Quantum Field Theory: Past, Present and Future

Video is now available of David Gross’s colloquium this past week at NYU, which had the title Quantum Field Theory: Past, Present and Future. It’s quite interesting to compare his current point of view to that of ten years ago. The earliest substantive post on this blog was this one, which reported on a similar sort of talk by Gross, of similar length, also here in New York.

If you look at that blog post, you’ll see that I found myself in strong disagreement with many of the main arguments Gross was making back in 2004. Remarkably, ten years later, there’s relatively little I would disagree with in his NYU talk on much the same topic. Back in 2004 he was predicting the imminent discovery of supersymmetry at the LHC, in the current talk supersymmetry was not mentioned at all. I think the negative LHC results have had a very real effect on his thinking.

His 2004 point of view on string theory was that it was a better, more fundamental replacement for QFT. His arguments for this weren’t very good then (see the old blog posting), and he seems to now have wisely abandoned them. Instead, the first hour of his talk was all about the story of our increasing understanding of the power of QFT. From there, he argued that there’s some larger framework that we don’t understand, which includes our current understanding of QFT, as well as things like quantum states that look like strings. He likes to refer to this conjectural new framework as QFT/string theory. Interestingly, there was no reference at all to “M-theory”.

Gross’s current vision of the future comes down to something close to mine: some yet undiscovered new ideas will tell us something new about the QFT framework, and this will show us how to make progress on quantum gravity and unification. I’d add something more specific, that previous progress came from understanding new ways of exploiting symmetries in QFT, so future progress may very well be of that same general nature. He pointed out that the story of past QFT progress was often that people had decided that something dramatically different was needed, but ended up realizing that they just needed to solve some very technical issues, not move to something very different (e.g. proper handling of renormalization and of gauge symmetry was needed, not new degrees of freedom).

In the question and answer period Gross made clear his distaste for the string theory landscape. About all he would say about anthropics was “Oy-vey”, and that it’s nothing but a cop-out. He characterized the supposedly finite number of “string vacua” with stabilized moduli and positive CC as likely irrelevant, since you don’t know what theory they are a solution to, and there’s an infinity of other solutions to the kinds of equations you’re considering.

All in all, I found watching this quite encouraging. Seeing one of the great elder statesmen of the field stop promoting failed ideas, challenge dubious received wisdom, and move on to a more promising take on where the field should be heading is cause for optimism. I hope younger theorists will pay attention.

Posted in Uncategorized | 4 Comments

Job Action at the Journal of K-theory

Back in 2007 I wrote here several times (see for example here and here) about the story of the resignation of the editorial board of the Springer journal K-theory in favor of a journal published by Cambridge, called the Journal of K-theory. For a detailed history of this, see Eureka Journal Watch.

At first this story fit in with the narrative of a group of mathematicians banding together to do something about high journal prices, but the actual story was much murkier. There never seemed to be any evidence that anyone had tried to negotiate a lower price with Springer. The editorial board resigned in January 2007, but the managing editor Anthony Bak had stopped sending papers to Springer in April 2006, and the resignation wasn’t made public until August 2007, a sequence of events that left some submitted and refereed papers in limbo.

The actual financial arrangements between Bak and Springer were never made public, and Bak was supposedly suing Springer for a significant amount of money, on grounds that also were never disclosed.

Wolfgang Lueck and Andrew Ranicki took over the task of dealing with the manuscripts in process at the Springer journal, and you can read Lueck’s account of that here.

When the new “Journal of K-theory” was started, there was a statement from the editors that:

The title of JKT is currently owned by a private company. This situation is only meant as a temporary solution to restart publication of K-Theory articles as soon as possible. It is the Board’s intention to create a non-profit academic foundation and to transfer ownership of JKT to this foundation, as soon as possible, but no later than by the end of 2009, a delay justified by many practical considerations.

(a more detailed version is here).

The non-profit foundation did get created, it’s the K-theory Foundation and one thing it does is sponsor conferences, and award every four years prizes for work by young mathematicians on K-theory, with the first two $1000 prizes awarded this year.

The latest news though is that there has been some sort of breakdown between management (the managing editor Bak), and the workers (much of the editorial board), leading to a strike (see news from Scott Morrison). The workers are demanding that the ownership of the means of production be transferred, as promised back in 2007, from Bak’s company (ISOPP) to the K-theory Foundation.

Morrison has more details here, and in the comments quotes a claim that Bak’s company has been been receiving 73-74,000 pounds per year, for services that Cambridge would normally pay 20-25,000 pounds per year for. So, this appears to not just be about the technicalities of ownership, but about significant sums of money coming in from publishing math papers. At Morrison’s site, Andrew Ranicki advises “Follow the money.”

It seems that removing control of the income thrown off by math journals from the clutches of Springer may not solve all problems. The editors on strike say that if Bak doesn’t fold, they start yet another journal.

Update: The text of a recent talk by Tony Bak describing the history of the journal is here.

From Scott Morrison, news last week was that:

As of a few hours ago, Tony Bak is no longer the President of the K-Theory Foundation, having been removed from the board by a unanimous (excepting abstentions) vote.

Editors of the Journal of K-Theory have begun contacting the authors of submitted papers to give them the opportunity to withdraw their papers, or to wait and consider the option of transferring to a new journal.

Posted in Uncategorized | 3 Comments

Various and Sundry

  • The MacArthur Foundation today announced “Genius” grants of $625,000 to 21 people, including two mathematicians, Jacob Lurie and Yitang Zhang. While there was a time these awards often went to mathematicians and theoretical physicists (the 1987 winners included string theorists Dan Friedan, David Gross, John Schwarz and Steve Shenker as well as mathematicians Robert Coleman and David Mumford), that has been much less common in recent years.
    Zhang, now a professor at the University of New Hampshire, is a perfect candidate for the award, unrecognized by academia (he worked at a Subway for a while) while he was doing brilliant and important work in number theory. Lurie is undeniably a genius, but kind of the opposite of Zhang, someone whose talents and work have been very well-recognized and rewarded already. He’s a Harvard professor and in November will be collecting a $3 million Milner-Zuckerberg Breakthrough Prize.
    The Wall Street Journal leads off their story about this with Lurie, characterizing him as “A mathematician offering his book free on the internet”, implying that’s what distinguished him for the award from the other possible genius candidates:

    This year’s winners span in age from 32 to 71 and include nine women and 12 men. A common thread: The winners reach their audiences in surprising places.

    “This year, we have several people who one might describe as being engaged to challenge the rest of us to be lifelong learners outside the traditional classroom,” said Cecilia Conrad, who directs the fellows program as a vice president of the foundation. “It’s new solutions to old problems.”

    Mathematician Jacob Lurie, who was honored for redefining models in algebraic geometry, negotiated with his publisher to make his book on math principles available for free download on his personal website. While academics sometimes place papers online free, putting a whole book online isn’t yet standard practice, according to the 36-year-old Harvard University professor. “From my point of view, the benefit of writing a book is for people to look at it. I would like as many people as possible to look at it,” he said.

    The book in question is the 2009 944 page Higher Topos Theory, available on Lurie’s web-site here. He has just put up on his website an updated version of his second book, the 1178 page Higher Algebra. For those mathematicians worried that they might have trouble reading these because of a lack of physics background, Lurie himself reassures people here that

    Since no knowledge of modern physics was required to write any of these books and papers, I can’t imagine that you need any such knowledge to read them.

    In other Lurie news, he has also just put up on his web-site an important new paper, a first draft of joint work with Gaitsgory on the proof of Weil’s Tamagawa number conjecture for function fields.

  • Skepticism about string theory and the multiverse abounds these days. A wonderful New York Times profile of Peter Higgs ends with

    This has led some theorists to propose that our universe is only one in an ensemble of universes, the multiverse, in which the value of things like the Higgs is random.

    Asked about that, Dr. Higgs lit up with a big grin. “I’m not a believer,” he said.

    “It’s hard enough to have a theory for one universe.”

    In Scientific American, George Ellis has a piece entitled Why the Multiverse May Be the Most Dangerous Idea in Physics.

    Meanwhile, from a Templeton Foundation-financed conference on the Philosophy of Cosmology in the Canary Islands, Sean Carroll reports via Twitter that string theorist Tom Banks is arguing that “string theory has failed as a theory of our world.”

  • Commenter Shantanu pointed out something I hadn’t realized, that David Gross will be here in New York this week, giving talks at NYU. Unfortunately I won’t be able to attend, in particular I have to teach at the time of his colloquium on Tuesday. Perhaps someone who can attend will report what he has to say.
  • In Grothendieck news, the English translation by Melissa Schneps of Winfried Scharlau’s book on the later period of Grothendieck’s life has started to appear, see some chapters here.

    A wonderful book of articles about Grothendieck’s mathematics, Alexandre Grothendieck: A Mathematical Portrait, edited by Leila Schneps, has recently been published (with a version of the articles also available here).

Update: Vigorous back-tweeting (see here, here, here, here and here) now going on from Sean Carroll and Tom Banks. Admitting that string theory unification has failed is just not done. Revised and extended remarks from Tom Banks add praise for the greatness of string theory and avoid the word “failure”:

Without string theory we would never have been in a position to understand anything serious about quantum gravity, but without going beyond the present understanding of string theory we can make no further progress.

This still though reads as “string theory is at a dead end”. Sean echoes the praise:

conventional string theory has given us enormous guidance toward quantum gravity.

and dismisses the failure issue as something obvious and not worth mentioning:

Need to go beyond is obvious.

I guess I should point out that it is not obvious that you need to go “beyond” string theory, in the sense of farther in the same direction. Might be that you need to abandon the direction that led you to a dead end, back-track, and try a different one.

Strange thing is that this discussion of a string theory dead end just seems to be about problems using string theory to do quantum gravity when there’s a positive cosmological constant (otherwise, according to Carroll, “String theory is great”). The idea of string unification seems to be so dead it’s not even worth mentioning.

Update: Shaun Hotchkiss has the latest news on Planck/BICEP2 and dust here. People are still scraping data off old Planck slides, with real Planck data on the BICEP2 patch rumored to be imminent for the past month or so:

Any day now we are to expect Planck’s paper revealing the non-conference-talk maps of the high frequency polarisation signal along BICEP2’s line of sight. These will just be images though, not raw data. The word on the street/corridor is that a fully written draft exists and has clearance to be submitted and nobody I’ve spoken to knows why it hasn’t been. The sort of phrases I’ve heard about what to expect from this is that “it will clarify a lot of things”, but “it won’t be conclusive”.

Update: Two more. I’ve been avoiding writing about the AMS-02 announcement about the positron excess, waiting to hear something sensible about its significance. Resonaances is on the job, giving an interesting take on the data, and claiming this has nothing to do with dark matter.

There’s a nice profile of Robbert Dijkgraaf here.

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Quick Links

Well worth reading is High Energy Colliding Beams; What Is Their Future, by Burton Richter. Richter is one of the pioneers of designing and building colliders, and he starts off by recounting some of the history. About proposals for a 100 TeV collider he comments on the challenges of doing this at high luminosity and the danger that the cost will be prohibitive (one thing I haven’t seen in these discussions is cost estimates), and asks why there is no large-scale program to develop low-cost high-Tc superconducting magnets.

He’s critical of the film Particle Fever on the same grounds discussed here (its portrayal of the only possibilities as being SUSY or the multiverse). About the multiverse, he writes:

There are two problems with the landscape idea. The first is a logic one. You cannot prove a negative, so you cannot say that there is no more to learn. The second is practical. If it is all random there is no point in funding theorists, experimenters, or accelerator builders. We don’t have to wait until we are priced out of the market, there is no reason to go on.

For some mathematics news, first there’s the announcement from the Flyspeck project of the completion of a formal proof version of the proof of the Kepler Conjecture by Thomas Hales. Hales is in Berkeley this week talking about something unrelated (the Langlands program) at an introductory workshop for this semester’s MSRI program on geometric representation theory. I’ve been watching some of the videos of the workshop talks, all of which have been quite good.

Also in Berkeley this semester is Peter Scholze’s course, with video of the first lecture here, notes here.

In yet more Berkeley news, in December they’ll host a mathematical physics workshop on Mathematical Aspects of Six-Dimensional QFTs. Better understanding the 6d N=(2,0) superconformal theory and its implications for various lower-dimensional phenomena is the main target here, a topic that will also be discussed here in the spring (where the 6d theory is called “Theory X”).

Posted in Langlands | 21 Comments

SUSY Bets

The LHC long shutdown (LS1) seems to be progressing on schedule, with physics collisions at 13 TeV planned for early April 2015. I’d guess the earliest 13 TeV results might appear at the summer 2015 conferences. The long term plan is to accumulate up to about 50 fb-1 of data per year for about 3 years of data-taking, ending in mid-2018. There will then be a year and a half shutdown (LS2), followed by data-taking at 14 TeV from 2020-2022. The plan is to end up with about 300 fb-1 before a long shutdown (LS3) starting in 2023.

Hopefully there will be much learned about the Higgs, and some unexpected discoveries. One of the main targets will continue to be SUSY searches, despite the negative results found so far at 8 TeV (and 25 fb-1). Something to watch will be how long it takes theorists heavily-invested in TeV-scale SUSY to give up and concede that this idea doesn’t work. For this, one thing to keep in mind is what precise bets theorists have made in the past.

There’s a new one this week. After Gordon Kane complained that he couldn’t find anyone willing to bet against SUSY, Marcelo Gleiser decided to take him up on it, with stakes a bottle of 15 year old Macallan (which goes for about $100). Marcelo seems to think he has a bet that will get him his Macallan if no SUSY is found in the run ending in 2018, but I fear he has been had. Kane specifies:

To have a meaningful bet the LHC has to work at an appropriate energy and luminosity. It is expected to take integrated luminosity of order 300 fb-1 at a total energy near 13 TeV in the next run, in less than two years after turning on in early 2015. Assuming those results, signals for gluinos and/or light neutralinos and/or charginos are expected, and that’s the appropriate bet.

The only problem with this is that the current LHC schedule foresees maybe 100 fb-1 two years after first physics in 2015, not 300 fb-1. For 300 fb-1 the schedule says the wait is likely to be until 2023, so Marcelo is going to have a very long wait for his fine Scotch.

Here’s the status of the other SUSY bets I know about, and I’d be curious to hear about any other known ones:

  • Back in 2000 some theorists at a conference in Copenhagen bet (stakes $50 cognac) about SUSY being found at the LHC by mid-2010. The losers welshed reneged on that bet, to be fair partly because the LHC was delayed, and didn’t really get going until 2010, at half design energy. A new version of the bet was made in 2011, with stakes raised to $100 cognac and a cutoff date in June 2016.
  • David Gross here announced back in 2012 that he had taken bets on SUSY, paying off once 50 fb-1 of data have been analyzed. This would likely at the earliest be in mid-2016, same time frame as the Copenhagen bet.
  • Garrett Lisi announced on Twitter back in 2009 that:

    Frank Wilczek just bet me $1000 that superparticles will be detected by July 8, 2015. Max Tegmark will arbitrate.

    At this point it seems that Wilczek is likely out $1000, since this date will only be 3 months into the run with results available for only a small amount of data if any.

  • Wilczek also has a 2013 bet with Tord Ekelof that gauginos will be found by end 2019. This one is just for some chocolate coins.
  • Jacques Distler made a $750 bet with Tommaso Dorigo based on the first 10 fb-1 of LHC data. This was more general, Jacques would win if either SUSY was found, or something else unexpected. Jacques paid up last year, see here.
  • Many theorists were highly skeptical of SUSY long before the LHC turned on. Back in 2008 Adam Falkowski assigned a probability of .1% to a SUSY discovery at the LHC, and gave Lubos Motl 100 to 1 odds for a bet about SUSY after 30 fb-1 of LHC data. Lubos still has his $100 since the LHC didn’t quite get to 30 fb-1, but he should be out the money probably sometime mid-next year.

If there are any others of these out there, let me know…

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Use the Moment Map, not Noether’s Theorem

For a fourth provocative slogan about quantum mechanics I’ve chosen:

Use the moment map, not Noether’s Theorem.

Pretty much every physics textbook these days explains the way symmetry principles work as:

  • Start with an action functional, invariant under a Lie group G.
  • Use Noether’s theorem to get a conserved charge (for each element of the Lie algebra of G).

There’s a short (slightly mystifying) calculation always given to derive this. I’d like to argue that this is really not the best way to think about the implications of having a Lie group act on a physical system, that for this it’s better to take the Hamiltonian point of view. There the way symmetry principles work is:

  • For a function on phase space (or on a general symplectic manifold) you get a vector field. This is just Hamilton’s equations, giving the vector field for time evolution corresponding to any Hamiltonian function.
  • The infinitesimal action of G on phase space gives a vector field for each element of the Lie algebra of G. The moment map takes an element of the Lie algebra to a function on phase space (the one corresponding to the vector field).

I’m ignoring some subtleties here having to do with the relation between vector fields and functions not being quite one-to-one.

All of the basic examples of conservation laws in physics come about this way. The action of time translation gives the Hamiltonian function, space translation the momentum, rotations give the angular momentum, and phase transformations give charge. You can get these either as moment maps, or using Noether’s theorem.

The moment map however gives you much more, with phase space providing structure that is not visible just from the action. A simple example is the harmonic oscillator in 3 variables. SO(3) rotations act on the configuration variables, preserving the action, so Noether’s theorem gives you 3 conserved quantities, the angular momentum variables. The moment map point of view however gives you much more. The phase space is 6 dimensional (3 positions + 3 momenta) and the Lie group Sp(6,R) of linear symplectic transformations acts on it, with a subgroup U(3) preserving the Hamiltonian. The U(3) includes the SO(3) rotations as a subgroup, but it is much larger (9 dimensions vs. 3), so the moment map gives you many more conserved quantities. After quantization, you learn that energy eigenstates are U(3) representations, telling you much more about them than what angular momentum tells you.

The moment map point of view also gives you quantities corresponding to the directions in Sp(6,R) that are not in U(3). In the quantum theory these act on the full state space (not preserving energy eigenstates) and your state space is a representation of (a double cover of) this group.

For the simplest possible harmonic oscillator, in one-dimension, Noether’s theorem doesn’t really tell you anything. The moment map point of view says that there is an Sp(2,R) acting on phase space, with a U(1) subgroup preserving the Hamiltonian. The moment map is just the Hamiltonian itself. In the quantum theory you find that the harmonic oscillator state space is a representation of (a double cover of) Sp(2,R), with the U(1) action on states characterized by integers, which correspond to the energy. This integrality is the essence of the “quantum” in “quantum mechanics”, and it’s quite invisible to Noether’s theorem, but a basic fact of the moment map point of view.

In some sense this is an argument for the Hamiltonian vs. Lagrangian point of view in general. The relation between the two is that, given a Lagrangian, one constructs a symplectic structure on the space of solutions of the variational problem, and thus a Hamiltonian formalism. Noether’s conserved quantities are then examples of moment maps. The problem is that typically this requires the use of constraints and the quite tricky constrained Hamiltonian formalism.

The positive argument for the Lagrangian point of view is that it comes into its own in the relativistic setting, making Lorentz invariance easy to handle by the Noether’s theorem method. This is quite true, with the standard version of the Hamiltonian formalism distinguishing the time direction and breaking Lorentz invariance. There is however a less well-known “covariant phase space” point of view, where one tries to work with the space of solutions of the equations of motion as one’s phase space. Only if one identifies a solution with its initial data at a fixed time does one distinguish the time direction. I’ve recently enjoyed reading Igor Khavkine’s review article, which in particular does a great job of explaining the history of this line of thinking.

The Lagrangian also comes with the extremely seductive point of view on quantization of the path integral. This point of view works very well for dealing with Yang-Mills theory, and I spent much of my early career convinced that all there was to quantization was figuring out how to make sense of integrating over the exponential of the action. I’m now much more aware of the advantages of the Hamiltonian point of view, especially in terms of understanding quantum theory as representation theory. In some sense what one really wants is to understand quantization in a way that takes advantage of both points of view, but the relationship between them is quite non-trivial.

The discussion here has been far too wordy for most people to make sense of. If you want to understand any of this, you need equations. Luckily, I’ve provided lots of them and many details here, see chapters 12 and 13 for the moment map, chapter 19-22 for the harmonic oscillator.

Posted in Quantum Mechanics | 25 Comments

Quick Links

Now back from the West Coast, here’s a list of things I’ve run across that may be of interest:

  • One piece of news from Berkeley is that Peter Scholze will be there this fall, giving a course describing new techniques for dealing with Langlands conjectures for number fields in an analogous manner to ones successfully used in the function field case. The course announcement (described by David Ben-Zvi as “shockingly futuristic”) is here.

    Since it’s never too early for Fields Medal predictions, I’m predicting an award for Scholze at the next ICM, in Rio, August 2018.

  • It also wouldn’t surprise me if Scholze gets one of the $3 million Milner/Zuckerberg breakthrough prizes in mathematics before then. I’m pleased to see that last week there was an announcement that some of the winners of this year’s math prize have banded together to fund graduate math fellowships in developing countries.
  • On the physics front, rumor from Peter Coles is that:

    I have it on very good authority that Planck’s analysis of the Galactic foregrounds in the BICEP2 region will be published (on the arXiv) on or around September 1st 2014.

    so next week we may (or may not…) find out if BICEP2 was seeing primordial gravitational waves or just dust.

  • In the meantime, this week there’s COSMO 2014, a cosmology conference going on in Chicago. The conference organizers have decided to have their public event mix artists and scientists, brought together around the topic of Multiverse: Fact, Fictions and Fantasies.
  • Also in the Chicago area, Fermilab last week hosted a Nature Guiding Theory workshop, considering the question of what to make of the failure of the LHC to find SUSY or other physics predicted by the “naturalness” paradigm. Some of the discussion was of the conventional SUSY or multiverse variety. For instance, see here, or Raman Sundrum’s Super-Natural vs. Other-Worldly in Fundamental Physics, which ends up arguing that:

    Naturalness, anthropic selection, Multiverse are Meta-theories. The collection of naturalness-related experiments – LHC, flavor, axion searches, tests of Inflation (e.g. BICEP2, …), Dark Matter search, form a Meta-experiment.

    I’m not sure what a “Meta-experiment” or “Meta-theory” is, other than that it’s not the conventional sort of science where you have the usual notions of how to make scientific progress.

    Some talks dealt with dumping the “naturalness” argument in favor of ideas about fundamental conformal invariance that I mentioned here. See for example here and here. By the way, Natalie Wolchover’s excellent article on these ideas has been given by Wired the title Radical New Theory Could Kill the Multiverse Hypothesis.

  • Going on at the same time as the cosmology conference in the Chicago area this week is a Fermilab workshop on Next Steps in the Energy Frontier. Today there were talks on future plans from Fermilab, CERN and the Chinese. The Chinese now have a Center for Future High Energy Physics directed by Nima Arkani-Hamed and are talking about a huge new machine that would start off as an electron-positron Higgs factory in 2028, then become a 50-90 TeV proton-proton collider in 2042. CERN is discussing similar plans, although they will have the high-luminosity LHC keeping them busy through 2035.

    On Thursday Arkani-Hamed will end the conference with a talk at 12:15 entitled “Go Big or Go Home…” (the US I guess already has decided to go home, with no plans for a big new machine). He really is ubiquitous at fundamental physics conferences in a variety of areas, since at 9-9:45 that morning he’ll be addressing COSMO 2014 up in Chicago on the topic of “Cosmological Collider Physics”. This Tuesday night he’ll be at the other public event there, a showing of the movie Particle Fever, which stars him discussing the multiverse.

  • There seems to be little scientific news anymore about string theory, but it is everywhere in popular culture. This past weekend it made Dilbert, it’s playing a big role in DC superhero comics, and a few weeks ago some peculiar British TV show designed to torment young “child geniuses” had string theorist Brian Wecht bringing an 11-year old to tears with questions on the subject.
  • John Horgan has put online an interesting interview with Carlo Rovelli, following up on one with George Ellis. Both Ellis and Rovelli criticize physicists for knocking philosophy. On the whole I’m quite sympathetic to what both Ellis and Rovelli have to say, although I think the problem with string theory is not really the “philosophical superficiality” that Rovelli sees as the problem. Jerry Coyne has a blog posting criticizing Rovelli as an “accomodationist”. Note that if you want to argue about religion, please do it at Coyne’s blog, not here.
  • For those interested in the metaphysical end of philosophy as applied to physics, Oxford is hosting a conference in October on The Metaphysics of Quantum Mechanics. It’s organized by the Power Structuralism in Ancient Ontologies and Metaphysics of Entanglement Projects. The first of these is funded by the European Research Council, the second by the Templeton World Charity Foundation (which I’ve never heard of before, and wonder about its relationship to the Templeton Foundation). Entanglement is the hot topic in fundamental physics these days. The Metaphysics of Entanglement project promises to bring quantum mechanics and theology together bu (Jerry Coyne is going to hate this…)

    bringing the research results of the above investigation to bear on our understanding of questions regarding the metaphysics of the incarnation and of the Trinity in philosophy of religion.

  • Anyone interested in making some easy money might want to contact Gordon Kane. He has a letter to the editor in the latest Scientific American arguing that string/M-theory predicts superpartners visible at the LHC and complains that:

    Predictions based on such theories should be taken seriously. I would like to bet that some superpartners will be found at the LHC, but I have trouble finding people who will bet against that prediction.

Update: One more. The New York Times today has a very good story about IMPA, the math institute in Rio de Janeiro.

Posted in Uncategorized | 11 Comments

Nature is Fundamentally Conformally Invariant

For a third slogan I’ve chosen:

Nature is fundamentally conformally invariant.

Note the weasel-word “fundamentally”. We know that nature is not conformally invariant, but the kind of thing I have in mind is pure QCD, where the underlying classical theory is conformally invariant, with quantization dynamically breaking conformal invariance in a specific way.

The group of conformal symmetries, its representations, and what this has to do with physics are topics I haven’t written about in the notes I’ve been working on. This is because I suspect we still haven’t gotten to the bottom of these topics, and properly dealing with what is known would require a separate volume. Part of the story is the twistor geometry of four dimensions, which Roger Penrose pioneered the study of, and which recently has found important applications in the calculation of scattering amplitudes.

As a more advanced topic, this slogan would normally have been put off until later, but I wanted to point to a new article by Natalie Wolchover in Quanta magazine which deals with exactly this topic. It describes several different efforts by physicists to rethink the usual story about the hierarchy problem, taking a conformally invariant model as fundamental. For the latest example along these lines, see this arXiv preprint. The whole article is well-worth reading, and it includes a quote from Michael Dine (whose work I’ve been critical of in the past) that I found heart-warming:

“We’re not in a position where we can afford to be particularly arrogant about our understanding of what the laws of nature must look like,” said Michael Dine, a professor of physics at the University of California, Santa Cruz, who has been following the new work on scale symmetry. “Things that I might have been skeptical about before, I’m willing to entertain.”

Perhaps particle theorists are beginning to realize that the landscape is just a dead-end, and what is needed is a re-examination of the conventional wisdom that led to it.

Posted in Quantum Mechanics | 33 Comments

Grand Unification of Mathematics and Physics

For a second slogan about quantum mechanics I’ve chosen:

Quantum mechanics is evidence of a grand unification of mathematics and physics.

I’m not sure whether this slogan is likely to annoy physicists or mathematicians more, but in any case Edward Frenkel deserves some of the blame for this, since he describes (see here) the Langlands program as a Grand Unified Theory of mathematics, which further is unified with gauge field theories similar to the Standard Model.

This week I’m in Berkeley and have been attending some talks at an MSRI workshop on New Geometric Methods in Number Theory and Automorphic forms. Number theory is normally thought of as a part of mathematics about as far away from physics as you can get, but I’m struck by the way the same mathematical structures appear in the representation theory point of view on quantum mechanics and in the modern point of view on number theory. For example, the lectures on Shimura varieties have taken as fundamental example the so-called Siegel upper-half space, which is the space Sp(2n,R)/U(n). Exactly the same space occurs in the quantization of the harmonic oscillator (see chapters 21 and 22 of my notes), where it parametrizes possible ground states. Different aspects of the structure play central roles in the math and the physics. In the simplest physics examples one works at a fixed point in this space, with Bogoliubov transformations taking one to other points, something which becomes significant in condensed matter applications. In number theory, one is interested not just in this space, but in the action of certain arithmetic groups on it, with the quotient by the arithmetic group giving the object of fundamental interest in the theory.

The workshop is the kick-off to a semester long program on this topic. It will run simultaneously with another program with deep connections to physics, on the topic of Geometric Representation Theory. This second program will deal with a range of topics relating quantum field theory and representation theory, with the geometric Langlands program a major part of the story, one that provides connections to the number theoretical Langlands program topics of this week’s workshop. I’ve got to be in New York teaching this semester, so I’m jealous of those who will get to participate in the two related MSRI programs here in Berkeley. Few physicists seem to be involved in the programs, but these are topics with deep relations to physics. I do think there is a grand unified theory of some kind going on here, although of course one needs to remember that grand unified theories in physics so far haven’t worked out very well. Maybe the problem is just that one hasn’t been been ambitious enough, that one needs to unify not just the interactions of the standard model, but number theory as well…

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Quantum Theory is Representation Theory

For a first slogan (see here for slogan zero) I’ve chosen:

Quantum theory is representation theory.

One aspect of what I’m referring to is explained in detail in chapter 14 of these notes. Whenever you have a classical phase space (symplectic manifold to mathematicians), functions on the phase space give an infinite dimensional Lie algebra, with Poisson bracket the Lie bracket. Dirac’s basic insight about quantization (“Poisson bracket goes to commutators”) was just that a quantum theory is supposed to be a unitary representation of this Lie algebra.

For a general symplectic manifold, how to produce such a representation is a complicated story (see the theory of “geometric quantization”). For a finite-dimensional linear phase space, the story is given in detail in the notes: it turns out that there’s only one interesting irreducible representation (Stone-von Neumann theorem), it’s determined by how you quantize linear functions, and you can’t extend it to functions beyond quadratic ones (Groenewold-van Hove no-go theorem). This is the basic story of canonical quantization.

For the infinite-dimensional linear phase spaces of quantum field theory, Stone-von Neumann is no longer true, and the fact that knowing the operator commutation relations no longer determines the state space is one source of the much greater complexity of QFT.

Something that isn’t covered in the notes is how to go the other way: given a unitary representation, how do you get a symplectic manifold? This is part of the still somewhat mysterious “orbit method” story, which associates co-adjoint orbits to representations. The center of the universal enveloping algebra (the Casimir operators) acts as specific scalars on an irreducible representation. Going from the universal enveloping algebra to the polynomial algebra on the Lie algebra, fixing these scalars fixes the orbit.

Note that slogan one is somewhat in tension with slogan zero, since it claims that classical physics is basically about a Lie algebra (with quantum physics a representation of the Lie algebra). From the slogan zero point of view of classical physics as hard to understand emergent behavior from quantum physics, there seems no reason for the tight link between classical phase spaces and representations given by the orbit method.

For me, one aspect of the significance of this slogan is that it makes me suspicious of all attempts to derive quantum mechanics from some other supposedly more fundamental theory (see for instance here). In our modern understanding of mathematics, Lie groups and their representations are unifying, fundamental objects that occur throughout different parts of the subject. Absent some dramatic experimental evidence, the claim that quantum mechanics needs to be understood in terms of some very different objects and concepts seems to me plausible only if such concepts are as mathematically deep and powerful as Lie groups and their representations.

For more about this, wait for the next slogan, which I’ll try and write about next week, when I’ll be visiting the Bay area, partly on vacation, but partly to learn some more mathematics.

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