Not Even Wrong

A few short items:

• No Higgs news on the LHC front, but on the BSM front today’s CERN talk Update on Searches for New Physics in CMS provides more evidence against the various exotic scenarios heavily advertised over the last twenty years. No sign of extra dimensions, SUSY, or other exotics. For more, see Resonaances. SUSY proponents now seem to be somewhere between the first stage of grief (“denial”) and the second (“anger”, see e.g. here) and on their way to the third (Bargaining – “I’ll do anything for a few more years”).
• From Sean Carroll’s Twitter feed I learned about this long description of a recent workshop bringing together philosophers and quantum field theorists. I guess my take on the “heuristic” vs. “mathematical” quantum field theory debate is that we need both, since there is still a huge amount we fundamentally don’t understand about QFT.
• For a finite number of degrees of freedom, quantum mechanics itself is, unlike QFT, rather well understood. For a nice recent review of some topics in the mathematics of quantization, see Ivan Todorov’s Quantization is a mystery.
• It’s not only string theorists producing over-hyped university press releases, there’s things like this. See more discussion here.

The latest CERN Courier book review section is out here. Besides a long review of Frank Close’s The Infinity Puzzle, there are some short reviews, including one for Stephen Heywood’s Symmetries and Conservation Laws in Particle Physics: An Introduction to Group Therapy for Particle Physics. That’s one I really want to see: I’m all for symmetries and conservation laws (see here), and Group Therapy for Particle Physics (at least for particle theorists) seems like an excellent idea.

This semester I’m not doing Group Therapy, but I am teaching group theory and representation theory. The class has started and I’m trying to write up lecture notes. One discouraging/encouraging thing is that looking around the web one finds several places other people have done this better, links are slowly getting added on the class web-page. The course is mainly aimed at mathematicians, hoping to provide our graduate students the background they need for several different areas, including number theory. It will however have a physics flavor, with more concentration on topics like spinors, geometric quantization, the Heisenberg algebra and oscillator representation than usual. The Dirac operator may even put in an appearance, we’ll see…

Update: Turns out there are more books on group therapy in particle physics. See here for J.F. Cornwell’s Group Therapy in Physics, Vol. 1. John Gribbin’s promotional In search of superstrings includes an appropriate appendix on Group Therapy for Beginners. Then there’s Terry Tomboulis’s Renormalization Group Therapy, which is something different.

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.