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In "Quantum field theory and the Jones polynomial" (Comm. Math. Phys. 1989 vol. 121 (3) pp. 351-399), Witten writes:

A representation Ri of a group G should be seen as a quantum object. This representation should be obtained by quantizing a classical theory. The Borel-Weil-Bott theorem gives a canonical way to exhibit for every representation R of a compact group G a problem in classical physics, with G symmetry, such that the quantization of this classical problem gives back R as the quantum Hilbert space. One introduces the "flag manifold" G/T, with T being a maximal torus in G, and for each representation R one introduces a symplectic structure ωR on G/T, such that the quantization of the classical phase space G/T, with the symplectic structure ωR, gives back the representation R. Many aspects of representation theory find natural explanations by thus regarding representations of groups as quantum objects that are obtained by quantization of classical physics. [page 372; emphasis added]

I'm fascinated by this idea — I haven't seen it before, but it seems natural, in that classical objects should not be linear, whereas quantum objects should be. I'm most interested in the last sentence: what examples can y'all come up with of representation-theoretic facts that can be "explained" by "physics" on G/T? (Besides, of course, Witten's application in the paper I quoted from.)

More generally, I've read the Wikipedia discussion of the Borel-Weil-Bott theorem, and done some random googling, but I haven't found an elementary description of the symplectic structure Witten refers to. Anyone want to pedantically spell out Witten's comment, please?

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I have a more general comment about why representation theory makes sense as study of quantum objects. How do you explain representations to people who don't know them? Well, one way would be to take a group of rotations of our world (which is deep down a 2-cover of SO(1;3)) as say: hey, a representation is a linear object that knows how to transform under this. Which is, essentially, a definition of a particle in quantum mechanics. –  Ilya Nikokoshev Nov 5 '09 at 20:26
    
The answer to this question has practical engineering implications too; per the question "[How does one geometrically quantize the Bloch equations?][1]" on Theoretical Physics StackExchange. [1]: theoreticalphysics.stackexchange.com/questions/551/… –  John Sidles Nov 21 '11 at 22:16

6 Answers 6

up vote 13 down vote accepted

Look at the orbits of G on g* the dual of the Lie algebra. These 'coadjoint orbits' have a canonical symplectic structure. Each of these orbits intersects the positive Weyl chamber exactly once; consider those intersecting it at a 'positive weight' (i.e. the the elements of g* that lift to characters on T, the maximal torus). The positive weights exactly classify the irreducible representations. At the same time, we can build a line bundle over the corresponding coadjoint orbit so that the symplectic form realises the Chern class. Sections of this bundle are automatically a representation of G, and the Borel-Weil-Bott theorem, in one of its friendlier guises, says that this is the representation you expect -- the one with the highest weight we started with.

Generically, a coadjoint orbit is just G/T. When it isn't, it's a deeper quotient, but you can pull back the canonical symplectic structure to G/T, and still do everything there. This is the symplectic structure ωR Witten is referring to.

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For actual applications, there are some; for example, the moment map image of G/B for the torus action is the convex hull of the weight diagram, and the weight multiplicities are approximated by the volume of fibers.

There's a similar business with tensor product multiplicities, where you look at the moment map image for G acting on the product of two coadjoint orbits. This one way of thinking about convexity of the support of tensor product multiplicities.

Also, if you look at the dimensions of the representations $V_{n\lambda}$ for some fixed lambda, you get a polynomial of order the dimension of the corresponding coadjoint orbit, and leading term its symplectic volume.

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A belated response, but appropriate nevertheless, I think: the locus classicus for this general theme about the deep connection between quantum particles and representation theory is given by E Wigner, "On unitary representations of the inhomogeneous Lorentz group," Annals of Mathematics 40 (1): 149–204. Wigner is the one who gave us Wigner's classification -- viz., the classification of most nonnegative energy irreducible unitary representations of the Poincare group. Since Wigner, it has become quite standard simply to define elementary particles as these irreducible representations.

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McGovern applies Borel-Weil-Bott to obtain a branching rule from G=Spin(7,C) to H=G2. The main point here is that there are parabolics P \subset G and Q \subset H such that G\P=H\Q. Then, via Borel-Weil-Bott, he reduces the branching problem to one between the Levi factors of these parabolics. The same idea applies to branching from SL(2n,C) to Sp(2n,C). The relevant paper is "McGovern, William M. A branching law for Spin(7,C)→G2 and its applications to unipotent representations. J. Algebra 130 (1990), no. 1, 166--175."

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I'll just elaborate for you the example mentioned by Scott: In the stereographic projection coordinates of S2, the symplectic 2-form is given by:

ω = dz^dz̄/(1+zz̄)2

Classically, one can construct three hamiltonian functions representing the generators of the Lie algebra su(2) which constitute of a subalgebra of the Poisson algebra corresponding to ω

TX = (z+z̄)/(1+zz̄)

TY = -i(z-z̄)/(1+zz̄)

TZ = (1-zz̄)/(1+zz̄)

Quantum mechanically, the representation of spin j of SU(2) is realized on a (reproducing kernel) Hilbert space generated by holomorphic sections of a line bundle whose expressions in the stereographic coordinates are 1, z, . . . , z2j. The su(2) Lie algebra can be realized on this space by means of the differential operators:

sX = -(1-z2)∂/∂z + 2jz

sY = -i(1+z2)∂/∂z - 2ijz

sZ = -2z∂/∂z -2j

Theories of geometric quantization offer systematic methods to make these constructions for a general compact Lie group for a concrete realization of the Bore-Weil-Bott theorem.

I would like to mention that many representation theoretical computations can be made using this realization of the representation theory of compact Lie groups. Also, this realization is connected to Perelomov's generalized coherent states.

There are some generalizations to representations on non-compact Lie groups. Also, the Borel-Weil-Bott theorem can be connected in many ways to supersymmetry.

The "linearization' of the classical mechanics is achievd through the realization of the quantum Hilbert space by sections of a "line" bundle. These sections also relates this realization to projective geometry via Kodaira's embedding thorem.

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The most concrete example of this is just SU(2)/S^1 = S^2. Depending on which point in su(2)* this S^1 is stabilising, the S^2 has various different volumes. When that volume is an integer k, there's an associated line bundle O(k), sections of which are the k+1 dimensional representation of SU(2).

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It is very important to point out that you have to work in the holomorphic world. SU(2)/S^1 is not just S^2 but it is CP^1. The line bundles O(k) are holomorphic line bundles, and we want to take holomorphic sections. –  Kevin H. Lin Oct 26 '09 at 10:07
    
Thanks Kevin, I'm not much of a geometer and tend to instinctively write the most familiar name for the topological space... :-) –  Scott Morrison Oct 26 '09 at 15:54

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