Not Even Wrong

The Simons Center for Geometry and Physics at Stony Brook is having a workshop this week on Mathematical Foundations of Quantum Field Theory. I was hoping to find time to go out there and hear some of the talks, but the beginning of classes has kept me here today and tomorrow, and later in the week I need to make a short trip to Toronto. But luckily there are high-quality videos, and today Witten gave an interesting talk on What one can hope to prove about three-dimensional gauge theory. What struck me most though was how little we still know about even simple questions about 3 and 4 dimensional gauge theory. Witten expressed hope that studying these questions is something that will get re-invigorated and draw new attention. I hope he’s right. Also worth watching is Arthur Jaffe’s summary of the history and state of the art of constructive field theory.

Also in the category of talks that I’d love to hear, but they’re a bit too far afield to get to this week are Jacob Lurie’s Whittemore Lectures at Yale, dealing with the Siegel Mass Formula, Tamagawa numbers and Non-abelian duality. I fear they don’t have video, but if anyone knows of a source of information dealing with what Lurie talked about, I’d love to hear about it.

Science publishing impresario John Brockman’s Edge web-site each year runs a “Question of the year” feature, with short pieces from a wide range of people providing their answer to the question. The past few years I’ve passed on their invitation to submit something, but this year the question was one that I couldn’t resist. It was “What is your favorite deep, elegant, or beautiful explanation?” and you can read people’s answers here.

There are quite a few answers from various physicists, with General Relativity, inflation and the multiverse getting a lot of attention. To me though, the most satisfying answer to the question involves the remarkable role of symmetry principles at the foundations of both our everyday laws of mechanics and our deepest ideas about quantum mechanics. Far more so than in classical mechanics, in quantum mechanics these principles are built into the fundamental structure of the theory. This makes it clear why quantum mechanics works the way it does, and indicates that the structure of quantum mechanics is likely to always be fundamental to our understanding of the physical world, not some approximation like the classical picture. In addition, it links together fundamental physical principles and a fundamental set of ideas that occur throughout modern mathematics, a veritable grand unification of the two subjects.

Here’s what I sent in:

Any first course in physics teaches students that the basic quantities one uses to describe a physical system include energy, momentum, angular momentum and charge. What isn’t explained in such a course is the deep, elegant and beautiful reason why these are important quantities to consider, and why they satisfy conservation laws. It turns out that there’s a general principle at work: for any symmetry of a physical system, you can define an associated observable quantity that comes with a conservation law:

1. The symmetry of time translation gives energy
2. The symmetries of spatial translation give momentum
3. Rotational symmetry gives angular momentum
4. Phase transformation symmetry gives charge

In classical physics, a piece of mathematics known as Noether’s theorem (named after the mathematician Emmy Noether) associates such observable quantities to symmetries. The arguments involved are non-trivial, which is why one doesn’t see them in an elementary physics course. Remarkably, in quantum mechanics the analog of Noether’s theorem follows immediately from the very definition of what a quantum theory is. This definition is subtle and requires some mathematical sophistication, but once one has it in hand, it is obvious that symmetries are behind the basic observables. Here’s an outline of how this works, (maybe best skipped if you haven’t studied linear algebra…) Quantum mechanics describes the possible states of the world by vectors, and observable quantities by operators that act on these vectors (one can explicitly write these as matrices). A transformation on the state vectors coming from a symmetry of the world has the property of “unitarity”: it preserves lengths. Simple linear algebra shows that a matrix with this length-preserving property must come from exponentiating a matrix with the special property of being “self-adjoint” (the complex conjugate of the matrix is the transposed matrix). So, to any symmetry, one gets a self-adjoint operator called the “infinitesimal generator” of the symmetry and taking its exponential gives a symmetry transformation.

One of the most mysterious basic aspects of quantum mechanics is that observable quantities correspond precisely to such self-adjoint operators, so these infinitesimal generators are observables. Energy is the operator that infinitesimally generates time translations (this is one way of stating Schrodinger’s equation), momentum operators generate spatial translations, angular momentum operators generate rotations, and the charge operator generates phase transformations on the states.

The mathematics at work here is known as “representation theory”, which is a subject that shows up as a unifying principle throughout disparate area of mathematics, from geometry to number theory. This mysterious coherence between fundamental physics and mathematics is a fascinating phenomenon of great elegance and beauty, the depth of which we still have yet to sound.

Last October there was a conference held in Paris to celebrate the 200th anniversary of the birth of Galois. Some of the talks were quite interesting, giving an overview of the current state of areas of mathematics in which Galois theory plays a role, in particular the Langlands conjectures. The videos of the talks are now available, see here.

The closing talk, by Alain Connes, has a certain amount of wild-eyed speculation about the “cosmic Galois group”, of the sort that I believe has inspired Arkani-Hamed recently (see the previous posting).

Roy Lisker reports (here, here and here) about his visit to Paris to attend the conference, as well as take a look at Galois’s manuscripts.

Update: It appears the Connes video is no longer there. But on his web-site you can find the slides for the conference talk, as well as another talk about Galois for a more general audience.

Most of the lectures from this year’s Jerusalem Winter School in Theoretical Physics are now available online. David Gross was the main organizer, and the choice of topics reflects his point of view on what is interesting these days in theoretical physics. The landscape and anthropics were pretty much completely suppressed (although Michael Dine did manage to slip in a mention, his slides are here). The idea of “string phenomenology”, i.e. getting the standard model out of a unified string theory and saying something about particle physics has fallen by the way-side.

In his summary talk (given at the beginning of the conference), Gross described the same point of view he has promoted for many years now. He thinks something is missing in our understanding of string theory (“we don’t know what string theory is”) which will somehow fix the failure of ideas about string theory unification. This failure though doesn’t matter anyway, since he now claims that string theory = QFT, based on gauge-gravity duality, and thus “string theory cannot be killed”. It’s very unclear how this equivalence claim fits together with the “we don’t know what string theory is” claim, since we do know what QFT is (we have a very good understanding of the Standard Model). It seems that Gross would like to define away much of the troubles surrounding string theory.

One idea that Gross has favored for a very long time is that string theory is telling us that we must give up our usual notions of space and time, only recovering them in some limit. Two of the series of lectures at the school were about rather grandiose attempts to do something along these lines. There were three lectures by Erik Verlinde on his ideas about “emergent gravity”. I continue to not be able to make much sense of this program. He invokes the reasonable idea that gravity is an effective long-distance force due to unknown fundamental degrees of freedom, which may very well be true. But he doesn’t seem to have anything specific to say about the fundamental degrees of freedom strong enough to get anything new out of this. There were some ideas at the end about dark matter and dark energy, but it was unclear to me whether these go anywhere.

Much more interesting was Nima Arkani-Hamed’s series of lectures on Scattering Amplitudes and the Positive Grassmannian, which he said could have been titled “How you get spacetime from the permutation group”. The first lecture started with a philosophical introduction that he said he would limit to 5 minutes, but which went on for 40. Only the first two lectures are now online.

Unlike Verlinde’s ideas, here there are very specific calculations involved. The starting point is recent progress in understanding perturbative amplitudes for N=4 SYM (and for more general gauge theories). These involve working in twistor space, and more generally working with variables such that locality and unitarity are no longer manifest. The basic mathematics and geometrical principles at work are quite different than the standard formulation of gauge theories in terms of local variables and gauge symmetry. As usual, Arkani-Hamed makes wildly enthusiastic claims. In this case, he claims to have finally found a remarkable new understanding of the subject, based on some combinatorial objects and dramatic new mathematical ideas. He does note that this still hasn’t been written up, and that it’s the third time in the past year that he has thought he had things understood, with the last two times not working out.

I’m quite curious to see where this all goes, although I confess that I’m planning on waiting a while to try and follow the details, since this is clearly very complicated work in progress, and a 4th or 5th iteration of the fundamentals may very well be on its way over the next few months. Arkani-Hamed is talking to mathematicians, including his colleague Pierre Deligne at the IAS, and says that he is moving from the old-style of Atiyah mathematics to a new-style of Grothendieck-based mathematics. I’m not sure what this means, but suspect that Atiyah’s ideas are still in there (Grassmanians, twistors and toric varieties are subjects he has been very much involved with), and that the Grothendieck business may be an artifact of talking to Deligne, who comes from that tradition. Grothendieck was the master of generality, so his ideas can be applied to a very large fraction of mathematics.

Also lecturing on amplitudes, from a much more down-to-earth point of view, was Zvi Bern. For more from both Bern and Arkani-Hamed, see the program of the recent Amplitudes 2011 conference in Michigan. Slides from Arkani-Hamed’s talk give a better idea of what he is up to, and Bern’s slides contain his summary comments about the state of the subject. He emphasizes the connections between amplitude calculations and other fields and seems to be arguing for more attention to practical results, less to “symmetry, beauty and aesthetics”. He worries that while “Today our field is one of the hottest ones around”, the long-term future is less clear: “Is our field just another (albeit long lasting) fad?”, so attention to results relevant to the rest of physics is important for long-term health.

Lots more “Amplitudes” conferences on the way, including Amplitudes 2012 in March, a conference at the Newton Institute in April, and one at the IHES in December.