User peter woit - MathOverflow

Answer by Peter Woit for Clifford Lie Algebras

2013-01-24T01:31:29Z

I don't know anyone else who calls this the "Clifford Lie Algebra". It is just one of the basic applications of Clifford algebras. Given the Clifford algebra of a quadratic form, the quadratic elements of the Clifford algebra give you the Lie algebra of the orthogonal group of that quadratic form.

There are many places to read about this, one of them would be Chapter 1.6 of "Spin Geometry" by Lawson and Michelson. I've written up some notes for a graduate course that include this, see here:

http://www.math.columbia.edu/%7Ewoit/LieGroups-2012/cliffalgsandspingroups.pdf

Answer by Peter Woit for Where does a math person go to learn quantum mechanics?

2012-12-16T20:50:20Z

I've just finished teaching the first semester of a year-long "Quantum Mechanics for Mathematicians" course. Some of the references I found most useful are

A good, clear, physics textbook. Shankar's "Principles of Quantum Mechanics" that many have mentioned fits the bill.
Faddeev and Yakubovskii, "Lectures on Quantum Mechanics for Mathematics Students" is short and to the point. Takhtajan's "Quantum Mechanics for Mathematicians" is at a higher level that I was aiming for, but quite good.
The first 60 or so pages of Folland's "Quantum Field Theory" are an excellent introduction to physics in general and QM in particular (and the rest of the book is a great QFT textbook).

Finally, I should point out that I've put up my course notes, which try to cover basic QM from a representation theory point of view, at the lowest level possible, they're here.

Answer by Peter Woit for The equivariant index of Dirac operator

2012-05-14T19:20:48Z

To see why the spinor bundle is the bundle $\Omega^{0,* }\otimes K^{1/2}$, you need to understand the relation between the spinor representation $S$ of $Spin(2n)$ and the exterior algebra representations $\Lambda^* (\mathbf C^n)$ of $U(n)$.

If you choose an orthogonal complex structure $J$ on $\mathbf R^{2n}$, this picks out a subgroup $U(n)\subset SO(2n)$, with double-cover $\widetilde{U(n)}\subset Spin(2n)$. The notion of a "square root of the top exterior power" $(\Lambda^n(\mathbf C^n))^{1/2}$ makes sense as a $\widetilde{U(n)}$ representation. One finds (by computing characters, or from your favorite construction of the spinor representation) that, restricted to $\widetilde{U(n)}$, the spinor representation $S$ of $Spin(2n)$ is $\Lambda^* (\mathbf C^n)\otimes (\Lambda^n(\mathbf C^n))^{-1/2}$

There's more about this in some of my class notes here

http://www.math.columbia.edu/~woit/LieGroups-2012/spinors.pdf

I learned this first from some beautiful lectures of Atiyah which explicitly discuss the Dirac/Dolbeault operator relation, see

"Classical groups and classical differential operators on manifolds" CIME, Varenna (1975).

Answer by Peter Woit for spin structures on full flag manifolds

2012-01-26T17:01:42Z

$G/T$ is a co-adjoint orbit in $\mathfrak g^*$ . The normal bundle to the inclusion $G/T\rightarrow \mathfrak g^*$ is trivial, so the tangent bundle of $G/T$ is stably trivial. This implies its Stiefel-Whitney and Pontryagin classes vanish.

(Argument stolen from Dan Freed "Flag Manifolds and Infinite Dimensional Geometry" in MSRI proceedings volume "Infinite Dimensional Groups with Applications".)

Answer by Peter Woit for What is significant about the half-sum of positive roots?

2011-11-08T04:02:47Z

One answer that hasn't appeared here yet is that $\rho$ is the highest weight of a spinor representation. Instead of the Dolbeault operator on the flag manifold, one can work with the Dirac operator. This leads to the subject of "Dirac Cohomology", see the book "Dirac Operators in Representation Theory". Here, instead of working with the Universal Enveloping Algebra and modules for it, one works with the tensor product of this with a Clifford algebra. Irreducible modules then acquire a spinor representation factor. This explains nicely the shift by $\rho$.

Hopefully this spring I'll finish this, which explains all this in more detail:

http://www.math.columbia.edu/~woit/brstdirac.pdf

Answer by Peter Woit for Suitable references for the the Stone-von Neumann Theorem

2011-05-17T13:53:42Z

An excellent reference explaining the history and significance of the Stone von Neumann theorem is Jonathan Rosenberg's

"A Selective History of the Stone-von Neumann Theorem"

available at

http://www-users.math.umd.edu/~jmr/StoneVNart.pdf

Note that one doesn't actually have uniqueness of representations of the position and momentum operators (since they're unbounded). The theorem applies to their exponentiated versions, which give the Heisenberg (mathematician's name) or Weyl (physicist's name) group.

Answer by Peter Woit for Standard model of particle physics for mathematicians

2011-03-12T22:33:09Z

The Folland book mentioned here is quite good. One of the most straight-forward physics references might be Pierre Ramond's "Field Theory: A Modern Primer", but it's still a long ways from mathematical rigor. Some comments about the other books and topics discussed here:

Weinberg's books are very good in their own way, but not really appropriate for mathematicians. The first one develops QFT not so much in terms of fundamental objects, but as a phenomenological framework forced upon us by principles such as special relativity and locality. The second one does gauge theory without using geometry, or coordinate-invariant notation, which is not a great idea for mathematicians. The third one is just about SUSY, concentrating on the parts of the subject not of much mathematical interest (the IAS volumes do the opposite).

About the IAS volumes, one should keep in mind that the main point of that exercise was to try to explain to mathematicians Seiberg-Witten theory as understood by physicists in terms of N=2 supersymmetric QFT. This has nothing to do with the Standard Model, and from what I remember the Standard Model doesn't appear in those volumes. They do contain a truly spectacularly good set of lectures by Witten on QFT (but not written up by him...), aimed at getting to the Seiberg-Witten story. This involves some heavy-duty use of non-perturbative supersymmetric quantum field theory, of the sort that is of mathematical interest in building TQFTs.

Besides not explaining the Standard Model, I don't think the IAS lectures really explain the use of supersymmetry to extend the Standard Model (the MSSM "minimal supersymmetric standard model"). This is a subject that has always been heavily advertised without much explanation of its significant problems, one of which is an extra 120 or so parameters. Initial results from the LHC rule out nearly half the most popular region in parameter space, chosen for simplicity and assuming that supersymmetry can be used to solve certain problems (dark matter particle, anomaly in measurement of muon magnetic moment). This still leaves the other half, as well as a lot of other less popular regions of parameter space. Over the next year or two I believe we'll see increasingly large regions of parameter space ruled out, but there is no way the LHC can rule out all of it. All it can do is change somewhat how physicists evaluate the likelihood of nature being described by conventional supersymmetric extensions of the Standard Model, a process which has started and will continue.

Answer by Peter Woit for Witten's QFT and Jones Poly paper

2011-02-21T21:14:24Z

As Konrad Waldorf noted, in this case G-bundles are trivializable (since $\pi_2(G)$ is trivial). So gauge transformations are just maps $$\phi:M\rightarrow G$$

and these have a homotopy invariant that can be non-trivial, the degree of the map. One way to compute this is as $$\int_M \phi^*\omega_3$$

where $\omega_3$ is a generator of $H^3(G)$. Or, as usual for a degree, just pick an element of G, and count points (with sign) in the inverse image.

Answer by Peter Woit for The influence of string theory on mathematics for philosophers.

2010-12-18T20:16:29Z

The Jaffe-Quinn manifesto really had little to do with string theory, but a lot to do with topological quantum field theory, especially 3d tqft. I remember Frank Quinn talking about this at length during a hike at the 1991 Park City summer school. He was lecturing there on topological qft, see

"Lectures on Axiomatic Topological Quantum Field Theory" in "Geometry and Quantum Field Theory, IAS/Park City Mathematics Series, Volume 1", edited by Daniel Freed and Karen Uhlenbeck.

The sort of thing that was worrying Quinn was:

Witten's great paper on "Supersymmetry and Morse Theory", which was published in a mathematics journal, the Journal of Differential Geometry.
Witten's Fields medal winning work on the Jones polynomial and Chern-Simons theory.

Quinn explained that at the beginning of his career he had been heavily influenced by the work of Thurston and Sullivan, but found that trying to emulate them had led him to lose track of what he precisely understood and what he didn't, requiring a painful period of getting back to a more rigorous way of working. He was worried that losing the distinction between works like Witten's and truly rigorous work would lead others to the problematic situation he had found himself in as a young mathematician. In the end, I think Atiyah's response won the day: he argued that mathematicians were fully capable of protecting their virtue while interacting with physicists. Shortly after this exchange, those topologists in the math community who were skeptical about the importance of what Witten was bringing to mathematics were conclusively won over by the Seiberg-Witten equations.

But the example set by Quinn of how to do TQFT in the end has largely won out. There was an attempt to teach mathematicians the actual QFT behind Seiberg-Witten at the IAS in 96/97, but I don't think it was very successful. These days both TQFT and the Seiberg-Witten equations remain very important ideas in topology, but they're pursued with conventional standards of rigor. Mathematicians have gotten used to taking physicist's QFT arguments and extracting and generalizing those parts that can be made rigorous and fit into the evolving mathematical tradition.

As others have mentioned, for the case of string theory, mirror symmetry is probably the best example of an idea coming out of it that has had a huge influence in mathematics. Yau's recent popular book "The Shape of Inner Space" contains lots of other examples of the interaction of math and physics surrounding Calabi-Yau manifolds.

Comment by Peter Woit

2012-07-08T18:04:44Z

There's a relatively straight-forward argument in Section I.6 of Stephen Kudla's "Notes on the Local Theta Correspondence", available at www.math.toronto.edu/~skudla/castle.pdf This may just be a restatement of the Rao argument. As mentioned elsewhere, it comes down to invoking non-triviality of the Hilbert symbol

Comment by Peter Woit

2011-06-01T20:06:13Z

Photons come from quantization of the infinite dimensional space of connections on a U(1) bundle unrelated to the Riemannian geometry. Matter particle come from quantization of a space of sections of a vector bundle, which may be the spin bundle tensored by one unrelated to the Riemannian geometry. I fear though that this question doesn't belong here, it's ill-defined and based on too many basic confusions.

Comment by Peter Woit

2011-06-01T19:58:43Z

On an n-dim Riemannian manifold, "tensor fields" are sections of the vector bundles constructed using the frame bundle and tensor products of R^n. Restricting attention to orthonormal frames, the frame bundle is a principal SO(n) bundle, with a distinguished (Levi-Civita) connection. Spinor fields come from the same construction, using a Spin(n) double cover the SO(n) bundle and replacing representations on tensor products of R^n by the spinor representation. Connections determine horizontal subspaces on the tangent bundle to the frame bundle, here take a lift to the double cover.

Comment by Peter Woit

2011-02-22T03:28:56Z

As Paul mentions, it's only for the case G=SU(2) that this is just the degree of the map.

Comment by Peter Woit

2011-01-25T03:35:01Z

This story clearly comes in many variants. In my favorite, the Langlands retort to "do you mind if I ask you a stupid question" is "That's two already."

Comment by Peter Woit

2011-01-07T01:25:19Z

Oops, in the above comment, the second occurrence of "Weyl denominator formula" should be "Weyl integral formula". Also, I just remembered that there's a nice discussion of the Atiyah-Bott fixed point calculation I mentioned, see section 14.2 of Pressley and Segal's Loop Groups.

Comment by Peter Woit

2011-01-06T23:50:49Z

The first formula is the Weyl dimension formula, the second the Weyl denominator formula. They both follow easily from the Weyl character formula, and I wrote up some notes that give the standard somewhat geometric proof of this using the Weyl dimension formula here: math.columbia.edu/~woit/notes12.pdf A more sophisticated geometrical way to get the denominator formula would be to apply the Atiyah-Bott fixed point formula to the index-theory version of Borel-Weil-Bott for the trivial representation, as an index of an operator on the flag manifold.