CERN: The View From Inside

Tommaso Dorigo has a new post up on Information control from CERN, where he discusses a Physics World interview by Matthew Chalmers of the head of communications at CERN, James Gillies.

Gillies addresses what CERN sees as a problem: information coming out first in blogs rather than from the CERN director through official press releases. One aspect of this is the release of information about experimental results, and Tommaso discusses this question on his blog. Unfortunately, I think CERN and the LHC are still quite a ways away from having any experimental results that need to be protected. For the rest of this year, the LHC will be getting a lot of attention from high energy physicists, but what they will be interested in is the question of how the machine is progressing towards the goal of colliding beams at a useful luminosity. For most of the history of the project, CERN’s information policy was remarkably open: the slides from presentations made before the technical committees guiding the project were posted in locations that, while not advertised, were easy to find and did not require a password to access. Anyone with a serious interest could follow along and get first-hand technically accurate information about what was happening.

Things changed rather dramatically after the accident last September 19th. Publicly accessible logbooks were edited to remove information, and public access to the websites of the technical committees was shut off:

Who ordered links to photos and some presentations to be password protected after they appeared on blogs?

[Aymar] wanted the CERN community to receive the news from him before it was made more widely available, so access to slides was temporarily restricted. People just hadn’t realized how much in the spotlight we are now.

Gillies doesn’t address the issue of why these websites have now been restricted, a policy that appears to be permanent and go beyond a “temporary” restriction. According to Chalmers:

CERN’s new director general [Rolf-Dieter Heuer] told staff on 12 January, that from now on people would hear about events first from him, not the press.

This kind of tight control of information about what is happening at the LHC seems to me to be a misguided policy. The best and most timely source of information for CERN staff about the LHC should be first-hand information from the engineers and physicists working on the project, not whatever has made its way up the chain of command and then been laundered for public consumption. Shutting off access by physicists to accurate technical information and making the DG the only source of news about what has happened at the LHC is likely to just encourage unchecked rumor.

Next week at Chamonix there will be an LHC Performance Workshop, and the slides are supposed to be publicly available here as the presenters post them over the next few days. These slides should give an accurate picture of where the project is and what a realistic proposed schedule for the rest of the year would look like. According to the Physics World interview, CERN’s plan is that “a realistic schedule will be announced” after the Chamonix workshop. Of course, by then, many people will have already have a good idea about what this schedule will be, that is, if the slides are not password-protected….

Posted in Experimental HEP News | 8 Comments

Lots of Unrelated Topics

I’m trying to finish writing up something about equivariant cohomology for the BRST project, slowed down by realizing there was something interesting about this that I didn’t understand. Soon that should be sorted out….

In the meantime, here are various other things that might be of interest:

The El Naschie/Elsevier saga continues, latest here.

There’s a new book out from Cambridge University Press entitled On Space and Time, which has chapters from different authors stretching from solid physics to theology, with lots of quantum gravity in between. The editor, Shahn Majid, is blogging here, on a site run by Cambridge.

Evidence for time travel has appeared in the British newspaper The Independent, which recently published an editorial by Mike Duff about string theory that appears to have come through a worm-hole connected to about 13 years ago.

There really are good reasons that theorists who insist on devoting their lives to absurdly speculative models of extra dimensions which have nothing to recommend them other than not being obviously inconsistent should stop promoting these things in the press. One of these reasons is that doing this tends to lead to articles like this one on Fox News.

British theorists at Durham are getting some new funding.

American scientists are lobbying for their piece of the stimulus pie that should be cooked and ready to serve within the next couple weeks. An editorial by David Gross and Eric Kandel is here, a letter from 49 Nobelists here. The latest news indicates that the NSF and DOE are still in line for massive short-term budget increases.

In France, President Sarkozy argued in a speech that French scientific research needs to be reformed, with the economic crisis that originated in the U.S. a good opportunity for the French to modernize and do things more the way they are done over here. Many French scientists are reacting with “shame and anger”, and planning on joining a general one-day strike this Thursday. I know little about the problems and virtues of the French research system, but perhaps scientists there should tell Sarkozy it’s a deal if he is willing to put up the sorts of cash the current U.S. administration is discussing.

There have been a few physics arXiv preprints that seemed worth a mention recently, although all of them have been discussed extensively by Lubos, who seems to be saner these days:

  • Smolin and Ellis argue here that if you take the landscape seriously you end up predicting a negative cosmological constant, falsifying the idea. This is along the same lines as other such wrong predictions (e.g. proton decay), and since their existence hasn’t slowed down the spread of landscape ideology, I doubt one more will do the trick.
  • Several authors here find 10668 as a lower bound when calculating the number of possible vacua in a certain class that might give the standard model at low energies. This is high enough to make getting any predictions (other than the wrong ones…) impossible, but along the same lines as previous estimates which didn’t slow down the landscapeologists. No reason to think this one will either.
  • Petr Horava has an interesting proposal for a new candidate sort of quantum gravity here, one with Lorentz breaking at short distances. I don’t know if this is any more testable than other such proposals. Even though his proposal has absolutely nothing to do with string theory, it’s rather amusing that the author finds it necessary to somehow invoke string theory:

    Given this richness of string theory, it might even be logical to adopt the perspective in which string theory is not a candidate for a unique theory of the universe, but represents instead a natural extension and logical completion of quantum field theory. In this picture, string theory would be viewed – just as quantum field theory – as a powerful technological framework, and not as a single theory.
    If string theory is such an apparently vast structure, it seems natural to ask whether quantum gravitational phenomena in 3 + 1 spacetime dimensions can be studied in a self-contained manner in a “smaller” framework. A useful example of such a phenomenon is given by Yang-Mills gauge theories in 3 + 1 dimensions. While string theory is clearly a powerful technique for studying properties of Yang-Mills theories, their embedding into string theory is not required for their completeness: In 3 + 1 dimensions, they are UV complete in the framework of quantum field theory.
    In analogy with Yang-Mills, we are motivated to look for a “small” theory of quantum gravity in 3+1 dimensions, decoupled from strings.

    So, the idea seems to be that now string theory is a “logical completion” of qft, although not needed to describe any of the forces we know about.

  • For two new survey talks at UCSB by Edward Frenkel about geometric Langlands, see here. He also gave an interesting talk there as part of the KITP string theory program, on recent work (summarized here) that has relations to both geometric Langlands and to the pure spinor formalism.

    Update: One more. The Chronicle of Higher Education has an article on SCOAP3, the plan to make the entire high physics literature open access by coming up with $14 million/year to pay off the publishers. I don’t really see this. The idea seems to be that the money is needed to get peer review, and the size of the literature is about 10,000 papers/year. So, cutting out the publishers, referees could be paid $1,400/paper to do peer review. This might dramatically increase the quality of refereeing, or at least the take-home pay of many physicists. Some quotes:

    But Mr. Mele says journals still play a crucial role in the professional life of scientists, even though readership has declined. “We do not buy journals to read them, we buy journals to support them,” he said. “They do something crucial, which is peer review.”

    Without journals, he asks, how would colleges evaluate the work of scientists to know whom to hire or whom to promote? And how would other scientists know which of the thousands of preprints contain the most important findings?

    “What we are really paying for here is for a service of peer review,” he said.

    but here’s the problem, at least in the U.S….

    The librarians praised the goals of the project, but some asked whether it was sustainable. After all, if the journals make their contents free online, why should college libraries use their shrinking resources to pay for them?

    Some librarians at public institutions say they cannot participate even if they want to. “Most states require that public funds allocated for purchasing have to be used to actually purchase something,” said Dennis Dillon, associate director for research services at the University of Texas at Austin. That is certainly the case in Texas, he said. “They can’t be used to pay for something that everyone already has for free.”

    Update: For more LHC-related hysteria generated by publicity-hungry academics, see this. It originates with a group at the Institute for the Future of Humanity, last seen promoting the idea that we live in a simulation.

    Posted in Uncategorized | 14 Comments

    The Worst Jobs in Science: Theoretical Physicist

    After the recent news that being a mathematician is the best job in the US, next month’s Popular Science magazine has come out with a list of the worst jobs, not overall, but in the sciences. “Theoretical Physicist” makes the list, right in between “Monkey-Sex Observer” and “Vermin Handler”. Here’s the text about this:

    For much of the past century, physics was an exciting, wide-ranging exploration. But to be a theoretical physicist today, you pretty much have to stake your career on one incredibly popular but pretty much unprovable notion: string theory. Since the idea that the universe is composed of small vibrating “strings” gained a following in the 1970s, the theory, which in some forms posits 10 dimensions and seeks a unifying “supersymmetry,” has captured the theoretical-physics community in the U.S. The easiest way to earn an appointment is to dive head-first into a branch of string theory, which dominates the top programs at Princeton, MIT and other influential institutions. The problem is, we simply have no idea if we’re on the right track, because the theory isn’t verifiable.

    Lee Smolin, a physicist at the Perimeter Institute for Theoretical Physics in Waterloo, Canada, who investigates quantum gravity and string theory, believes that the physics monoculture is stifling. “Science has become too risk-averse, and its progress is being hurt as a result,” he says. When CERN’s Large Hadron Collider restarts later this year, however, it could end the waiting, helping to confirm parts of string theory — or dash it altogether. If supersymmetric particles called sparticles are bashed into existence: yay! But if the W boson particle does not react as hoped, that damages a central pillar of the theory. Across the U.S., whole careers are boiling down to the chance that a big box comes up with something.

    It’s true that, for string theorists, a lot is hanging on the question of whether sparticles are found at the LHC. If none are seen, I suspect that will pretty conclusively finish off in most theorist’s minds the idea that string theory unification can be connected in any way with observations. The business about string theory and W-bosons is utter nonsense, presumably coming from this.

    As mentioned here repeatedly, claims that hiring in particle theory is dominated by string theory are behind the times. String theorists are now yesterday’s fad, with terrible job prospects if they don’t have a permanent position. Today’s fads are LHC phenomenology and cosmology (news from the rumor mill about two new jobs is that UCSB wants “candidates with interests in phenomenological aspects of particle physics and related areas of astrophysics and cosmology”, Rutgers wants “a focus on LHC physics, broadly conceived.”) String theory is on its way out in American universities it seems, but the long-standing pattern of fad-driven hiring isn’t. Which is one thing that makes the idea of trying for a career in theoretical physics these days about as appealing to many smart young people as the idea of going into the vermin handling business…

    Posted in Uncategorized | 54 Comments

    Quick Posting, European Edition

    It’s semester break and I’m in Paris this week, but I have a few moments to post on some topics that may be of interest:

    People here in Europe seem quite normal, but Lubos has just won an award for Best European Blog.

    Some details of the proposed new US stimulus package have just been released. Science is one of the main beneficiaries, with the DOE getting:

    $1.9 billion for basic research into the physical sciences including high-energy physics, nuclear physics, and fusion energy sciences and improvements to DOE laboratories and scientific facilities.

    This should dramatically change the situation at Fermilab, which has been operating so far this year under a low continuing resolution level of funding. For the immediate future, the funding situation in the US for HEP and science in general looks bright, although this presumably will come to a screeching halt whenever the federal government stops financing everything by printing a trillion dollars a year.

    Travis Brooks of SPIRES has produced Top Cites lists for 2008. He has a blog posting about this here, and the list of most heavily cited papers during 2008 is here. For the first time in many years, a new hep-th fad is having visible impact, with Bagger-Lambert at number 37, and three other papers on the same topic in the top 50.

    Over at the Edge web-site, there’s an interview with Frank Wilczek.

    Posted in Uncategorized | 9 Comments

    Science Channel Inks Deal With Physicist Michio Kaku

    Fresh from his leading role in the History Channel’s Parallel Universes (if you missed it on TV, the DVD is available here), according to a press release today, Michio Kaku will now be appearing regularly on Discovery’s Science Channel:

    The Science Channel has signed a multi-year agreement with theoretical physicist Dr. Michio Kaku. As part of the deal, the Discovery-owned network will produce a 10-part series based on his New York Times bestselling book Physics of the Impossible and has exclusive television rights to Kaku’s other works for adaptation in series and specials.

    In addition, Kaku will become the host of “Sci Q Sundays,” the Sunday programming block that explores scientific news and topics.

    The show has a web-site here, including instructions on how to build a time-machine.

    Posted in Uncategorized | 28 Comments

    Dyson on Birds and Frogs

    Next month’s Notices of the AMS has an essay by Freeman Dyson entitled Frogs and Birds, which was written for his planned Einstein Public Lecture. In it, he divides mathematicians up into two species: birds, who “fly high in the air and survey broad vistas” (i.e. seek abstraction, unification and generalization), and frogs, who “see only the flowers that grow nearby” (i.e. study the details of specific examples).

    Dyson himself is resolutely a frog, but writes that “many of my best friends are birds”, and argues that both birds and frogs are needed to do justice to the breadth and depth of the subject of mathematics. Frog that he is, his essay covers a variety of quite different special topics that have drawn his attention, linked together only weakly by the bird/frog theme. These include a discussion of the roles of complex numbers and linearity in quantum mechanics, a proposed idea about how to attack the Riemann hypothesis (try and enumerate 1d-quasicrystals, since the zeros of the zeta function have this structure), and a collection of profiles and anecdotes about various mathematicians and physicists (Besicovitch, Weyl, Yang, Manin, von Neumann).

    Personally I suppose I fit well into Dyson’s bird category, but among the best mathematicians that I know, the frog/bird distinction is often unclear. Many of them make their reputation by proving rather abstract and general theorems, but these proofs are often the result of a huge amount of detailed investigation of examples. I agree with Dyson that both points of view are needed, and see the most successful cases of progress in mathematics coming from mathematicians who avoid the temptation to fly too high into arid abstraction, or sink too deep into irrelevant detail.

    Dyson includes a long section on string theory, which I’ll include here:

    I would like to say a few words about string theory. Few words, because I know very little about string theory. I never took the trouble to learn the subject or to work on it myself. But when I am at home at the Institute for Advanced Study in Princeton, I am surrounded by string theorists, and I sometimes listen to their conversations. Occasionally I understand a little of what they are saying. Three things are clear. First, what they are doing is first-rate mathematics. The leading pure mathematicians, people like Michael Atiyah and Isadore Singer, love it. It has opened up a whole new branch of mathematics, with new ideas and new problems. Most remarkably, it gave the mathematicians new methods to solve old problems that were previously unsolvable. Second, the string theorists think of themselves as physicists rather than mathematicians. They believe that their theory describes something real in the physical world. And third, there is not yet any proof that the theory is relevant to physics. The theory is not yet testable by experiment. The theory remains in a world of its own, detached from the rest of physics. String theorists make strenuous efforts to deduce consequences of the theory that might be testable in the real world, so far without success.

    My colleagues Ed Witten and Juan Maldacena and others who created string theory are birds, flying high and seeing grand visions of distant ranges of mountains. The thousands of humbler practitioners of string theory in universities around the world are frogs, exploring fine details of the mathematical structures that birds first saw on the horizon. My anxieties about string theory are sociological rather than scientific. It is a glorious thing to be one of the first thousand string theorists, discovering new connections and pioneering new methods. It is not so glorious to be one of the second thousand or one of the tenth thousand. There are now about ten thousand string theorists scattered around the world. This is a dangerous situation for the tenth thousand and perhaps also for the second thousand. It may happen unpredictably that the fashion changes and string theory becomes unfashionable. Then it could happen that nine thousand string theorists lose their jobs. They have been trained in a narrow specialty, and they may be unemployable in other fields of science.

    Why are so many young people attracted to string theory? The attraction is partly intellectual. String theory is daring and mathematically elegant. But the attraction is also sociological. String theory is attractive because it offers jobs. And why are so many jobs offered in string theory? Because string theory is cheap. If you are the chairperson of a physics department in a remote place without much money, you cannot afford to build a modern laboratory to do experimental physics, but you can afford to hire a couple of string theorists. So you offer a couple of jobs in string theory, and you have a modern physics department. The temptations are strong for the chairperson to offer such jobs and for the young people to accept them. This is a hazardous situation for the young people and also for the future of science. I am not saying that we should discourage young people from working in string theory if they find it exciting. I am saying that we should offer them alternatives, so that they are not pushed into string theory by economic necessity.

    Finally, I give you my own guess for the future of string theory. My guess is probably wrong. I have no illusion that I can predict the future. I tell you my guess, just to give you something to think about. I consider it unlikely that string theory will turn out to be either totally successful or totally useless. By totally successful I mean that it is a complete theory of physics, explaining all the details of particles and their interactions. By totally useless I mean that it remains a beautiful piece of pure mathematics. My guess is that string theory will end somewhere between complete success and failure. I guess that it will be like the theory of Lie groups, which Sophus Lie created in the nineteenth century as a mathematical framework for classical physics. So long as physics remained classical, Lie groups remained a failure. They were a solution looking for a problem. But then, fifty years later, the quantum revolution transformed physics, and Lie algebras found their proper place. They became the key to understanding the central role of symmetries in the quantum world. I expect that fifty or a hundred years from now another revolution in physics will happen, introducing new concepts of which we now have no inkling, and the new concepts will give string theory a new meaning. After that, string theory will suddenly find its proper place in the universe, making testable statements about the real world. I warn you that this guess about the future is probably wrong. It has the virtue of being falsifiable, which according to Karl Popper is the hallmark of a scientific statement. It may be demolished tomorrow by some discovery coming out of the Large Hadron Collider in Geneva.

    I don’t know where Dyson got the estimate of ten thousand string theorists; my own estimate would be more like one to two thousand (with the number strongly dependent on how you decide who is a “string theorist”). The large yearly Strings200X conferences that bring together a sizable fraction of active string theory community tend to draw roughly 500 people.

    The Princeton-centric assumption that there are lots of string theory jobs embedded in his question “And why are so many jobs offered in string theory?” is quite problematic, as any young string theorist on the job market could explain to him. There actually aren’t a lot of string theory jobs out there, and a lot of Ph.D.s in the subject being produced, leading to a lot of ex-string theorists now working in the financial industry and elsewhere. These days, if you are going to choose your field based on where the jobs are, you become an LHC phenomenologist or a cosmologist. If you want to be a string theorist, you better be a string phenomenologist or a string cosmologist. Also rather unrealistic is Dyson’s “it could happen that nine thousand string theorists lose their jobs”, due to tenure in the academic system. Even if a consensus develops over the next few years that string theory was all a big mistake, twenty years from now there will still be a cadre of (older) people working in the field.

    Dyson’s idea, that 50-100 years from now, a new revolution in physics will show how string theory fits in may be right. It also may be that this has already happened, as much of the field has moved into the study of gauge-string dualities, where string theory provides a useful approximation for strongly coupled systems, and the idea that it unifies particle physics is falling by the wayside.

    Posted in Uncategorized | 14 Comments

    Best Job in the US

    According to the latest JobsRated listing released today, the best job in the US is that of mathematician. Pay is good, stress is low, and you don’t have to get your hands dirty, but can sit in front of a computer monitor all day. Nice work if you can get it. The job of physicist is significantly less desirable: down at number 13, not quite as good as working as a philosopher (number 12), but a bit better than being a parole officer (number 14).

    Posted in Uncategorized | 17 Comments

    New Year Links

    There’s a conference going on in Jerusalem now on the topic of Particle Physics in the Age of the LHC. Some slides and other talk materials are here, video may start appearing here. Not clear when the “Age of the LHC” is; unfortunately we’re still a ways away from first collisions, even farther from new physics. Next year, starting in May, the KITP will be running a program on The First Year of the LHC, which may also be jumping the gun a bit, at least to the extent that the topic is LHC physics results. Last year’s LHC program, Physics of the Large Hadron Collider, has a web-site that still begins with the counter-factual “The Large Hadron Collider (LHC) will begin operation by the end of 2007.”

    Also next year, the KITP will be running another supposedly LHC-related program, entitled Strings at the LHC and in the Early Universe. I wonder what the KITP director thinks of this, since he’s on record as thinking it unlikely that the LHC will have anything to say about string theory. A much less dubious KITP program about string theory is the one starting today, with the title Fundamental Aspects of String Theory. This program focuses on the current lack of understanding of what string theory really is:

    Over the last decade, string theory has seen important conceptual and technical advances on a host of long-standing problems involving non-pertur-bative and strongly-coupled physics. However, the fundamental ingredients of superstring theory and M-theory are still not well understood, and this five month program will be directed at these open questions.

    The first week will be devoted to introductory talks about string field theory and the pure spinor formalism, two quite different attempts to give a new and different formulation of string theory.

    Also starting today is the big annual meeting of the AMS, held this year in Washington, DC. One of the important features of this meeting is that many institutions, especially smaller ones, do their initial interviews for next year’s jobs at the meeting. This coming hiring season promises to be an exceptionally brutal one for job candidates, with financial problems leading to freezes and reduced hiring at many places. One resource for young mathematicians on the job market is the web-site of the Young Mathematicians Network.

    I wrote about Witten’s talk on quantum Yang-Mills theory at the Yau birthday conference here. A write-up of the talk is now available as a preprint here.

    There’s a new book coming out this month that I’m looking forward to reading, Graham Farmelo’s biography of Dirac, entitled The Strangest Man: The Hidden Life of Paul Dirac, Quantum Genius. Nature Physics has a review here.

    This week’s Science Saturday featured John Horgan and George Johnson discussing the state of science journalism and what it has to do with blogging. As science journalists, they take exception to the point of view common among scientists that their job is just to try and accurately transmit to the public the claims being made by scientists.

    Posted in Uncategorized | 8 Comments

    Dis-moi qui tu aimes (je te dirai qui tu hais)

    A colleague has very helpfully provided me with a copy of the murder mystery set at the IHES that I wrote about recently here, and I’ve just finished reading it. Since I’m not much of an afficionado of this genre of fiction, I can’t really evaluate how good a murder mystery it is. But as a memoir of the IHES during the 1980s, it is excellent. A claim at the beginning of the book that “any resemblance to real persons is just coincidence” seems to be one of the few things in it (besides the murder) that is fiction. As far as I can tell, the descriptions of all characters correspond precisely to someone at the IHES during that period, with only the names changed. I’m guessing that all or most of the anecdotes about these characters also correspond to reality.

    It’s a roman a clef, so here’s the key for the major characters:

    Andre Grusin = Leon Motchane
    Henrik Dekker = Nicolaas Kuiper
    Charles Bouleaux = Marcel Berger
    Antoine Fleuret = Alain Connes
    Jacob Zuram = Barry Mazur
    Boris Grekov = Mikhael Gromov
    Jacques Chevalier = Pierre Deligne

    Among the minor characters, I suspect

    Joe Bub = Dennis Sullivan
    David Amir = Ofer Gabber
    Albert Toudy = Adrien Douady

    I don’t think I’ll be giving away too much of the plot to mention that, since the novel was written nearly twenty years ago, back when string theory was a hot topic, one of the plot twists involves string theory. There’s a discovery that “superstring theory is renormalizable and predicts that gluonic interactions are colorless”.

    Posted in Book Reviews | 7 Comments

    Notes on BRST IX: Clifford Algebras and Lie Algebras

    Note: I’ve started putting together the material from these postings into a proper document, available here, which will be getting updated as time goes on. I’ll be making changes and additions to the text there, not on the blog postings. For most purposes, that will be what people interested in this subject will want to take a look at.

    When a Lie group with Lie algebra [tex]\mathfrak g[/tex] acts on a manifold [tex]M[/tex], one gets two sorts of actions of [tex]\mathfrak g[/tex] on the differential forms [tex]\Omega^*(M[/tex]). For each [tex]X\in \mathfrak g [/tex] one has operators:

  • [tex]\mathcal L}_X: \Omega^k(M)\rightarrow\Omega^k(M),[/tex] the Lie derivative along the vector field on [tex]M[/tex] corresponding to [tex]X[/tex]
  • and

  • [tex]i_X:\Omega^k(M)\rightarrow\Omega^{k-1}(M)[/tex], contraction by the vector field on [tex]M[/tex] corresponding to [tex]X[/tex]
  • These operators satisfy the relation

    [tex]di_X+i_Xd={\mathcal L}_X[/tex]

    where [tex]d[/tex] is the de Rham differential [tex]d:\Omega^k(M)\rightarrow \Omega^{k+1}(M)[/tex], and the operators [tex]d, i_X, \mathcal L_X[/tex] are (super)-derivations. In general, an algebra carrying an action by operators satisfying the same relations satisfied by [tex]d, i_X, \mathcal L_X[/tex] will be called a [tex]\mathfrak g[/tex]-differential algebra. It will turn out that the Clifford algebra [tex]Cliff(\mathfrak g)[/tex] of a semi-simple Lie algebra [tex]\mathfrak g[/tex] carries not just the Clifford algebra structure, but the additional structure of a [tex]\mathfrak g[/tex]-differential algebra, in this case with [tex]\mathbf Z_2[/tex], not [tex]\mathbf Z[/tex] grading.

    Note that in this section the commutator symbol will be the supercommutator in the Clifford algebra (commutator or anti-commutator, depending on the [tex]\mathbf Z_2[/tex] grading). When the Lie bracket is needed, it will be denoted [tex][\cdot,\cdot]_{\mathfrak g}[/tex].

    To get a [tex]\mathfrak g[/tex]-differential algebra on [tex]Cliff(\mathfrak g)[/tex] we need to construct super-derivations [tex]i_X^{Cl}[/tex], [tex]{\mathcal L}_X^{Cl}[/tex], and [tex]d^{Cl}[/tex] satisfying the appropriate relations. For the first of these we don’t need the fact that this is the Clifford algebra of a Lie algebra, and can just define

    [tex]i_X^{Cl}(\cdot)=[-\frac{1}{2}X,\cdot][/tex]

    For [tex]{\mathcal L}_X^{Cl}[/tex], we need to use the fact that since the adjoint representation preserves the inner product, it gives a homomorphism

    [tex]\widetilde{ad}:\mathfrak g \rightarrow \mathfrak{spin}(\mathfrak g)[/tex]

    where [tex]\mathfrak{spin}(\mathfrak g)[/tex] is the Lie algebra of the group [tex]Spin(\mathfrak g)[/tex] (the spin group for the inner product space [tex]\mathfrak g[/tex]), which can be identified with quadratic elements of [tex]Cliff(\mathfrak g)[/tex], taking the commutator as Lie bracket. Explicitly, if [tex]X_a[/tex] is a basis of [tex]\mathfrak g[/tex], [tex]X_a^* [/tex] the dual basis, then

    [tex]\widetilde{ad}(X)=\frac{1}{4}\sum_a X_a^*[X,X_a]_{\mathfrak g}[/tex]

    and we get operators acting on [tex]Cliff(\mathfrak g)[/tex]

    [tex]{\mathcal L}_X^{Cl}(\cdot)=[\widetilde{ad}(X),\cdot][/tex]

    Remarkably, an appropriate [tex]d^{Cl}[/tex] can be constructed using a cubic element of [tex]Cliff(\mathfrak g)[/tex]. Let

    [tex]\gamma= \frac{1}{24}\sum_{a,b}X^*_aX^*_b[X_a,X_b]_{\mathfrak g}[/tex]

    then

    [tex]d^{Cl}(\cdot)=[\gamma, \cdot][/tex]

    [tex]d^{Cl}\circ d^{Cl}=0[/tex] since [tex]\gamma^2[/tex] is a scalar which can be computed to be [tex]-\frac{1}{48}tr\Omega_{\mathfrak g}[/tex], where [tex]\Omega_{\mathfrak g}[/tex] is the Casimir operator in the adjoint representation.

    The above constructions give [tex]Cliff(\mathfrak g)[/tex] the structure of a filtered [tex]\mathfrak g[/tex]-differential algebra, with associated graded algebra [tex]\Lambda^*(\mathfrak g)[/tex]. This gives [tex]\Lambda^*(\mathfrak g)[/tex] the structure of a [tex]\mathfrak g[/tex]-differential algebra, with operators [tex]i_X, \mathcal L_X, d[/tex]. The cohomology of this differential algebra is just the Lie algebra cohomology [tex]H^*(\mathfrak g, \mathbf C)[/tex].

    [tex]Cliff(\mathfrak g)[/tex] can be thought of as an algebra of operators corresponding to the quantization of an anti-commuting phase space [tex]\mathfrak g[/tex]. Classical observables are anti-commuting functions, elements of [tex]\Lambda^*(\mathfrak g^*)[/tex]. Corresponding to [tex]i_X, \mathcal L_X, d[/tex] one has both elements of [tex]\Lambda^*(\mathfrak g^*)[/tex] and their quantizations, the operators in [tex]Cliff(\mathfrak g)[/tex] constructed above.

    For more details about the above, see the following references

  • A. Alekseev and E. Meinrenken, The non-commutative Weil algebra, Invent. Math 139, 135-172 (2000), or arXiv:math/9903052
  • E. Meinrenken, Clifford algebras and Lie groups, 2005 Toronto lecture notes
  • G. Landweber, Multiplets of representations and Kostant’s Dirac operator for equal rank loop groups, Duke Mathematical Journal 110, 121-160 (2001), or arXiv:math/0005057
  • B. Kostant and S. Sternberg, Symplectic reduction , BRS cohomology and infinite-dimensional Clifford algebras, Ann. Physics 176, 49-113 (1987)
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