Notes on BRST I: Representation Theory and Quantum Mechanics

Posted on November 5, 2008 by woit

This is the first posting of a planned series that will discuss the BRST method for handling gauge symmetries and related mathematical topics. I’ve been writing a more formal paper about this, but given the substantial amount of not-well-known background material involved, it seems like a good idea to first put together a few expository accounts of some of these topics. And what better place for this than a blog?

Many readers who are used to my usual attempts to be newsworthy and entertaining, often by scandal-mongering or stirring up trouble of one kind or another, may be very disappointed in these posts. They’re quite technical, hard to follow, and of low-to-negative entertainment value. You probably would do best to skip them and wait for more of the usual fare, which should continue to appear from time to time.

Quantum Mechanics and Representation Theory

A quantum mechanical physical system is given by the following mathematical structure:

  • A Hilbert space [tex]\mathcal H[/tex], the “space of states”. A state of the physical system is determined by a vector [tex]|\psi\rangle\in \mathcal H[/tex], with unit norm (i.e. [tex]||\psi||^2=\langle \psi|\psi\rangle=1[/tex]).
  • An algebra [tex]\mathcal O[/tex] that acts on [tex]\mathcal H[/tex]. To each physical observable corresponds a self-adjoint operator [tex]O\in\mathcal O[/tex]. Eigenvectors in [tex]\mathcal H[/tex] of this operator correspond to states where the observable has a well-defined value, which is the eigenvalue.

    • If a physical system has a symmetry group [tex]G[/tex], there is a unitary representation [tex](\Pi, \mathcal H)[/tex] of [tex]G[/tex] on [tex]\mathcal H[/tex]. This means that for each [tex]g\in G[/tex] we get a unitary operator [tex]\Pi(g)[/tex] satisfying

    [tex]\begin{displaymath}\Pi(g_3)=\Pi(g_2)\Pi(g_1)\ \text{if}\ g_3=g_1g_2\end{displaymath}[/tex]

    i.e. the map [tex]\Pi[/tex] from group elements to unitary operators is a homomorphism. The [tex]\Pi(g)[/tex] act on [tex]\mathcal O[/tex] by taking an operator [tex]O[/tex] to its conjugate [tex]\Pi(g)O(\Pi(g))^{-1}[/tex].

    When [tex]G[/tex] is a Lie group with Lie algebra [tex](\mathfrak g, [\cdot,\cdot])[/tex], differentiating [tex]\Pi[/tex] gives a unitary representation [tex](\pi, \mathcal H)[/tex] of [tex]\mathfrak g[/tex] on [tex]\mathcal H[/tex]. This means that for each [tex]X\in \mathfrak g[/tex] we get a skew-Hermitian operator [tex]\pi(X)[/tex] on [tex]\mathcal H[/tex], satisfying

    [tex]\pi(X_3)=[\pi(X_1),\pi(X_2)]\ \text{if}\ X_3=[X_1,X_2][/tex]

    i.e. the map [tex]\pi[/tex] taking Lie algebra elements [tex]X[/tex] (with the Lie bracket in [tex]\mathfrak g[/tex]) to skew-Hermitian operators (with commutator of operators) is a homomorphism. On [tex]\mathcal O[/tex], [tex]\mathfrak g[/tex] acts by the differential of the conjugation action of [tex]G[/tex], this action is just that of taking the commutator with [tex]\pi(X)[/tex].

    The Lie bracket is not associative, but to any Lie algebra [tex]\mathfrak g[/tex], one can construct an associative algebra [tex]U(\mathfrak g)[/tex] called the universal enveloping algebra for [tex]\mathfrak g[/tex]. If one identifies [tex]X\in \mathfrak g[/tex] with left-invariant vector fields on [tex]G[/tex], which are first-order differential operators on functions on [tex]G[/tex], then [tex]U(\mathfrak g)[/tex] is the algebra of left-invariant differential operators on [tex]G[/tex] of all orders, with product the composition of differential operators. A Lie algebra representation is precisely a module over [tex]U(\mathfrak g)[/tex], i.e. a vector space with an action of [tex]U(\mathfrak g)[/tex].

    So, the state space [tex]\mathcal H[/tex] of a quantum system with symmetry group [tex]G[/tex] carries not only a unitary representation of [tex]G[/tex], but also a unitary representation of [tex]\mathfrak g[/tex], or equivalently, an action of the algebra [tex]U(\mathfrak g)[/tex]. [tex]X\in \mathfrak g[/tex] acts by the operator [tex]\pi(X)[/tex]. In this way a representation [tex]\pi[/tex] gives a sub-algebra of the algebra [tex]\mathcal O[/tex] of observables. Most of the important observables that show up in practice come from a symmetry in this way. An interesting philosophical question is whether the quantum system that governs the real world is purely determined by symmetry, i.e. such that ALL its observables come from symmetries in this manner.

    Some Examples

    Much of the structure of common quantum mechanical systems is governed by the fact that they carry space-time symmetries. In our 3-space, 1-time dimensional world, these include:

  • Translations in space: [tex]G=\mathbf R^3, \mathfrak g =\mathbf R^3[/tex], Lie Bracket is trivial.
    For each basis element [tex]e_j\in \mathfrak g[/tex] one gets a momentum operator [tex]\pi(e_j)=iP_j[/tex]
  • Translations in time: [tex]G=\mathbf R, \mathfrak g =\mathbf R[/tex]. If [tex]e_0[/tex] is a basis of [tex]\mathfrak g[/tex], [tex]i\pi(e_0)=H[/tex], the Hamiltonian operator. The fact that this operator generates time-translations is just Schrodinger’s equation.
  • Rotations in 3-space: [tex]G=SO(3)[/tex], or its double cover [tex]G=Spin(3)=SU(2), \mathfrak g = \mathbf R^3[/tex], with bracket given by the vector product. For each basis element [tex]e_j\in \mathfrak g[/tex] one gets an angular momentum operator [tex]\pi(e_j)=iJ_j[/tex]. These operators do not commute, so cannot be simultaneously diagonalized.

    • Another example is the symmetry of phase transformations of the state space [tex]\mathcal H[/tex]. Here [tex]G=U(1), \mathfrak g=R[/tex], and one gets an operator [tex]Q_e[/tex] that can be normalized to have integral eigenvalues.

    This last example also comes in a local version, where we make independent phase transformations at different points in space-time. This is an example of a “gauge symmetry”, and the question of how it gets represented on the space of states is what will lead us into the BRST story. Next posting in the series will be about gauge symmetry, then on to BRST.

    If you want some idea of where this is headed, you can take a look at slides from a colloquium talk I gave recently at the Dartmouth math department. They’re very sketchy, the postings in this series should add some detail.

    Posted in BRST | 63 Comments

    The Circus Begins

    Posted on November 2, 2008 by woit

    Friday’s arXiv posting of the paper by CDF about the multi-muon anomaly they are seeing has already generated three different conjectural explanations of what physics might be responsible for this. Undoubtedly many, many more are on the way.

  • Some members of the CDF collaboration have posted a paper entitled Phenomenological interpretation of the multi-muon events reported by the CDF collaboration. This explains the large numbers of muons with a rather baroque mechanism, conjecturing the production of a 300 GeV heavy particle decaying through a chain of 3 lighter particles, the last of which is supposed to be the long-lived (20 picosecond) one. This interpretation was part of the original draft PRL from last June/July. The CDF collaboration as a whole seems to have decided not to support the draft PRL and this interpretation, instead releasing just a PRD paper that describes the anomaly without trying to interpret its significance. It also seems that only two-thirds of the collaboration put their names on the PRD paper, and the interpretation paper was put out just by a small group. The whole story is somewhat reminiscent of the “Superjet” affair (see Tommaso Dorigo’s multi-part discussion here), which also involved a PRD publication about an anomaly signed by the collaboration, and an interpretation (in terms of squarks) signed by a much smaller group led by Paolo Giromini.
  • An hour or so after the Giromini et. al. paper came in on Friday, a group of string theorists had posted the 40-page Towards Realistic String Vacua on hep-th claiming to explain the CDF results with a class of string vacua:

    We also describe model-independent physical implications of this scenario. These include the masses of anomalous and non-anomalous U(1)’s and the generic existence of a new hyperweak force under which leptons and/or quarks could be charged. We propose that such a gauge boson could be responsible for the ghost muon anomaly recently found at the Tevatron’s CDF detector.

    If the Giromini et. al. explanation invoking 4 new particles is baroque, it’s hard to know what the right word is for the far more complicated constructions that are described in this paper.

  • There’s a new version this evening of the 3-week old Arkani-Hamed/Weiner LHC Signals for a SuperUnified Theory of Dark Matter, in which they claim to have a new signature for supersymmetry, with a large fraction of all SUSY events looking exactly like what CDF described. Oddly enough, the changes to the paper don’t include a mention of the CDF result. This paper also invokes a rather baroque mechanism, involving both the supersymmetric extension of the standard model and a whole new complicated dark sector.

    • This last paper is also supposed to explain the PAMELA data, and papers with other explanations of this are starting to flood hep-ph.

    So far, all the explanations of the anomaly seen by CDF look suspiciously complicated, which may be one reason that many members of CDF are so skeptical about the whole thing that they were unwilling to sign on to the PRD submission. But I’m sure that many more proposals for how to explain the anomaly are being drafted at this very moment, and maybe one of them will be more convincing.

    Update: Over at Tommaso Dorigo’s blog there’s a short posting about Giromini et. al., and an exchange with Nima Arkani-Hamed, who claims to have had no inside knowledge of the CDF “lepton jets” when he wrote his paper with Weiner predicting them. He also explains how the exact mechanism discussed in that paper is unlikely to explain the CDF result since their observed rate is too high for this.

    Update: New Scientist has the story, emphasizing the possible relation to the work of Arkani-Hamed and Weiner:

    So what could it be? As it happens, Weiner and Nima Arkani-Hamed of the Institute for Advanced Study in Princeton, New Jersey, and colleagues have developed a theory of dark matter – the enigmatic stuff thought to make up a large proportion of the universe – to explain recent observations of radiation and anti-particles from the Milky Way.

    Their model posits dark matter particles that interact among themselves by exchanging “force-carrying” particles with a mass of about 1 gigaelectronvolts.
    The CDF muons appear to have come from the decay of a particle with a mass of about 1 GeV. So could they be a signature of dark matter? “We are trying to figure that out,” says Weiner. “But I would be excited by the CDF data regardless.”

    CDF spokesperson Jacobo Konigsberg is quoted as saying:

    we haven’t ruled out a mundane explanation for this, and I want to make that very clear

    Update: Then there’s Slashdot, where the hypothetical CDF particle is advertised as accounting for the Arkani-Hamed et. al. theory of dark matter.

    Update: Another story, at Physics World, which has more from various people at CDF. Again, that Arkani-Hamed/Weiner “predicted a CDF–like signal”, although the problem with the rate being too low is mentioned.

    Also Nature, where one learns:

    Theorists are already coming up with ideas about what might be producing the excess muons. One possibility is that they stem from the decay of a heavier, yet-to-be-discovered particle — perhaps related to dark matter, an unseen material that is believed to make up some 85% of matter in the Universe.

    Another idea from string theory evokes seven-dimensional ‘branes’ — theoretical surfaces that are inhabited by exotic particles manifested as strings. These higher-dimensional branes might be home to force-carrying particles that interact weakly with our three-dimensional world and create a faint, but traceable, signal in the data.

    But Adam Falkowski, a theorist at CERN, Europe’s particle accelerator laboratory near Geneva, Switzerland, says that the explanations need some work, and cautions against attempting to force the data to fit into particular theories.

    Update: More press stories here and here.

    Posted in Experimental HEP News | 24 Comments

    Discovery of a New Particle?

    Posted on October 30, 2008 by woit

    Except for the excitement surrounding first beams in the LHC, particle physics has been an all-too-quiet subject recently. It looks like that may be about to change, with a dramatic new result announced by the CDF experiment this evening, in a preprint entitled Study of multi-muon events produced in p-pbar collisions at sqrt(s)=1.96 TeV.

    The CDF result originates in studies designed to determine the b-bbar cross-section by looking for events where a b-bbar pair is produced, each component of the pair decaying into a muon. The b-quark lifetime is of order a picosecond, so b-quarks travel a millimeter or so before decaying. The tracks from these decays can be reconstructed using the inner silicon detectors surrounding the beam-pipe, which has a radius of 1.5 cm. They can be characterized by their “impact parameter”, the closest distance between the extrapolated track and the primary interaction vertex, in the plane transverse to the beam.

    If one looks at events where the b-quark vertices are directly reconstructed, fitting a secondary vertex, the cross-section for b-bbar production comes out about as expected. On the other hand, if one just tries to identify b-quarks by their semi-leptonic decays, one gets a value for the b-bbar cross-section that is too large by a factor of two. In the second case, presumably there is some background being misidentified as b-bbar production.

    The new result is based on a study of this background using a sample of events containing two muons, varying the tightness of the requirements on observed tracks in the layers of the silicon detector. The background being searched for should appear as the requirements are loosened. It turns out that such events seem to contain an anomalous component with unexpected properties that disagree with those of the known possible sources of background. The number of these anomalous events is large (tens of thousands), so this cannot just be a statistical fluctuation.

    One of the anomalous properties of these events is that they contain tracks with large impact parameters, of order a centimeter rather than the hundreds of microns characteristic of b-quark decays. Fitting this tail by an exponential, one gets what one would expect to see from the decay of a new, unknown particle with a lifetime of about 20 picoseconds. These events have further unusual properties, including an anomalously high number of additional muons in small angular cones about the primary ones.

    The exciting possibility here is that a new, relatively long-lived particle has been observed, one that decays in some way that leads to a lot more muons than one gets from Standard Model states. It should be remembered though that this is an extraordinary claim requiring extraordinary evidence, and the possibility remains that this is some sort of background or detector effect that the CDF physicists have missed. It should also be made clear that this paper is not a claim by CDF to have discovered a new particle, rather it is written up as a description of the anomalies they have found, leaving open the possibility that these come from some standard model processes or detector characteristics that they do not yet understand.

    The overwhelming success of the Standard Model during the past 30 years has meant that essentially all claims from accelerator experiments to see some new, non-SM physics have turned out to be mistaken. As a result, collaborations like CDF are now extremely careful about making such claims and will only do so after the most rigorous possible review. It’s a remarkable event that this one has gotten out, signed off on by the entire collaboration (although from what I understand, people can drop their names from the publication list of a specific paper if they disagree with it, maybe one should check this author list carefully…).

    What would really be convincing would be a confirmation of this from D0, the other Tevatron detector. The D0 collaboration would not only be working with a detector that has somewhat different characteristics, but would also have some motivation to find a problem with the result from their competition. If they also see it, that would be pretty extraordinary evidence. Another sort of extraordinary evidence would be to see evidence for the same kind of new particle in other channels.

    This will undoubtedly unleash a flood of papers from theorists promoting models that extend the Standard Model in ways that would produce something with the observed experimental signature. This is not a signature characteristic of supersymmetry or any of the other known heavily-studied classes of models. If real, as far as I’m aware it’s something genuinely unexpected. Perhaps phenomenology experts can point to some less well-known models with this kind of signature. The only such thing I’m aware of is a very recent paper from three weeks ago by Arkani-Hamed and Weiner, entitled LHC Signals for a SuperUnified Theory of Dark Matter. They discuss a theory of dark matter involving a new hidden gauge symmetry, broken near the GeV scale, saying that this is “motivated directly by striking Data from the PAMELA and ATIC collaborations”. In these models there can be Gev-scale Higgs and gauge particles decaying to an anomalously large number of leptons. They discuss the question of whether the parameters of such models can be adjusted to give large decay lengths, and predict the observation of events that “contain at least two “lepton jets”: collections of n > 2 leptons, with small angular separations and GeV scale invariant masses”, pretty much just what CDF sees . Since the CDF paper undoubtedly has been the topic of intense discussion among the 450 or so physicists in the collaboration for many months now, the most likely explanation for the appearance of a new theory paper a few weeks ago discussing exactly the signatures in question is that news of what’s in the paper got out to some theorists early. Even if this particular result goes away, this gives some indication of what sorts of things are likely to happen once LHC data starts being collected and analyzed.

    The bottom line though is that for the first time in quite a while, there’s some very exciting and potentially revolutionary news in particle physics. It’s coming not out of the LHC, which is still a hope for the future, but from a currently functioning machine which is producing more data every day. If this result holds up, this data contains a wealth of information about some new physics which will likely revolutionize our understanding of elementary particle physics. Particle physics may already have started to move out of its doldrums.

    Update: This evening I came across an unexpected source of information about this, one which might explain why news of this result may have leaked out a while ago. More about this later.

    Update: The results from PAMELA mentioned here, and which are listed as motivation for the Arkani-Hamed/Weiner paper, are now officially out. For discussion of this from someone much better informed than me, see this posting at Resonaances.

    Update: There’s an excellent detailed posting about the paper from CDF’s own Tommaso Dorigo. If you’re interested in understanding exactly what is going on here, that’s where you should start.

    Update: For entertainment, there’s always Lubos.

    Update: I’m now free to explain what I was alluding to when I earlier mentioned an “unexpected source of information”. Yesterday evening while I was trying to find out more about the CDF result, its relation to previously published experimental results, and to possible phenomenological models, I was running some Google searches on relevant terms. One of these turned up something very surprising: the first result was a summary of CDF’s internal review of drafts of PRL and PRD papers on the subject, the second was the PRL draft. Both of these documents were from early July, and part of a publicly accessible directory containing all the materials from a review of the draft at that time. Investigating further, it became clear that the CDF web-server was seriously misconfigured, allowing directory listings and public access to a wide array of their work materials.

    I wrote an e-mail to people at CDF warning them about the problem, heard back quickly, and have just checked that they have fixed it, so I don’t think I’ll cause them a problem other than embarassment by telling this story here. It seems likely that these materials have been publicly accessible and indexed by Google probably for several months now.

    I confess to reading these documents, figuring that anything really interesting that is Googleable is fair game. Since they clearly were not intended for public consumption I won’t disseminate information about them beyond this story about their existence and the fact that the PRL draft contained tentative material interpreting the data in terms of new physics, the sort of thing the released paper avoids. One thing I can say is that it is very impressive to see the amount of effort and very serious scientific work behind a review of this kind. A lot went into this, and presumably a lot more has gone into it since last July. I understand why CDF does not make this kind of thing public, but it actually would be a wonderful example for the public of how science is done to do so.

    Update: See Resonaances for a discussion of the dark sector model building.

    Update: Another CDF blogger heard from: John Conway.

    Posted in Experimental HEP News | 42 Comments

    Job Posting

    Posted on October 25, 2008 by woit

    This is the time of year during which a large number of physicists and mathematicians must turn their attention to the problem of finding employment for next fall. Physics Today has a jobs site here, which has a new posting that may interest some of my readers. Thanks to Michael Williams for pointing this out to me.

    If applicants need a PD18 form, it’s available here.

    Posted in Uncategorized | 17 Comments

    Quantum Physicist Offers Solution To Global Market Meltdown

    Posted on October 16, 2008 by woit

    I briefly met John Hagelin when I was an undergraduate. At the time he was a Harvard particle theory graduate student, soon to get his degree and start a quite respectable career in the subject. His biography on his web-site claims that:

    His scientific contributions in the fields of electroweak unification, grand unification, super-symmetry and cosmology include some of the most cited references in the physical sciences. He is also responsible for the development of a highly successful Grand Unified Field Theory based on the Superstring. Dr. Hagelin is therefore at the pinnacle of achievement among the elite cadre of physicists who have fulfilled Einstein’s dream of a “theory of everything” through their mathematical formulation of the Unified Field—the most advanced scientific knowledge of our time.

    Hagelin moved on from doing physics research, first to help run the Maharishi International University (now the Maharishi University of Management) in Fairfield, Iowa, then to run for president as the candidate of the Natural Law Party (it seems that “Joe the Plumber” was a member of this). Nowadays, the Maharishi has passed away, and Hagelin has moved to New York, where he is Executive Director of the Global Financial Capital of New York.

    Soon after Hagelin moved to New York, his organization bought an impressive building down near Wall Street (chosen because “It’s one of the very few buildings in all of New York City that’s oriented due east”), with the goal “to inspire financiers to come forth to support the creation of Heaven on Earth”. They also are buying up land near Princeton “where, with the support of the township, a university for 5,000 Yogic Flyers is to be established.” Part of the plan seems to be to raise $3 billion or so from the financial industry, by offering 10% interest on 15 year loans.

    Last year they held a news conference to explain how the a “surging stock market” was one

    of the concrete signs of the success of the Invincible America Assembly in Iowa—the largest-ever scientific demonstration project to document the effects of large group meditations on the economic and social trends of the nation, according to Dr. John Hagelin, world-renowned quantum physicist, executive director of the International Center for Invincible Defense, and President of the Global Union of Scientists for Peace, who is leading the Assembly.

    This year the story is a bit different, with a recent press release entitled Quantum Physicist Offers Solution to Global Market Meltdown explaining that:

    a group of nearly 2,000 advanced experts is now in place at Maharishi University of Management in Fairfield, Iowa. He said the influence of coherence generated by this group is helping to calm the nation in the midst of the global crisis, but a larger, more powerful group of 8,000 experts (the square root of one percent of the world’s population) is needed to neutralize worldwide fears and re-establish confidence in the global markets.

    The cost to establish this group on a permanent basis, Dr. Hagelin said, would be negligible compared to what has been lost in a single hour during the current financial crisis.

    In a recent Open Letter to the Yogic Flyers of the Nation, Hagelin, the Raja of Invincible America, urges them on as follows:

    I am writing you from Wall Street, where I am living and working with members of my national team at the Global Financial Capital of New York, one block from the New York Stock Exchange. We are starting to make remarkably good progress in our efforts to bring Maharishi’s knowledge of enlightenment and invincibility to leaders here whose thinking and actions vitally impact the whole world. However, ultimately, our success is dependent on you. We will be successful when the leaders are receptive to our message. But their openness is 100% dependent upon the numbers in the Domes. Why? Because a rise in national consciousness directly translates into a rise in openness among leaders of business, health, education, defense, government, etc.

    At this critical time I urge everyone to fly together in large groups

    Maharishi said we need 2500 experts flying together to guarantee invincibility for America. This requirement is because the turbulence in the collective consciousness of one country can easily spread like a wildfire to create a similar turbulence in another country. And this is why, at this critical time, with the economic stability of the world hanging in the balance, I urge everyone in Iowa and everyone in the country to recommit to fly together in large groups.

    Update: Today was a beautiful day, and on a bike ride downtown I stopped to take a look at 70 Broad Street, the headquarters of Global Financial Capital of New York. Looking in the windows of the below ground level and the lobby level, the place looked pretty much abandoned. The building has some floors upstairs, maybe there is some activity up there.

    They did just put out this press release, announcing a press conference to be held in the lobby at 11am on Tuesday and webcast. Hagelin will be announcing a $1 billion plan to fund 1000 experts in New York and 8000 experts in Iowa who will “create coherence for the whole world—the basis of an invincible, prosperous global economy.” No word in the press release about where the $1 billion was going to come from.

    Update: Video of the press conference is here. The plan is simple: just give Hagelin and his followers $1 billion, and they’ll have this financial crisis all sorted out soon, by means of experts using the unified field.

    Posted in Uncategorized | 87 Comments

    Princeton Companion to Mathematics

    Posted on October 15, 2008 by woit

    I just recently got my hands on a copy of the new Princeton Companion to Mathematics, and I fear that this is likely to seriously impact my ability to get things done for a while, as I devote too much time to happily reading many of its more than 1000 pages.

    The book is an amazing document (and physically, a beautiful, if weighty object), unlike anything else I know of. Its coverage of mathematics and mathematical culture is very wide and sometimes deep, but it makes no attempt to be comprehensive. Thus, the accurate title “Companion to” rather than “Encyclopedia of”. The most remarkable aspect of the book is the extremely high quality of the contributions from a large number of different authors. It includes many wonderful long expository articles, mostly at a level that a good undergraduate math student could hope to appreciate, with much of the book accessible to an even wider audience. The articles are often written by some of the best researchers and expositors around. For example, one can find Barry Mazur writing on Algebraic Numbers, Janos Kollar on Algebraic Geometry, Cliff Taubes on Differential Topology, Ingrid Daubechies on Wavelets, Persi Diaconis on Mathematical Statistics, and many, many others of similar quality. The table of contents is available here.

    The book also includes extensive articles on historical topics in mathematics and short biographies of a large number of mathematicians, as well as coverage of applications and a section largely devoted to describing the art of problem-solving and how mathematics really gets created. This section includes a beautiful set of five essays called “Advice to a Young Mathematician”, which give five different equally fascinating perspectives from some of the best in the subject about how they achieved what they did, as well as what they have learned from years of helping students become researchers. The authors of these pieces are Michael Atiyah, Bela Bollobas, Alain Connes, Dusa McDuff, and Peter Sarnak. Luckily for all young (and old) mathematicians, this chapter is freely available here.

    The person most responsible for this is clearly the editor (and author of some of the pieces), Fields Medalist Timothy Gowers, who had help from many others, including fellow Fields Medalist Terry Tao. Gowers has a weblog, and he has written about the book in these entries (and there’s a podcast interviewing him on the book web-site at PUP). Terry Tao has a posting about the book here.

    If you’re looking for a gift for someone with a serious interest in mathematics, no matter what their background, you won’t do any better than this.

    Posted in Uncategorized | 23 Comments

    The Landscape at Princeton and Harvard

    Posted on October 15, 2008 by woit

    String theory in general seems to have gone very quiet recently, but attempts to intensively promote the string theory landscape view of fundamental physics show no signs of slowing down at all. The Princeton Center for Theoretical Science is running a year-long program Big Bang and Beyond, partially funded by the D.E. Shaw hedge fund. Next month there will be a program on “String Landscape: Examining how the string landscape alters approaches to fundamental physics and cosmology”, featuring a public lecture by Leonard Susskind.

    A couple weeks ago at Harvard, Frederik Denef gave a colloquium promoting Landscape research. The talk concentrated on making an analogy between the string theory landscape and condensed matter phenomena, relating this to recent attempts to use duality to study 3d CFTs of interest in condensed matter physics. Frederik also described a web-site (2ndcheek.com) which explains how string theory proves the Bible is right. Unfortunately this web-site no longer seems to be in operation.

    Also at Harvard, or at least across the street, landscapeologists and other string phenomenologists have taken over the Clay Mathematics Institute this week for a workshop on Stringy Reflections on LHC, yet another attempt to make the case that string theory has something to say about the LHC, despite strong evidence against this (see for instance David Gross’s talk at Strings 2008). Some of the slides from the talks have begun to appear on-line, and Michael Dine’s landscape talk is here.

    Update: David Berenstein has a report from the conference in Cambridge here.

    Posted in Uncategorized | 7 Comments

    News From Various Fields

    Posted on October 8, 2008 by woit

    The hypothetical field with one element (known as Fun) now has its own blog, ceci n’est pas un corps. Among the many things of interest, there’s a link to a video of Alain Connes explaining his recent paper with Consani and Marcolli on the subject. This is part of a project at the Journal of Number Theory to get authors to produce video introductions to their papers.

    Over at the n-category Cafe, there’s a discussion of Alan Weinstein’s new paper on The Volume of a Differentiable Stack. A stack is a sort of replacement of a quotient space, of a sort that allows one to keep track of the fact that different points may correspond to different stabilizers. One would like to naturally assign as many properties of spaces as possible to stacks, and one interesting question is how to count sizes, for instance how to define volumes. If everything is finite, it turns out the thing to do is to count points by dividing by the size of the stabilizer, but something much more subtle is required in the differentiable case, where the sets involved are all infinite.

    The last year has seen a flurry of activity on hep-th that goes under the name of the “M2 mini-revolution”, or maybe AdS4/CFT3. I’d been waiting for some expository accounts of this to appear to read more about it, and there’s a new one here. It includes graphs that show a total of 157 papers on the topic with 223 authors. Things didn’t really get started until this past spring, with the number of papers peaking at 40/month in July, with 53 new authors joining the game that month. Since then, the trend is downwards, with 21 paper in September. Thomas Klose does a good job of surveying the subject and what has been learned about it. The only applications mentioned are in the last line of the last page, which refers to possible future uses in condensed matter physics.

    The 2009 fiscal year has started already, with no federal budget in the US. The idea now seems to be to wait until it’s half over next March or so, then have the new administration put together a 2009 budget and a 2010 budget simultaneously. In the meantime, a continuing resolution has been passed, under which money can be spent at the level of the 2008 budget. This is pretty bad news for US HEP, since the relevant 2008 level was one involving serious budget cuts. The effect of these was mitigated by a later supplemental appropriation, and that can evidently be used to keep the Fermilab budget at a level such that layoffs can be avoided (see discussion from Oddone here). But it remains completely unclear what will happen to the lab a few short months from now.

    This past weekend the Skeptics Society at Caltech cosponsored with the Templeton Foundation a conference on Origins: the BIG Questions. The afternoon session was devoted to discussing God, the morning featured physicists promoting anthropics. An account of the talks is here, including this about Lenny Susskind’s talk

    The conference began with a real bang – the Big one of course, and a lesson on what preceded that singularity as best understood today by physicists. Susskind condensed his Stanford undergraduate cosmology course into a beautiful one-hour primer on the universal constants (Planck’s, gravitational constant, speed of light…) that support life. It turns out that life can only evolve and survive in a narrow window of values for these constants, a fact that Christians have recently embraced as proof of an intelligent designer. But Susskind explained how quantum mechanics support the existence of a multiverse that regularly spawns new universes with different sets of constants, making it inevitable that our comfy universe should appear. (I asked him whether a future day Dr. Strangelove could create the conditions that spawn a new universe in our own – he said no, but without a compelling explanation.)

    and this about Sean Carroll’s:

    Caltech physicist Sean Carroll delivered a great talk on time’s arrow – how time fits into the universe and how it cannot exist without fluctuations in entropy. He explained how the physical constants give our universe just the right amount of clumpiness so that time can flow, and he presented an alternative theory – consistent with quantum mechanics – on how universes can bear “babies” with differing constants.

    After Carroll, Caltech biologist Christoff Koch explained how:

    consciousness may in fact entail a new force not yet discovered by physicists.

    Update: Sean has a posting about the “Origins” conference, mainly devoted to explaining what was wrong with the presentation of a crackpot who was speaking as part of the religion segment.

    Posted in Uncategorized | 14 Comments

    2008 Physics Nobel Prize

    Posted on October 7, 2008 by woit

    Half of this year’s Nobel Prize in physics has been awarded to Yoichiro Nambu for his work on spontaneous symmetry breaking, the other half to Kobayashi and Maskawa for the CKM matrix as an explanation of CP breaking. A detailed explanation of the scientific context of the prizes is here.

    The prize for Nambu is well-deserved and rewards one of the deepest, surprising, and most important ideas in particle theory, that of spontaneous symmetry breaking. In 1960 Nambu realized that non-invariance of the vacuum state under an axial symmetry could explain the existence of pions and determine many of their properties. He also realized that this phenomenon is closely related to what happens in various condensed matter models (including the BCS model of superconductivity), where the ground state is not invariant under a symmetry of the theory. Nambu has often been mentioned as a candidate for the award, and it’s surprising that it has taken nearly 50 years for it to come about. I’m a bit curious about how often Nobel Prizes in other fields are awarded for work done a half-century ago. Unfortunately the rather difficult times particle physics has found itself in the last few decades may mean that the best way for a particle theorist to get the prize is to have been around during the field’s heyday, and to remain in good health.

    The Kobayashi-Maskawa award is a somewhat less obvious one, since it didn’t involve a surprising breakthrough, and it is about something completely different than Nambu’s work. In the early-mid 60s, starting with work by Nicola Cabibbo, it had been shown that (in modern language…) flavor mixing matrix for two generation models of the weak interactions is governed by a single angle, now known as the Cabibbo angle. In 1972, Kobayashi and Maskawa pointed out that for three generations the matrix is determined by three angles and a complex phase. The complex phase makes this a possible source of CP violation, and it seems to be the main if not only source of observed CP violation. The Nobel committee text makes the claim that:

    This is in fact a result known in mathematics since around 1950, but the contacts between mathematics and physics were not great around 1970.

    This is a rather odd description of the situation, since the mathematical facts involved here are quite simple ones about which 3×3 unitary matrices satisfy certain conditions, and presumably could have been derived by mathematicians who first worked with such matrices in the 19th century if this particular condition had come up. I can’t think of anything mathematicians learned around 1950 that is needed to solve the problem, and, once stated physicists could easily solve it without help from mathematicians (who perhaps would have been more likely to just have been a hindrance in this case…)

    The quark mixing matrix embodies most of the unknown parameters of the Standard Model, and as such is a crucial object for experimental studies of particle physics. Because of this, the KM paper is the second most highly cited paper in particle physics (after Weinberg’s “A Model of Leptons”), but this ranking doesn’t reflect the depth or importance of the ideas.

    Lots of other blogs are also covering this. Tommaso Dorigo points out that it’s a bit anomalous that a Nobel Prize for the CKM (Cabibbo-Kobayashi-Maskawa) matrix doesn’t include Cabibbo, who is still around. Lubos Motl thinks that what is important here is that Nambu is also a string theorist, and that one of the people on the committee making this choice was a string theorist, Lars Brink.

    Update: Lots of coverage of this in the press. Michio Kaku has an editorial in Forbes, where, like Lubos, he sees the real significance as being that Nambu is a string theorist:

    And there may even be a theory beyond the Standard Model, a true theory of everything that can unify all forces. It’s called string theory.

    Not surprisingly, one of the founders of string theory is Professor Nambu, a newly minted Nobel Laureate who has years of research ahead of him. The best is yet to come.

    Posted in Uncategorized | 32 Comments

    Deal Breaker

    Posted on September 30, 2008 by woit

    Discussion of quantum gravity at a level similar to that of parts of the professional physics community makes it to prime-time TV:

    Update: I should have credited where I first saw reference to this (no, I don’t normally watch this show…), it was at Capitalist Imperialist Pig. If you are having trouble with the embedded link to the video, you might have better luck following his link to the cbs web-site and seeing what you can find there….

    Update: Lubos is claiming credit for this, implying that he’s the model for Sheldon. He has a point. I always thought that the string theory community should worry about the implications of having Lubos represent them on the web, now there’s prime-time TV to think about.

    Posted in Uncategorized | 27 Comments