The Templeton-funded FQXI organization has announced today the awarding of 30 grants totalling more than $2 million dollars for foundational research in physics. On the one hand I’ve always been dubious about this organization since it is funded by a foundation dedicated not to scientific research but to bringing science and religion together. On the other hand, given the sad state of some of current theoretical physics research, the idea of an organization with a different perspective coming in with new funding and the ability to encourage new ideas that are not getting attention seems highly promising.

The proposal summaries for the successful grants are often so vague that it’s hard to tell what they are actually about, although presumably the full proposals give much more detail. FQXI seems to have succeeded in keeping the Templeton religious agenda at bay, with none of the grants trying to bring religion into science. But I have to confess I find the list of grants rather discouraging. FQXI will be funding several well-known string theorists, a group that has not exactly been starved of funding or attention in recent years. Some of the grants are for “multiverse” research, again something that I don’t think physics desperately needs more of right now.

Almost completely missing from the list of topics awarded grants is high energy physics, or any foundational research into the standard model. Also very hard to find is any interest in further research into the new mathematical ideas that have come out of quantum field theory research during the last thirty years. In brief, what seems to me the most promising way forward for foundational research in physics, working on better understanding the standard model QFT and its mathematical context, doesn’t seem to be something on the FQXI agenda. To be fair, I have the depressing suspicion that if I had to go through all the grant proposals submitted to them, I might not have been able to do much better in terms of coming up with promising things to fund.

Last week an interesting semester-long program on Non-commutative Geometry began at the Newton Institute in Cambridge, and some of the talks have already begon to appear on this web-site. The program will include a Templeton-sponsored workshop in early September on the topic of Fundamental Structures of Space and Time. Like FQXI, the workshop mostly seems to be free of religious influence, although there will be a public panel discussion on The Nature of Space and Time which will feature two clergymen.

Over at Cosmic Variance, Sean Carroll, who is at least as dubious about Templeton as I am, has a much more positive take on the FQXI grants. In the comment section FQXI associate director Anthony Aguirre points to a new mission statement at Templeton. At their web-site you can also watch a rather long video about this if you’re so inclined, or see a list of upcoming conferences they sponsor on topics in science and religion (they’re especially interested in cosmology).

**Update**: There’s a story about this at Inside Higher Ed.

Thanks for your take on this. In terms of the particular projects funded, the truth is that FQXi does not have any particular scientific “agenda” (e.g., pro- or anti-string theory, etc.) or criteria other than what was stated in the Request for Proposals. Another panel would likely make different decisions, but I suspect not *very* different ones. If there are foundational, unconventional projects (or researchers) in high-energy theory that you see languishing for lack of funds, please encourage them to apply for the next round!

best,

Anthony

Peter,

There do appear to be a couple of funded projects focused on gauge theory, including this highly interesting and unusual one involving gauge theory and the structure of the standard model:

Deferential Geometry

Of course, my opinion may be biased.

Also, it looks like the institutional links under the researcher names actually point to the researcher home pages.

There are quite a few funded projects on unconventional approaches to quantum mechanics — that seems an interestingly different point of interest than the standard fare.

Hi Garrett,

I should have mentioned that yours was the one proposal among the 30 about geometry and the standard model. Good luck with it!

Thanks. And even more than luck, help is welcome.

If there are foundational, unconventional projects (or researchers) in high-energy theory that you see languishing for lack of funds, please encourage them to apply for the next round!Is this a joke, or what? As with the mainstream, a system based on mutual recommendation eliminates unconventional research by definition.

Would you not say the two experimental proposals to look at the ‘constancy’ of the fine structure constant are to do with HEP?

And in theory we have Donoghue’s ‘emergent gauge symmetry’ which would be very much at the heart of HEP.

Thomas,

By HEP= high energy particle physics I meant the conventional meaning of studying particle interactions at high energies = short distances. The question of variation of constants on cosmological scales is a very different subject.

The Donoghue proposal is also more or less the exact opposite of the kind of mathematical investigation of the structure of gauge theories that I was talking about. He wants to throw away that whole structure as not fundamental.

The emergence of gauge bosons (as well as the emergence of Fermi statistics and gravitons) has been an active research area since 1987 in condensed matter theory. Many results were obtained, such as a unification of gauge interaction and Fermi statistics.

Just google “emergent gauge bosons”.

I can’t believe what people will pay for. FQXI (sounds like a southern radio station – “QXI in Dixie!”) is the IKEA of theory. Hidden variables in cosmology? Why not apply the Bell inequalities to colliding galaxies? Apparently this dude Lisi has figured it all out anyway so why bother?

The real question people should be studying is how the academic world collapsed under the dehumanizing weight of sex, drugs, and rock and roll.

-drl

Funding projects that look at possible time variations in fundamental constants is an extremely good idea. If the fine structure constant or the electron-proton mass ratio turn out to be time dependent, then this is an unambiguous indication of new physics. Moreover, unlike say, dark matter, time varying constants are relatively difficult to incorporate into the conventional frameworks that we use to describe physics. So the discovery of time variations would tell us a great deal about what kinds of changes are required if we are to understand the fundamental physics of the universe.

IKEA of Theory, indeed! I would suggest FQuiXi as a more commercial audiologo, though IT IS a southern radio station of sorts, if you think about it. Though that kind of money can surely buy a lot of sex, drugs, rock and roll and surf boards, I would say Templeton has managed to make science so silly and way out there that religion looks a lot more attractive! Not surprisingly, perhaps, most of these people work in institutions (my own included) that would insure them decent funding for better refereeing than what this list brings out without selling off their scientific honesty! No? Maybe I am the one who’s being naive…

In reply to Wen

http://www.math.columbia.edu/~woit/wordpress/?p=436#comment-14002

Xiao-Gang Wen,

thanks for joining this discussion. Congratulations on your Foundational Questions award!

http://dao.mit.edu/~wen/

I agree with Smolin’s position that more support should go to those with the courage and intellectual independence to pursue approaches to basic problems that are not commonplace: in effect allocating to the individual track-record instead of to the entrenched program. Your condensed-matter type approach to the emergence of spacetime and fundamental interactions is an example of vigorous investigation off the beaten track.

Hopefully the awards will have a “multiplier” effect when departments at major universities see the FQX spotlight picking out individuals committed to bold original lines of research. Conceivably, some departments will be influenced to shift policy in awarding postdoctoral fellowships, and provide graduate students wider choice of topic.

D.R. Lunsford, any physics department in the US would be lucky to get “the dude Lisi” as a research fellow

http://arxiv.org/abs/gr-qc/0511120

Clifford bundle formulation of BF gravity generalized to the standard modelthat is exactly the kind of research dude they should be looking for.

Just looking down the list of awardees, I see they are almost all attached to major academic institutions and it is spread out—not much clumping.

Michigan, Berkeley, Kansas, Massachusetts, London-Imperial, Yale, SantaBarbara, Perimeter-Waterloo, Illinois, Barnard, Harvard, Tufts, Louisiana, Oxford, Cornell….etc. …

(and of course MIT where X-G Wen is)

I really like the spread. It feels like a “democracy of the excellent” instead of a prestige pecking-order. And if they wanted to have a “multiplier effect” on the way departments think, then a wide spread was well calcuated. Congratulations to those responsible (Tegmark, Aguirre, and all who helped.) Lots more could be said—which I’ll leave to others.

Who,

massive shrug – quaternion disease. Most of us get past that stage while undergraduates.

None of the proposals listed on FQXI has a shred of physical interest. Big surprise.

-drl

Thanks Who.

DRL,

The relationship:

fermions -> spinors -> Clifford algebra -> quaternions

seems pretty clear. Which step do you dislike?

I’m kind of surprised to be the only guy on the list with a weird TOE. Makes me feel lonely. I’m curious what the 20 proposals that made the first cut from 172 to 50, but not the second cut from 50 to 30 were.

Garret,

What you have is algebraic logic chopping with no real dynamical principle, and so no theory. But congrats on getting people to pay you for it. I looked at similar ideas in the early 90s and rejected them as a waste of time. V. Fock did similar work with the Dirac algebra in GR as far back as the mid 30s. That work is still valuable, but offers no new ideas – only Dirac on a Riemannian manifold.

-drl

drl,

That’s essentially correct — except this happens to be unusually successful algebraic logic chopping. And if I had the dynamics worked out perfectly, I wouldn’t need time to work on it. In any case, this comment thread isn’t the best place to discuss it. I’d be happy to talk elsewhere.

Best,

Garrett

Garrett,

Yes not here. “The true metaphysics of y5 is hard.”

-drl

In replay to Garrett,

I do not know very much about the 20 proposals that did not make the second selection … but I do know “something” about some of those that did not make the first selection … in fact my project “Algebraic Quantum Gravity: Spectral Space-time via Non-commutative Geometry” (by the way: this was a project in “nc-geometry”+operator algebras+category theory) was immediately rejected without any explanation, apart from mentioning the fact that there were so many other much better proposals.

I am actually very pleased to see that (at least for Quantum Gravity and TOE research) the funds went to some of the best experts in the field (some of them like C. Isham, L. Crane, F. Morkopoulou, O. Dreyer are some of the researchers that I admire the most).

I am a bit disappointed to see that the grants (apart from Garrett’s case of course!) actually went to people that are extremely famous, mostly with tenures and whose research programs do not need any particular help to get visibility, credibility … and most of all … funds …

Considering the fact that the main aim of the Fqxi “foundation” was to support “out of box thoughts”, “unconventional approaches” and even “research that would not be otherwise possible to fund” … I really find it difficult to see any relation with these words and “most” of the actual selected programs. The only natural pleasing answer is that … apart from those well-known names, already publishing heavily in the field, there are really no new thinkers that deserve to be supported.

On one side, this is a really good news for universities in USA or UK … since they already got all the “interesting” people with the good ideas!

On the other side, sincerely, I find it difficult to believe that there are no “original thinkers” in China, Japan, India, Korea (or “God forbid” … in some “developing” country …) that applied for a grant … and I find even more difficult to believe that they “do not” need support more that those selected!!

I would have liked to know in advance that when I submitted my “low profile” project I was going to compete with Nobel prize level people looking for extra sources, through “mutual recommendation” … it could have saved me (and to many others) some precious time and probably some face!!

In replay to the following phrase from Aguirre:

“If there are foundational, unconventional projects (or researchers) in high-energy theory that you see languishing for lack of funds, please encourage them to apply for the next round!”

For someone that is working (and probably trying his best to do some research) in difficult conditions of almost total lack of funds, heavy teaching load (with no real leaves, nor sabbatical) and in complete isolation (apart from an internet connection), it is a bit depressing to get this kind of answers … anyway thanks!

In replay to Woit (and all the other people that have problems accepting funds from religious institutions … do they?):

I really fail to understand the point here: I always considered spiritual and religious parts of life as natural as those related to art, literature, science and as a normal byproduct of human life (like breathing, eating, sleeping, enjoying conversation, play, music, knowledge …. ).

I see that most of the complaints actually come from people with permanent positions or anyway with good jobs in American Academies and it seems strange that those people fail to recognize that they are actually sponsored by public funds coming from a vast majority of a “very religious” population and from an administration that is officially promoting not only “innocent” religious ideas, but also very much “radical and fundamentalist” approaches to religious life (let alone political practices!) … I would take their words seriously when they will change job … or change ideas!

Well … good luck to Garrett and to the other winners of the Fqxi awards.

And forgive me for a replay that is a bit “resentful”, but I could not resist

Sincerely, Paolo

Although my sympathies (and congratulations on funding) lie with Garrett, my better judgment agrees with D R Lunsford that he’s off in the weeds. I think he suffers not so much from “quaternion disease” as from Lagrangian disease — the impression that what matters about a field theory is its (geometric and/or internal) symmetry algebra and its nominal Lagrangian as written in an introductory text. I recall getting a rude awakening from this illusion when I first tried to go from reading about non-Abelian gauge theories to trying to calculate anything in one.

I’m not a working physicist (or mathematician) and don’t claim to understand QFT even to the degree that a freshly minted Ph.D. from a second-rate university does, let alone people who can compute accurate QCD backgrounds for collider experiments. But you don’t really have to understand functional determinants and renormalization group flow to realize that the Lagrangian of an effective field theory may not look much like its Planck-scale antecedent. Take the example of Faddeev-Popov ghost fields (for mathematicians, more or less the Maurer-Cartan form on (the identity component of) the group of QCD SU(3) gauge transformations). They’re “unphysical” in the sense that they violate the spin-statistics theorem (among other things), so you rarely see them included in the headline SM Lagrangian — but good luck expressing anything quantitatively in QCD without them or their equivalent.

Is the entire fermion structure of the Standard Model just as “unphysical”, arising not from “algebraic logic chopping” of some set of geometrical objects but from something deeper such as the need to calculate past a wider redundancy in Fock space? I lack the techniques to state this question rigorously, let alone to answer it, but I can use it (along with other foundational questions about the QFT formalism) as a litmus test for ToE candidates. I’m looking for some combination of predictive and explanatory power, and I’m convinced that neither is to be found along lines of reasoning that focus narrowly on the symbols under the action integral.

So at best I think Garrett has wandered off in a different direction from the string theorists, and perhaps not as far (he is presumably not long-term dependent on “faute de mieux” grants for his livelihood). With apologies to Douglas Adams, the fundamental design flaws of his approach are completely hidden by its superficial design flaws. I say this having spent an unfortunate proportion of my life chasing similar dreams, and in hope that he will use some of the liberty that money buys to broaden his knowledge of how QFT calculations are actually made.

Cheers,

– Michael

Paolo wrote:

On the other side, sincerely, I find it difficult to believe that there are no “original thinkers” in China, Japan, India, Korea (or “God forbid” … in some “developing” country …) that applied for a grant … and I find even more difficult to believe that they “do not” need support more that those selected!!I would have liked to know in advance that when I submitted my “low profile” project I was going to compete with Nobel prize level people looking for extra sources, through “mutual recommendation” … it could have saved me (and to many others) some precious time and probably some face!!

Dear Paolo,

You summarize my impressions when I saw the list of awardees. I am somewhat relieved that my decision to not submit my project to the FQXi grants was a good one (at least I was able to spare myself of further frustrations), although I was confident that my project did meet their criteria. Well, let’s keep on going. You have my sympathy.

Best wishes,

Christine

Thank you, Paolo. I couldn’t have said it so well myself. Of course I wish all the awardees the best in their researches, but I was stunned when I saw the list. My proposal was on pseudomonads in a quantum topos style unification. I was rejected on the first cut because “the subject matter of the proposal didn’t meet the criteria of the fqxi guidelines”. I prefer people to be honest.

Kea wrote:

Of course I wish all the awardees the best in their researchesYes, I also wish them all the best of course. I look forward to learn about the new developments and results that these investigations will lead.

Christine

Dear FQXi applicants, well-wishers, and naysayers:

Your comments on this forum are valuable to us. A few comments in return.

First, believe me, we would have liked nothing more than to have had $20m to give away, and to fully fund all of the initial proposals we received. Cutting these requests tenfold was very difficult for the panels, and they did their best using exactly the same criteria provided to the applicants in the RFP document. Note also that some very prominent researchers had unsuccessful applications — and we hear from them too!

Second, the two-part application process was devised in part to save the community a substantial amount of time in preparing lengthy and ultimately-unsuccessful applications. This succeeded, but other than additional money there is no way to mitigate the frustration of most applicants going unfunded.

Third, it should be kept in mind that even for established researchers, getting funding for foundational projects is not — at all — easy. Usually this sort of work is done “on the side” while supported by other more conventional funding, and part of FQXi mission is to bring this side-work into the fore.

Fourth, since from the outset we took care to make FQXi open to researchers worldwide, we were actually disappointed in the rather small number of applications from outside North America and Europe. Certainly there was no bias against them — in fact, the favorable buying power of the dollar alone gives such applications (from India, for example) a substantial advantage in terms of cost-effectiveness. We think many more are out there, and we will redouble our efforts to get the word out to such researchers in future grant cycles.

Finally, we are indeed aware of the difficulty of judging the value of truly original research proposals — as we *must* do in order to make funding decisions. We are working on ideas and strategies for this, and welcome constructive ideas along these lines.

Garrett,

Judging from the comments to an earlier posting (“Various and Sundry”, July 21, 2006), I may owe you an apology for assuming (based on the paper you linked above) that you were only interested in the stuff under the action integral. I agree with your comment there that BRST is of “foundational importance”. The construction you are looking for, in which the BRST operator is extended to form the exterior derivative on a gauge bundle, is being discussed in the comments to “P. University Press”, and there’s a link there to a paper by Schuecker which may help kick-start a literature search.

When I get a chance, I will try to summarize concisely where the Faddeev-Popov ghost field fits (a standard result; see for instance Peskin & Schroeder) and how it can be extended to the Maurer-Cartan form on the space of right-invariant vector fields on the gauge bundle (probably also nothing new, but I haven’t found it in the literature).

Cheers,

– Michael

Paolo,

Sorry about your proposal. I was also surprised to see so many familiar names on the list — I figured there would be more dark horses from all over the world. The only consolation I might offer is that they don’t appear to have funded any category theory projects. It’s not hard to see the category theory star rising though, so don’t give up on it.

Michael,

If the biggest flaw with my theory is having a nice Lagrangian, I’ll be very happy. And Faddeev-Popov ghosts are at the heart of what I’m working on, and I am indeed very interested in the Maurer-Cartan BRST construction. I’ve seen it here and there, and even used it in my own work, but my understanding is sketchy. I’ll go look in the other comment thread immediately. Also, it would be great to talk about this at

physics forums

which I think would be a practical and appropriate place to discuss it. I’d love to see your summary there. Please send me an email so we can talk about it:

gar at lisi dot com

I’ve been away at my sister’s wedding for a few days, but I’m back now for awhile.

Anthony,

Thanks!!! If you do check back on this thread, I have a question:

When do you think FQXi will be opening up its discussion forum?

Garrett and Michael Edwards mention ghosts, BRST, and Maurer-Cartan.

A useful paper about that is

Geometrical reinterpretation of Faddeev-Popov ghost particles and BRS transformations

by Jean Thierry-Mieg (J. Math. Phys. 21 (12) December 1980 (2834-2838)

whose abstract states:

“… A classical geometrical interpretation of the ghosts fields is presented. BRS rules follow from the Cartan-Maurer fibration theorem. The statistics of ghosts are explained and the effective quantum Lagrangian is derived without factorizing the volume of the gauge group. Topologically nontrivial ghost configurations are defined. …”.

The body of the paper says in part:

“… the connection 1-form … describes at the same time both the Yang-Mills gauge particle and the Faddeev-Popov ghost particle. With respect to a section, i.e., a gauge being chosen, the connection actually splits into the sum of two components:

the gauge field … which is horizontal

and

the ghost field … which is normal to the section.

The exterior differential … also splits, and its component normal to the section is … the BRS operator.

… the Cartan-Maurer structural theorem, which states the compatibility of the connection with the fibration, implies the BRS transformation rules of the gauge and ghost fields.

Moreover, the ghost does not contribute to the curvature 2 form (field strength) and may be thus eliminated from the description of the classical theory. …”.

Tony Smith

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

Garrett:

The forums are on our current to-do list, and should hopefully be up in time for the fall.

best,

Anthony

Tony,

I think it’s more accurate to say that the most fundamental form of the connexion on a principal bundle – a separation of the tangent space at each point into horizontal and vertical subspaces – is expressed as the Maurer-Cartan form (vertical projection) on the _bundle_. Its restriction to the vertical subspace is identified with the Faddeev-Popov ghost, because infinitesimal gauge transformations are the vertical subalgebra of the Lie algebra E of right-invariant vector fields on the bundle. (“Right-invariant” here means invariant under the right action of the bundle structure group, i. e., the gauge group.)

The connexion 1-form on the bundle pulls back on a local section to the “gauge” connexion 1-form on the base space. Because this pullback is defined using the connexion itself (it’s dual to the “pushforward” of vector fields on the base space to “horizontal” vector fields on the bundle), you could say that it “splits out” the portion of the Maurer-Cartan form that acts on the horizontal subspace. But I would hesitate to use the word “normal” in this context because it doesn’t have anything to do with metric structures or inner products. _Any_ connexion 1-form on the bundle is a projection from E to its vertical subspace, which is in turn the kernel of the “projection” (pushforward, really) of E to the tangent bundle of the base manifold.

Similarly, one can define an exterior derivative d_E: \Lambda^k -> \Lambda^(k+1) on the space of local-polynomial-valued alternating forms on E without essential use of a connexion on the principal bundle. The special case of a 0-form Q (e. g., a Lagrangian polynomial) yields the exact 1-form d_E Q, whose restriction to the vertical ideal of E is -s Q, where s is the BRST operator (I believe that the minus sign is the historical sign convention, if G&S and P&S are any guide). That’s what is meant by the statement that -s is the “component normal to the section” of d_E, although “normal” is again somewhat misleading; the vertical ideal of E is well-defined irrespective not only of any metric-like structure but also of independent of the connexion.

The space \Lambda of local-polynomial-valued alternating forms on E, and the corresponding graded Lie algebra of derivations on \Lambda, is a powerful formalism for expressing the calculus of variations. The derivations on \Lambda can be extended to an even larger operator space via the wedge product and the Frölicher-Nijenhuis bracket. If I had to bet on a dark horse in the Theory of Everything stakes, I think this would be the mathematical formalism in which it ought to be expressed.

Cheers,

– Michael

Nope. OK, Michael, can you send me an email and then I’ll start a Physics Forums thread for this?

This is starting to feel inappropriate for Peter’s blog, but I thought I should at least explain my comment on the Frölicher-Nijenhuis bracket and provide a reference.

Another term for a graded Lie algebra is “superalgebra”, and the sheaf of alternating forms on a vector bundle is a classic example of a supermanifold. The subspace \Omega^1_* of \Lambda consisting of (vert E)-valued horizontal alternating forms on E can be given a Lie superalgebra structure via the Frölicher-Nijenhuis bracket, which coincides with the Lie bracket on 0-forms (i. e., polynomials taking values in vert E).

Better yet, the entire graded Lie algebra Der \Omega(vert E) of derivations on \Lambda, including (for instance) the BRST operator, can be identified with the direct sum of two copies of \Omega^1_*, one acting by contraction (the generalization of the inner derivative) and the other by Lie derivation (the generalization of the Lie derivative). See, for instance, this paper by Cap et al.:

http://citeseer.ist.psu.edu/cap94frlichernijenhuis.html

I think you might find that Der \Omega(vert E) is the Lie superalgebra of a supergroup of physical interest. This isn’t the same thing as (the physicist’s idea of) supersymmetry, because it doesn’t involve spinors and doesn’t establish any preferred relationship between particular odd and even elements of the algebra. One could try to mix form degrees of freedom of multiple ranks to get spinor fields via the Kähler route, but I think that’s probably a dead end. Instead, I would expect you could exploit the isomorphism between (portions of) the _operator_ algebras on forms and on spinors in the course of BRST quantization.

I suppose that where this is headed is a generalization of the BRST quantization procedure to second class constraint systems. I wonder whether that would have any physical uses.

Cheers,

– Michael

Garrett,

I’m not really in the habit of online forums (I tend to drop in for a short while, use up my little store of insight, and go back to lurking) but feel free to write me at m.k.edwards at that webmail service that Google runs. Don’t expect great things, though; I’m almost out of things to say already. The rest is on the “P. University Press” thread, along with some pointers into the literature from Peter and Thomas that may take me quite some time to digest.

Cheers,

– Michael

Michael Edwards mentions

“… Another term for a graded Lie algebra is “superalgebra” …”

and

mentions “… alternating forms … \Lambda …”

as

“… the mathematical formalism in which … the Theory of Everything … ought to be expressed.

ought to be expressed …”.

I don’t disagree with the general ideas, but as to more specific details I note that :

1 – Some graded Lie algebras are NOT “superalgebras”, but are interesting Lie algebras that combine vectors, bivectors, and spinors, such as by the 5-grading

g = g(-2) + g(-1) + g(0) + g(1) + g(2)

where

g = E6 ————————————— the total Lie algebra

g(0) = spin(8) + R + R —————— the bivectors

dimR g(-1) = 16 = dimR g(1) ——— the (complexified) spinors

dimR g(-2) = 8 – dimR g(2) ———– the (complexified) vectors

2 – If you want to include spinors, it is nice to extend beyond the exterior algebra \Lamda of alternating forms and go to Clifford Algebras and look at Clifford Modules, which are closely related to Dirac operators.

Also, with respect to my comment about Maurer-Cartan forms, BRS, etc,

my comment consisted of direct quotes from the paper of Jean Thierry-Mieg

at J. Math. Phys. 21 (12) December 1980 (2834-2838),

so any praise or criticism should be of Thierry-Mieg, and not me.

Tony Smith

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

Sorry for following up to myself (again) but this was too stupid to let stand:

> The subspace \Omega^1_* of \Lambda consisting of (vert E)-valued horizontal alternating forms on E …

This was a silly thinko. Horizontal forms on E annihilate vert E. I meant to write, \Omega^1_* is \Lambda(vert E, vert E), which is isomorphic to \Lambda(E, vert E) / Hor \Lambda(E, vert E). The horizontal forms constitute a subalgebra of \Lambda(E, vert E) and are defined independent of any connexion; presumably a choice of connexion can be used to select a unique representative of \Lambda(E, vert E) corresponding to an element of \Omega^1_*. Hey, look, another fiber bundle!

Anyway, I found the “standard” name for that connexion 1-form on the fiber bundle:

http://en.wikipedia.org/wiki/Ehresmann_connection

Whoever “Silly rabbit” at Wikipedia is, he’s doing a bang-up job of overhauling their connection-related entries.

Cheers,

– Michael

Tony,

No personal criticism of you (or Thierry-Mieg) intended; it’s just that the word “normal” gives me hives. I also have this irrational aversion to Clifford algebras, second only to my distaste for div, grad, curl and all that. This is part of the reason that I never made it as a physicist. But if you can find an elegant way to extend the Frölicher-Nijenhuis construction to Clifford algebras, I’m all ears.

Cheers,

– Michael

Michael Edwards said “… But if you can find an elegant way to extend the Frölicher-Nijenhuis construction to Clifford algebras, I’m all ears. …”.

For one example, consider hep-th/0112263

Geometric (Pre)Quantization in the Polysmplectic Approach to Field Theory

by Igor V. Kanatchikov

After using the Frolicher-Nijenhuis theorem with respect to a (Poisson-)Gerstenhaber algebra, Kanatchikov says

“… it was found suitable to work in terms of the space-time Clifford algebra valued operators and wave functions,

rather than in terms of nonhomogeneous forms and the graded endomorphism valued operators acting on them.

In general, a relation between the two formulations is given by the “Chevalley quantization”

map from the co-exterior algebra to the Clifford algebra …”.

For details, see the paper itself.

Tony Smith

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

Tony,

On a first reading, the Kanatchikov paper is about as clear as mud to me. I fail to see why you would want to “generalize the cotangent bundle” in the direction he chooses, I have no idea which objects are defined solely in terms of the bundle structure and which ones require a connexion or a local section or a metric or a complex structure or a Minkowski background or God knows what else for their definition, and the presence of the word “heuristically” in the abstract is a major turn-off.

I could tolerate all that in a model that reproduces real live phenomenology, but I refuse to participate in any universe whose fundamental dynamics involve the kind of hocus-pocus which permeates this paper – including “Chevalley quantization”, which seems to be nothing but replacing a symbol that has a well justified commutation property with one that doesn’t. Given that there is so much that I don’t yet understand about the calculus of variations on objects whose geometrical significance I can grasp, I am going to have to leave the work of “Cliffordization” to someone else for the present.

Which is not to say that I’m not still all ears for a Clifford-Frölicher-Nijenhuis construction that I have a prayer of understanding.

Cheers,

– Michael

Garrett, Michael, Tony,

I’m loathe to discourage people from discussing things related to BRST, it’s exactly the sort of thing I think physics needs more of. But I think you’d be better off discussing this privately, for one thing you could exchange properly TeXed documents…

Peter,

No worries. I sent my little notation crib sheet (basically a drastically condensed summary of parts of G&S with extensions in the Frölicher-Nijenhuis direction) over to Garrett already, and we’ll see how far we can get privately on BRST-ish lines of reasoning. If anyone else is interested in playing with these ideas, write at m.k.edwards at google’s webmail.

Perhaps we’ll hare off into the direction of perturbative cohomology. Applying BRST quantization to the Hodge theory of elliptic complexes sounds like fun. Some elliptic complexes, including the de Rham complex, can be defined without reference to a metric. However, you also need an elliptic operator to select a unique harmonic representative of each cohomology class, and its definition involves an arbitrary choice of metric. So the theory lives on a GL^+_n principal bundle; varying this metric to recover diffeomorphism invariance requires a BRST counterterm.

Anyone know offhand what happens to Atiyah-Singer when the Laplacian is actually a hyperbolic operator?

Cheers,

– Michael

I should have known that Ed Witten was there first:

http://projecteuclid.org/Dienst/UI/1.0/Summarize/euclid.cmp/1104161738

But he doesn’t seem to have known at the time quite how to formally define “instanton moduli space” as the quotient space of cohomology classes and replace the singular Green’s function of the elliptic operator with a metric-fixing term and its ghost counterterm. If I were a real mathematical physicist I would try to pick up where Witten left off.

Cheers,

– Michael

Amazing. I start reading Witten’s paper, I see a lot of index slinging and heuristics and wildcat guesses, I start to question his reputation for mathematical brilliance – and then he proposes an elegant resolution of the problem posed for Hodge theory by working in the Minkowski signature component of the frame bundle: relate two inner products on the _Hilbert_ space, one positive definite and one Lorentz invariant, via the time reversal operation.

I won’t clutter up Peter’s blog with the implications but they’re quite enlightening, as is the central assertion of the paper: a generally covariant quantum field theory is one in which the stress tensor is a BRST commutator. In fiber bundle language, such a theory lives on the unreduced GL^+_n frame bundle of an orientable manifold; the infinitesimal form of general covariance states that the Lie derivative of the Lagrangian with respect to any vertical vector field on the frame bundle is zero. That simply means that dL is a horizontal form on the frame bundle; i. e., the restriction of dL to the vertical ideal (aka the BRST commutator of L) is zero.

So even a totally heuristic approach to writing down a Lagrangian candidate will succeed in producing a “generally covariant” (under infinitesimal changes of metric, not general diffeomorphisms) theory, as long as its energy-momentum tensor (the variation of L with respect to a change of metric, calculated the hard way) coincides with the BRST commutator of some element \lambda of the exterior algebra on the fiber bundle. Although its index-slinging formula may be totally unrecognizable, \lambda will differ from L only by a BRST-closed form in \Lambda, i. e., one whose exterior derivative is a horizontal form.

That brings me back on topic for Peter’s blog. A sufficiently brilliant mind wandering in the string theory wilderness for forty years is likely to discover that some candidate manifolds and Lagrangians are less intractable than others. This may result in isolated flashes of insight like Witten’s that unleash a flood of interesting mathematics. The problem is not string theory but the string theory monoculture in which the majority of sufficiently brilliant minds are all wandering in the same wilderness. My mind is not in that class, so I selfishly wish there were more people of Witten’s caliber wandering in my bundle-theoretic-topology wilderness, in hope that one of them will lead me out someday.

Cheers,

– Michael

Oh, now that’s a neat way of looking at it. QFT and GR are the same theory of general covariance, just written in “Dirac-Feynman” gauge (gauge-fixing operator d+\delta) and “Einstein-Hilbert” gauge (gauge-fixing term R). I bet that someone sufficiently clever could parameterize this to interpolate between them, the same way that in QED one can interpolate between Landau gauge and ‘t Hooft-Feynman gauge using the \xi parameter whose name I can’t remember.

The FQXi grant awardee pages now have links to technical abstracts, which are much more revealing.

Having commented here on the Frölicher-Nijenhuis calculus, I thought I would point out that there’s a good exposition available online. The PDF of a book-length follow-on to Kriegl and Michor’s work (as exemplified by the Cap et al. paper), published by Springer as “Natural operations in differential geometry”, is at http://www.emis.de/monographs/KSM/. It does a very thorough job on the Frölicher-Nijenhuis bracket and friends, including applications on gauge bundles.

The angle from which the authors approach Lagrangians is different from Schücker’s, and they do not draw his connection between the BRST operator and the restriction to the vertical ideal of the exterior derivative on the bundle. Consequently, they do not seem to have applied the Frölicher-Nijenhuis calculus to the space of local-polynomial-valued alternating forms.

Extending BRST in this direction is proving quite enlightening. The BRST operator seems to be more closely related to Lie derivation with respect to the (connexion-dependent) vertical projection than to the (connexion-independent) bundle exterior derivative. In a quite rigorous sense it is complementary to the “exterior covariant derivative”, i. e., Lie derivation with respect to the horizontal projection on vect[P]. This makes it very useful for expressing variational principles on principal bundles, since the generalized Stokes theorem applies properly to the bundle exterior derivative, not the exterior covariant derivative.

Cheers,

– Michael