these two terms interchangably, which is unfortunate, and can lead to confusion. The Pati-Salam model with SU(4)xSU(2)xSU(2) gauge symmetry is parity invariant. To achieve this invariance one must extend the “usual” parity symmetry by a Z_2 which interchanges the two SU(2) factors in the gauge group. However, as correctly pointed out by Jacques, the PS theory is not chirally invariant because the fermions are not in a real representation. Since Jacques showed that the embedding used by Lisi gives a real fermion representation, he is correct in saying that the Lisi embedding does not contain the PS model. Lee is incorrect in saying the opposite (…) One can (and should) break the parity symmetry in PS spontaneously, but this does not suddenly generate chiral fermions from non-chiral fermions. The fermions were chiral to begin with.
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Lee Smolin on Dec 15th, 2007 at 7:11 pm

(…)
What is usually called the Pati-Salam model was defined in two papers, one of which is accessible through the KEK archive and is Jogesh C. Pati, Abdus Salam, LEPTON NUMBER AS THE FOURTH COLOR.Phys.Rev.D10:275-289,1974. (I should note this paper has 2700+ citations, so this is not an obscure work.) Another clear paper is H.S. Mani, Jogesh C. Pati, Abdus Salam, ‘NATURALNESS’ OF ATOMIC PARITY CONSERVATION WITHIN LEFT-RIGHT SYMMETRIC UNIFIED THEORIES. Phys.Rev.D17:2510,1978. In these papers it is clearly explained that their theory is parity invariant and parity is broken only spontaneously. They are explicit that for every left handed current in their model there is a parity related right handed current with equal coupling constants. Hence before spontaneous symmetry breaking the fermion rep for Pati-Salam must be parity invariant. More specifically, in the first paper above, just after eq 3, they state that the fermions are in the representation (4-bar,2,1)+(4-bar,1,2) of SU(4) x SU(2)_L x SU(2)_R, which is parity invariant. (Note that parity exchanges the left and right handed SU(2)’s.)

In Distler’s post he asserts to the contrary that in the Pati-Salam model the fermions are in the representation he calls R_ps= (4,2,1)+(4-bar,1,2) of SU(4) x SU(2)_L x SU(2)_R, which is not parity invariant. (Parity changes this to (4,1,2)+(4-bar,2,1) which is not equivalent to R_ps.) This disagrees with what is stated in the above paper, by the omission of a single bar. This small change has a major impact on the discussion because it turns a parity symmetric theory into a parity non-symmetric theory which is not the Pati-Salam model.

It seems to me this invalidates Distler’s discussion of Pati Salam and by extension suggests that his second post on Lisi is incorrect. While Lisi’s scheme is not quite the same as the above, because the electroweak gauge symmetry is unified first with local lorentz, then with G2, the moral is the same because Lisi breaks E8 to a version of electro-weak unification which is parity symmetric. This is ok for the same reason Pati-Salam is ok, because parity can be broken spontaneously.

(…)

Ps to HIGGS.

One does not need a special extension of parity to switch SU(2)_L with
SU(2)_R, because they couple to left and right handed currents in the usual sense, so parity switches them. And, in case there is any confusion, the above discussion is not affected by any terminological confusion as to the meaning of chiral.
……………………………………………….

Jacques Distler on Dec 15th, 2007 at 7:39 pm

(…)

Start by writing everything in terms of left-handed 2-component Weyl fermions (whose Hermitian conjugates are right-handed Weyl fermions).

The quantum number of a generation of fermions are as I stated in my post (and as can be confirmed by a myriad of contemporary sources). To check, it suffices to look at what happens when you break the Pati-Salam group down to the Standard Model gauge group to see that what I wrote down give the correct Standard Model content, and that what you wrote down does not.

In fact, the representation you wrote down is anomalous, so could not possibly be correct.

H-I-G-G-S stated the distinction between Parity and Chirality correctly. The names of the two SU(2) groups in Pati-Salam are just names.
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H-I-G-G-S on Dec 15th, 2007 at 7:46 pm

(…)
In the Pati-Salam paper what they say is that \psi_L is in the (2,1 \bar 4) and \psi_R in the (1,2 \bar 4). It is common
in the study of grand unification to write all fermions as left-handed fermions. \psi_R^*, after multiplication by a suitable matrix, is left-handed. Let us call it \psi’_L. We then have

\psi_L: (2,1, \bar 4)

\psi’_L: (1,2, 4)

Note that complex conjugation does not change the SU(2) rep because the 2 is pseudo-real. The representation in which all fermions are left-handed is the one that Jacques was correctly using and it agrees with Pati-Salam. The relevance of the reality of the representation when written this way is that a real representation allows a gauge invariant mass term whereas a complex one does not. (…)

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Lee Smolin on Dec 15th, 2007 at 7:49 pm

Jacques,

You are claiming 1) that the original Pati-Salam paper I refer to is incorrect about what their own theory is and 2) even though their big point is that the fundamental dynamics could parity invariant that they should have based the model on a parity non-invariant representation (4,2,1)+(4-bar,1,2) instead of the parity invariant one they specify?
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H-I-G-G-S on Dec 15th, 2007 at 7:57 pm

Lee,

(…) Jacques agrees with PS. He is simply using a basis in which all fermions are left-handed. PS are not. But they have exactly the same degrees of freedom and are
describing the same theory. (…)

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Lee Smolin on Dec 15th, 2007 at 8:36 pm

Dear HIGGS

I see, if it is then just a terminological mixup that is of course fine for this issue. (…)

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Lee Smolin on Dec 17th, 2007 at 11:47 am

Dear HIGGS,

(…)

“3) He has nothing new to say about dynamics so I expect the theory to be non-renormalizable, to the degree the theory is even defined.

5) The E8 symmetry is not a symmetry at all since there is no limit proposed in which it is restored.”

These two points are addressed in my paper, by the giving of an action which is fully E8 gauge invariant and by the proposal of a spin foam quantization based on that E8 invariant action. While we do not yet know if that leads to an ultraviolet finite quantum theory, it is known that related forms of spin foam actions are uv finite, for reasons that may apply here. In particular, spin foam models for just general relativity are ultraviolet finite and well defined. So there is a solid basis to proceed to investigate a quantum theory of an E8 unified theory. (Note that my paper applies to a general class of theories that include Lisi’s E8 proposal.)

(…)

“4) He starts with all couplings equal but does no RGE analysis to see whether they take reasonable values at low-energies. This seems unlikely since the low-energy couplings do not unify without superpartners or some other new structure at low energies.”

I agree it would be good to a RG analysis but I also hope that the first paper proposing a new theory does not already have to present a full renormalization group analysis. (…) is it not true that in any spontaneously broken gauge theory, broken at a scale M, the couplings must unify above the scale M? Is it not also true that there must be many ways to fill in the desert so as to explain a unification of couplings at a scale M, the MSSM gives just one way to do this? The MSSM is sufficient, but not necessary to complete grand unified theories to give a unification of gauge couplings.

“1) It does not contain the standard model fermions in a chiral rep. In fact it contains fermions in real reps, so they will presumably have large masses and one will not get the chiral structure of the SM.”

This is claimed (…) Let me ask two questions, related to this issue in the whole class of models I study including E8. (…) In these models a semi-simple gauge group G with exact G invariant dynamics is spontaneously broken to a subgroup

H= local-lorentz x Y,

where Y is a yang-mills gauge group. The definition of chirality you use involves treating the left and right handed parts of fermion fields differently, so it cannot be applied directly to reps of G. Now do you know the answer to the following question: what property does a representation of G have to have so that it gives rise, after spontaneous symmetry breaking, to spacetime spinors which are chiral in Y? (…)

Second, (…) Distler’s post (…) seems to involve two distinct claims. First that the particular gauge groups Distler, in an earlier post, claims are subgroups of a parrticular non-compact form of E8, are not. Second that the construction Lisi gives of fermions arising from certain pieces of the adjoint of E8 do not give rise to chiral reps (in the senses that you use it) after the spontaneous symmetry breaking. (…) In particular, if Distler is making the first claim, why is it at all necessary to bring up the issue of chiral fermion reps?

(…)

“2) It mixes bosons and fermions with (…) BRST but no definite proposal about what the mathematical structure is that lies behind this. For example, are the physical states defined by BRST cohomology classes?”

Lisi is definite about the mathematical structure he is referring to, he is using a certain definition of a “BRST extended connection”, but he is not using it the way BRST is often used to construct gauge invariant amplitudes for yan-mills theory. In any case, as I indicated in my paper, in case Lisi’s approach fails there is another way fermions could arise in such a theory. This does not in particular, limit them to coming from generators in the adjoint of E8. This is one reason why I am interested in the more general question I raised above.

(…)

I do not have a big stake in how the issue turns out with Lisi’s fermions because I have a different proposal for how fermions can arise in the kind of gravity-gauge theory unifications his proposal fits into. This proposal does not limit them to arising from certain generators in the adjoint, as Lisi’s does. (…) I am happy either way, I just want to understand what is true.

(…)

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Jacques Distler on Dec 17th, 2007 at 1:16 pm

I find the mixing of these two issues confusing. In particular, if Distler is making the first claim, why is it at all necessary to bring up the issue of chiral fermion reps?

(…) Even though the particular real form of D_4xD_4, chosen by Lisi, does not embed in any real form of E_8, there are other real forms of D_4xD_4 which do. So a thorough analysis would study those other real forms as well, to see what one can obtain with them.

That is what I did (I fact, I looked at what you get for all five cases where G can be embedded as a subgroup, though I only reported the result for the two “Pati-Salam”-like cases.

[W]hat property does a representation of G have to have so that it gives rise, after spontaneous symmetry breaking, to spacetime spinors which are chiral in Y?

In a nutshell, if you look at the piece of the adjoint representation that transforms as a “2″ (as opposed to a “2bar”) of SL(2,C), then that should be a complex representation of SU(3)xSU(2)xU(1). Ideally, it should contain 3 copies of the (3,2)_{1/6}+(3bar,1)_{1/3}+(3bar,1)_{-2/3} +(1,2)_{-1/2}+(1,1)_1.

……………………………………………….

Lee Smolin on Dec 17th, 2007 at 1:29 pm

Jacques,

Thanks re the first. Re the second, yes, I knew that, the question is, is there some general property of a rep of G that guarantees that or for which this is never the case? Can it ever be true for a real rep?

Let me add a third question, is the issue of embedding of Lie algebras-leaving the fermion question aside-clear for the euclidean case? That is does SO(4) + SU(3)+ SU(2) + SU(2) + U(1) embedd in the real form of the lie algebra of E8, using the decomposition suggested by Lisi? Is there any sense in which the lie algebra SL(2,C) + SU(3)+ SU(2) + SU(2) + U(1) fits in a complexification of the lie algebra E8?

Finally, what is the best reference to understand why there are only these two non-compact forms of E8 and where the results you quote come from? At the level of the lie algebra it would seem there might be more freedom. Is it easy to say why there is not?

(…)

……………………………………………….

Jacques Distler on Dec 17th, 2007 at 1:41 pm

is there some general property of a rep of G that guarantees that or for which this is never the case? Can it ever be true for a real rep?

See here for an argument that applies quite generally to the “Euclidean” (compact real form) case of any group (not just E_8). It is almost, but not quite a proof for the Minkowskian (noncompact real form) case. There are some potential loopholes there. (At least, they seem like potential loopholes to me; someone who knew more about the representation theory involved could probably complete the proof pretty quickly.)

Finally, what is the best reference to understand why there are only these two non-compact forms of E8 and where the results you quote come from?

See the article by Marcel Berger that I linked to in my post. To define a noncompact real form, you need an involution of the Lie algebra that acts as +1 on the compact generators, and as -1 on the noncompact generators. Such an involution also defines a symmetric space structure, and Berger classified those.

……………………………………………….

Jacques Distler on Dec 17th, 2007 at 1:51 pm

Actually, to find the noncompact real forms, you need to classify Riemannian symmetric spaces. That classification long predates Berger’s paper.

See the textbook by Helgason.

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Lee Smolin on Dec 17th, 2007 at 4:12 pm

Dear Jacques,

(…) Following your post, the Pati-Salam fermions are in a rep given by

R_ps = (2,r) +(2-bar, r-bar)

where these refer to their transformation properties under the sum of the spacetime lorentz algebra so(3,1) and H, where H is the Pati-Salam algebra

H=su(4)+ su(2)+su(2)

And r = (4,2,1)+(4-bar,1,2)

Now we established before that r is not equivalent to its complex conjugate, so R_ps is in the standard terms chiral. It is also the case that parity takes R_ps to itself. Now, R_ps is also pseudo-real, ie it is equivalent to its complex conjugate, as it is the sum of itself and its complex conjugate. So suppose there were a bigger lie algebra G that contained so(,1)+H as a subalgebra. Could not R_ps arise from the decomposition of a pseudo-real representation of G?

(…)

……………………………………………….

Jacques Distler on Dec 17th, 2007 at 4:29 pm

Dear Lee,

Perhaps it would be best if we retreated, for a moment, to the “euclidean” case, where we are embedding Spin(4), instead of SL(2,C). Then

R = (2,1, r) + (1,2, rbar)

which is still complex, if r is a complex representation of H.

In any case, Lisi’s game (interpreted most broadly) is to embed things in the adjoint representation of some real form of some Lie group. That is always a real representation.

(…)

……………………………………………….

Lee Smolin on Dec 17th, 2007 at 4:43 pm

Jacques,

But in the lorentzian case where complex conjugation exchanges the left and right factors,

R_ps = (2,1, r) + (1,2, rbar)

of SU(2)_L+SU(2)_R+H

is equivalent to its complex conjugate.

(…)

……………………………………………….

Jacques Distler on Dec 17th, 2007 at 5:24 pm

Lee,

That’s the point! A priori, in the Lorentzian case, R is a real (or pseudo real) representation. So you might think that you could get a chiral spectrum (a net number of copies of R). But, for any embedding of SL(2,C) in the noncompact real form, which is related by Wick rotation to an embedding of Spin(4) in the compact real form, this never happens. I have just explained to you why it never happens.

More broadly, there are other embeddings of SL(2,C) in the noncompact real form, not related by Wick rotation to embeddings of Spin(4) in the compact real form, which also necessarily yield a nonchiral spectrum. Specifically, I have looked at embeddings that proceed via SL(4,R) and via SU(2,2).

What I have not done is show that these are the only remaining possibilities. So there’s still a challenge outstanding, to any readers of my blog, to close that gap.

……………………………………………….

Jacques Distler on Dec 18th, 2007 at 2:15 am

I wrote

What I have not done is show that these are the only remaining possibilities. So there’s still a challenge outstanding, to any readers of my blog, to close that gap.

But let me be very clear that this “potential loophole” is irrelevant to your paper, which concerned the Euclidean case (embedding Spin(4) in the compact real form of some Lie group), and to Lisi’s paper, which concerned an embedding via D_4xD_4.

Neither of these can ever lead to a chiral spectrum (let alone to 3 Standard Model generations).

(…)

……………………………………………….
Lee Smolin on Dec 18th, 2007 at 10:17 am

Dear Jacques,

I am not sure I understand your last point. Let me review. We have been discussing the situation encountered unifying the Pati-Slam gauge symmetry with the local lorentz transformations. This leads us to consider

R_ps = (2,r) +(2-bar, r-bar),

with r = (4,2,1)+(4-bar,1,2)

as described in my 149 above.

We have established 1) that r is chiral, in the sense that it is in a complex rep of the YM gauge symmetry that acts on left handed spinors, 2) that R_ps is parity invariant and 3) that R_ps is equivalent to its complex conjugate.

Thus, if there were a unification of gravity and Yang-Mills in terms of the connection of a larger G which has as a subalgebra local lorentz+H_ps, along the lines of Peldan, myself and others, R_ps could arise from the decomposition of a rep of G which is also equivalent to its complex conjugate.

Do I understand correctly that you agree with this but are arguing that in the particular case of if G=E8, R_ps cannot arise from decomposition of the adjoint of E8 ?

Note: the above is all assuming lorentzian signature. It is true that in my paper I worked with Euclidean signature, but only for simplicity.

(…)

……………………………………………….

Jacques Distler on Dec 18th, 2007 at 10:45 am

(…)

If the embedding of SL(2,C) in the noncompact real form of G is related by Wick rotation to an embedding of Spin(4) in the compact real form of G, then R is nonchiral.

This has nothing to do with the specific choice G = E_8.

——

(…)

For concision, I will phrase things in terms of embeddings of Lie algebras, instead of groups.

Let g_C be the complex Lie algebra, which has (at least) two real forms: a compact real form, g_e, appropriate to the Euclidean case, and a noncompact real form, g_l, appropriate to the Lorentzian case. Let h be the lie algebra of the SM or Pati-Salam, and h_C the corresponding complex Lie algebra (of which h is the compact real form).

We are interested in embeddings of

so(4) x h ⊂ g_e

and

so(3,1) x h ⊂ g_l

I say “these embeddings are related by Wick rotation” if they stem from the same embedding of complex lie algebras

d_2 x h_C ⊂ g_C

by choosing different real forms.

……………………………………………….

Lee Smolin on Dec 19th, 2007 at 12:39 pm

(…)

What we established (…) was that the fermions are in a rep r which has all the following properties:

1) It is chiral in the standard sense (see 99 and 151)

2) When taken in full as a rep R_ps, as defined in previous posts, of lorentz+H where H=su(4)+su(2)+su(2) it is parity invariant.

3) R_ps is also pseudo-real.

It was important to establish these points, because they bear on Lisi’s proposal and more generally for model building of this type. Pati-Salam is not a trivial point to be filled in, it is the key to whether Lisi succeeds or fails. This key point was not commented on by Distler or anyone here until I brought it up.

I also asked what properties a Lie algebra G has to have so that it has a subalgebra of lorentz + H, with a representation that on breaking to the subalgebra gives R_ps. My question, central to understanding if anything like Lisi’s proposal can work, was based on the fact that the situation was rather different than the case of the usual standard model where 2 and 3 don’t hold.

Distler did in 165 offer a general statement, “If the embedding of SL(2,C) in the non-compact real form of G is related by Wick rotation to an embedding of Spin(4) in the compact real form of G, then R is non-chiral.” So far as I can tell this is false because G=SO(3,1)+H is itself a counterexample to it, given the above. Were there further discussion this would be the place to start.

(…)

……………………………………………….

Jacques Distler on Dec 19th, 2007 at 1:17 pm

(…)

1. Pati-Salam is not a vector-like theory. It is inherently chiral.
2. (…) You can never turn a vector-like theory into a chiral theory, “leaving chiral fermions at low energies” by spontaneously breaking parity.

In #102, you said

In Distler’s post he asserts to the contrary that in the Pati-Salam model the fermions are in the representation he calls R_ps= (4,2,1)+(4-bar,1,2) of SU(4) x SU(2)_L x SU(2)_R, which is not parity invariant.
…
It seems to me this invalidates Distler’s discussion of Pati Salam and by extension suggests that his second post on Lisi is incorrect.

which (…) was (…) wrong.

In the sense you are using it (viewing the complete fermion representation as (2,r)+(2bar,rbar) of SL(2,C)xH), property 3 holds for the Standard model and, indeed, for every unitary quantum field theory (…).

What is true of Pati-Salam, which is not true of the SM, is that there is a definition of “parity” such that property 2 holds.

But that is completely irrelevant (as far as I can tell) to the rest of your argument.

(…)

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Brother Love on Dec 19th, 2007 at 4:39 pm

Perhaps I can help resolve confusion, just in case anyone is, by digressing. The PS model has chiral fermions (so P \Psi \neq \Psi) in the fundamental rep and has a left-right symmetry (under P the Langrangian looks the same). The SU(2)_R group factor may be broken to a U(1) (isospin) via a Higgs in the fundamental representation of SU(2)_R acquiring a VEV. (Maybe when we are seeing this referred to as a vector model somewhere above, it is referring to this Higgs being in the fundamental representation of the SU(2).) Consequently, the L-R symmetry is now broken. However, the fermions were always chiral.

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Metaphysical ‘karma’

A finite set of mathematical relations is the foundation of reality.

The objetive of science is to find those relations. As a consequence, science also attempts to clarify the ontological status of those mathematical relations. That is, to bring the metaphysical frontier in which they are deeply buried closer into the physical frontier.

If mathematics were reality itself, then reality would be an infinite substance, or an infinite number of substances, which would reflect in the impossibility of building physical laws.

(Ok, I want to believe that).

In any case, somehow, reality emerges to us as a deeply constrained thing. At the same time, science seems to have intrinsic limits to probe reality. Whether the former is a reflection of the latter, or vice-versa, seems to be our inescapable metaphysical ‘karma’.

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See also previous posts (here and here).

Shut up and calculate?

Tegmark’s new paper… I have no comments for the moment. I didn’t read it.

Update: Just finished reading it. Bad philosophy. Sorry.