Toeplitz operators provide a natural language with which to do geometric quantization. I don't want to really understand them, and I don't need them in full generality. I'm looking for some references that will provide formulas with which to compute products of Toeplitz operators, and specifically formulas for the asymptotics of such products as Planck's constant h → 0. I will give some background (also so the experts can correct any errors I might have), and then sketch the type of calculation I would like to perform.


A quantization of a commutative Poisson algebra A (a commutative algebra A is Poisson if it comes equipped with a bilinear bracket {,}: AAA that is a derivation in each coordinate (i.e. Leibniz rule) and a Lie bracket (i.e. antisymmetry and Jacobi)) is a smooth bundle of algebras over the real line (or some open subinterval thereof), with certain properties. Think of the bundle as a family Ah of (noncommutative) algebras, where h is my real variable. The conditions are that A0 = A; that we have given linear isomorphisms φh : A0Ah; and that limh→0 φh-1( h-1 [ φh(a) , φh(b) ] ) = {a,b}, where [,] is the commutator bracket and {,} is the Poisson bracket. Quantizations are not determined by their Poisson algebras: for example, take any smooth map RR that sends 0 to 0 and has derivative 1 at 0, and pull your bundle of algebras back along this map.

(A typical example of a quantization is the universal enveloping algebra of a finite-dimensional Lie algebra. Indeed, if G is a finite-dimensional Lie algebra (sorry, I don't know how to make fraktur letters here), then the symmetric algebra S_G_ is the algebra of polynomials on the dual space G*, and inherits a Poisson bracket by {f,g}(p) = <p,[df(p),dg(p)]>, where <,> is the pairing G*GR, and since G* is a vector space, I can canonically identify T*pG* = G; [,] is the Lie bracket on G. Anyway, let Gh be the Lie algebra G with the rescaled bracket [a,b]h = h[a,b]. Then the universal enveloping algebra U_G_h is a quantization of the Poisson algebra S_G_.)

((Another parenthetical: most people actually do quantizations of C, and then ask that complex conjugation deform well. They also sometimes decorate their formulas with _i_s. This has to do with the ability to find good representations noncommutative algebras in terms of bounded operators on Hilbert spaces. I'll skip such parts of the definition.))

Quantizations were originally invented to describe quantum mechanics on Rn, and this is the situation I'm trying to understand. Let A be an algebra of functions on the cotangent bundle T*Rn = R2n. If we take the algebra of polynomials on R2n, it is generated by {p1,..., pn, q1,..., qn}, and the Poisson bracket is defined by {pi,qj} = δij, the Kronecker delta. Here are two quantizations:

  • The QP quantization. For h nonzero, let Ah be the algebra of differential operators on the polynomial ring R[q1,..., qn]. Construct the maps φh by sending qiA to (multiplication by) q in Ah, and send piA to the partial derivative operator δi. For a more complicated monomial, first write it with all qs to the left and all ps to the right, and then apply the above maps letter-by-letter.
  • The Z Z-bar quantization. Complexify A, and change variables so that zj = qj + ipj and wj = qj - ipj. Write every monomial with the zs to the left and the ws to the right, and let the monomials act on the polynomial algebra C[z1,..., zn] analogous to in the QP case. I believe that you can reincorporate by real structure by recording the action of "complex conjugation".

So far, I've been playing with the QP quantization. This has a natural extension to the algebra of functions on T*Rn that are polynomial in the p variables but smooth in the q variables (i.e. CRnR[p1,..., pn]).

What I've been told is that Toeplitz operators quantize (at least) the algebra of analytic functions on T*Rn, (actually, they work much more generally to quantize symplectic manifolds, but I don't need them to), and that the quantization corresponds to the Z Z-bar quantization above.

The calculation I'd like to do

Starting with a Lagrangian on Rn, it's relatively straightforward to write down the formal Feynman path integral (a power series in h), and I know some things about the derivatives of this formal power series with respect to the physical variables. This series is a putative solution to Schroedinger's equation. When the Lagrangian is quadratic in velocity, I can compute the Hamiltonian explicitly, and sure enough Schroedinger's equation is satisfied. When the Lagrangian is not quadratic, the Hamiltonian is still well-defined, but I don't have explicit enough formulas, and in general it is not polynomial in momentum. I do have various differential equations satisfied by the Hamiltonian, and I'm hoping that with these and the right formulas for the asymptotics of products of Toeplitz operators, I can check the Schroedinger equation (asymptotically) without knowing precisely what the Schroedinger operator is.

  • $\begingroup$ @Scott: I much prefer posts with lots of background --- I vote up the ones that have it. It also helps me think of exactly my question if I go through the background right before asking it. $\endgroup$ – Theo Johnson-Freyd Oct 25 '09 at 5:21

You probably need some conditions to restrict the kind of Schroedinger operators you want to work with. Have a look at the work of Charles and Vu-Ngoc on Toeplitz operators and the semiclassical limit, in particular theorem 1 of this http://people.math.jussieu.fr/~charles/Articles/BerToep.pdf and section 1.3 of this http://people.math.jussieu.fr/~charles/Articles/Half1.pdf and for spectral asymptotics in more concrete examples (a non-degenerate potential well) see this http://hal.archives-ouvertes.fr/docs/00/06/73/18/PDF/birkhoff.pdf


A good reference for your question is the article: arXiv-math-ph/0405065v1 "Quantization Methods: A Guide for Physicists and Analysts". Chapter 6 treats the case of R^2n.


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