You may have read somewhere today that Columbia professor strips down to underwear in bizarre lesson to help baffled students learn quantum mechanics (first-hand sources here and here). That wasn’t me, but I have been talking to my class for the last couple weeks about quantizing fermionic variables, and some simple quantum mechanical examples of supersymmetry. Notes on the supersymmetry stuff are here, earlier notes on the fermionic version of quantization and what it has to do with Clifford algebras and spinors are on this page.
These notes are not quite finished, mainly because I’ve been trying to sort through the hairy issue of sign conventions that comes up when you start dealing with a Hermitian inner product on anti-commuting variables, something you need to do to get unitary representations. There’s a detailed treatise on the subject by Deligne and Freed, who are very smart and sensible, but I’d like to understand this better. The choices they make end up leading to odd self-adjoint operators having eigenvalues proportional to a square root of i, which is consistent, but not exactly intuitively clear. The best source for finding details of the mathematics used in SUSY is probably the IAS volume that Deligne/Freed is part of. The first part includes valiant efforts by Bernstein, Deligne, Freed, and Morgan to get the mathematics right, including the signs (they say “Writing this has been an absolute cauchemar de signes!”). One sign they get wrong is a typo on page 91 (equation 4.4.5).
The parallel stories of bosonic and fermionic oscillators are among the deepest things in theoretical physics, and involve just spectacularly intricate and deep mathematical ideas (symplectic geometry, rotation and spin groups, Heisenberg groups, the metaplectic representation, Clifford algebras and Weyl algebras, spinors, etc., etc…). I hope the course notes I’ve been writing give a little insight into this and the way Lie groups, Lie algebras, and their representations are involved. Generically, “supersymmetry” refers to generalizing the notion of a Lie algebra to include odd generators, and thus get a “super” Lie algebra, sometimes acting in an interesting way that mixes even and odd variables. In the notes I describe two very simple examples, showing how one gets a “square root” of the Hamiltonian operator.
There are all sorts of interesting structures one can get by looking for supersymmetrical versions of QFT, and the IAS volume describes a lot of them. One wonderful example is the N=2 susy gauge theory that gives a TQFT with observables four-manifold invariants. This is an unphysical theory, but tantalizingly close to physical theories. It involves a “twisting” mixing the space-time and internal symmetries which might be the sort of thing needed to avoid the problem of the kind of “superpartners” that is deadly for SUSY extensions of the standard model.
Perhaps the most compelling example though is the way the fact that the Dirac operator is a square root of the Laplacian can be thought of as an example of SUSY. This is one of the deepest ideas in mathematics, something whose implications I suspect we still don’t completely understand.