Quick Links

  • This month’s Physics Today has a long article by Wojciech Zurek, Quantum Darwinism, classical reality, and the randomness of quantum jumps. I’m not sure if there’s anything new there, but it’s a very clear exposition of what seems to me the most penetrating point of view on the measurement problem in quantum mechanics, one that gets far too little attention in the press.

    I’d like to know what this makes me in terms of various ideologies of the interpretation of QM. Am I a quantum Darwinist, or maybe a Zurekian?

  • At another extreme, getting lots of media attention while not saying anything substantive, there’s the multiverse of the Many Worlds interpretation. The media campaign to promote this is still in high gear. Recent examples include Brian Cox: ‘Multiverse’ makes sense at BBC News, this week’s New Scientist, which has a bunch of things including Multiverse me: Should I care about my other selves?, and an upcoming program here in New York that tells us that:

    We may live in a multiverse in which every possibility happens and with each new possibility the universe branches off into another of many worlds.

    The New Scientist article has Don Page pointing out that this explains the problem of evil. God likes the idea of everything possible happening all the time so much he’d rather not be bothered to stop bad things from happening:

    “God has values,” he says. “He wants us to enjoy life, but he also wants to create an elegant universe.” To God the importance of elegance comes before that of suffering, which, Page infers, is why bad things happen. “God won’t collapse the wave function to cure people of cancer, or prevent earthquakes or whatever, because that would make the universe much more inelegant.”

    For Page, that is an intellectually satisfying solution to the problem of evil. And what’s more, many worlds may even take care of free will. Page doesn’t actually believe we have free will, because he feels we live in a reality in which God determines everything, so it is impossible for humans to act independently. But in the many-worlds interpretation every possible action is actually taken. “It doesn’t mean that it’s fixed that I do one particular course of action. In the multiverse, I’m doing all of them,” says Page.

  • On the math front, I just noticed that Pieter Belmans has a blog. One of the many nice things there is his “atlas” for Spec Z[x].
  • Over at Persiflage, anyone interested in how NSF grant applications in mathematics are evaluated can find an extensive and well-informed discussion.
  • Videos from last week’s Heidelberg Laureate Forum (which features Fields Medalists and others) are available here.
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Thirtieth Anniversary of the First Superstring Revolution

Today’s about the date that I’d pick for the 30th anniversary of the First Superstring Revolution. Witten’s paper Some Properties of O(32) Superstrings arrived at the journal Physics Letters on September 28,1984, so presumably was finished and sent out around September 25.

The effect of this paper on the field was a bombshell. Witten was at the time far and away the most influential person in the field, regularly producing staggeringly original work that was having a huge impact. The arrival that fall of a preprint from him announcing that he had stopped work on everything else, and now had what looked like a viable, consistent unified theory of everything, one that he claimed was determined by a single parameter and made predictions (“It predicts axions and stable Nielsen-Olesen vortex lines”) was the true First Superstring Revolution.

I wrote about this in some detail ten years ago, for the 20th anniversary, so won’t repeat what is here and here, supplemented by comments from Larry Yaffe. For something more recent along the same lines, see here.

Ten years ago the 20th anniversary of the First Superstring Revolution was celebrated with a symposium at Aspen, but as far as I know, no one has organized a 30th anniversary celebration. There are now many, many known ways of trying to get unification out of strings, with the original 1984 hope that anomaly cancellation gave a more or less unique possibility long gone. As for unification itself, thirty years later Witten remains a true believer in the vision that came to him in September 1984 (see here and here), although he now seems to see little hope for vindication during his lifetime.

Posted in Uncategorized | 31 Comments

Horgan Interview With Witten

Back in 1996 John Horgan’s The End of Science appeared, which included material from a fascinating 1991 interview of Edward Witten. I had mixed feelings when reading this. On the one hand, Horgan was doing something truly remarkable, challenging Witten in a way that no one else dared. This was 7 years after the “First Superstring Revolution”, and it was starting to become clear that string theory was not working out as hoped. No journalist other than Horgan though was talking about this, or willing to confront someone of Witten’s stature with difficult questions. Pretty much every other story in the press stuck to the simple narrative that Witten was a genius, and superstring theory a great success. On the other hand, Horgan did use his author’s freedom to edit and frame the interview to make Witten look bad (today he admits the Witten profile was “pretty snarky”), so he was landing some low blows, against a rather gracious opponent.

This year Witten won the Kyoto Prize, and I was shocked to hear that Horgan was the person chosen to interview him. Witten rarely gives interviews and I would have thought that Horgan would be the last person in the world he’d agree to an interview with, given his past experience. The interview is now available here.

This time around Horgan avoids the snark, and asks some straightforward questions about whether Witten’s views have changed since 1991, and what he now thinks about string theory, the multiverse, anthropics, etc. I have to admit that I find Witten’s answers depressing, in contrast to Witten’s advisor David Gross’s current take on these issues (discussed here). About anthropics, Witten’s “I don’t like it, but may be the way to go” contrasts with Gross’s “cop-out”, and his insistence on string theory as the way forward contrasts to Gross’s emphasis on the fertility of quantum field theory.

Back in 1996, after the appearance of Horgan’s book, Gross and Witten wrote in to the Wall Street Journal (reproduced here) to argue that Horgan was wrong, since string theory would be tested by finding SUSY at the Tevatron, or, failing that, definitely at the LHC. We all know how well that has worked out, and Gross seems to have learned a lesson from this. Witten on the other hand has moved on to even more dubious testability claims (e.g. that the string theory landscape can be tested by “seeing a signature of a prior phase transition in the CMB”). From the 1996 claim that vindication would come “in the next decade”, he now is talking about “200 years from now”. His one point of close agreement with Gross is that both agree that not knowing what string theory is when time-dependent effects are large is a big problem, one that has seen no progress.

By a couple years from now, the idea of making progress in our lifetime by seeing SUSY at the LHC, then going on to use this to learn about string theory should be finally finished off. Already Gross seems to have evolved from the 1991 point of view to a more promising one, perhaps Witten at some point will start to do the same.

: The AMS has something similar, a Mathematical Moment with Witten. Pretty much everything said about string theory is exactly the same as thirty years ago, only change is that the story used to be that string theory would get some vindication at the LHC, now it’s:

The verification of superstring theory is probably a long way off, but could be found here on Earth, using particle accelerators (possibly much more powerful than those of today)

consistent with the “200 years” estimate from the Horgan interview.

Posted in Uncategorized | 26 Comments

Planck: It’s Just Dust

The Planck paper with results on dust in the BICEP2 patch of sky is now out, see here. I’m sure experts will weigh in soon and I’ll link to such discussions, but my non-expert take is that Planck is saying that what BICEP2 saw is likely just dust. See section 6 of the paper, especially figure 9 which appears to show that BICEP2’s claimed value of r=.2 is just what you’d expect from dust.

: More details from Natalie Wolchover and Sean Carroll.

Looks like Scientific American will have to pulp this month’s magazine, with its Lawrence Krauss cover story about how BICEP2 is experimental evidence for quantum gravity and the multiverse.

Update: For more press coverage, see Nature, New Scientist, BBC News, The Guardian, the Washington Post, the Daily Mail and the New York Times.

The best explanation for all this that I’ve seen of course is from a blogger, Sesh Nadathur at Blank on the Map.

Update: Jester has a sensible take on this fiasco here. It now seems that release of the full Planck polarization results has been pushed back from October to “late November”, just before the early December conference planned long ago to discuss the results. The joint analysis of BICEP2/Planck data that will show if there’s any evidence of something besides dust is supposed to be released at the same time.

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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 | 1 Comment

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


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