Second-order term of the Fedosov quantised product

In Fedosov's version of quantisation of functions on a symplectic manifold, the product is given in terms of a symplectic connection. I have looked through Fedosov's book in deformation quantisation, and can't find the second order term for the product of two functions (which will involve the curvature), though I can find other expressions to second order. Am I being very unobservant (very likely), or if there really is no expression given there, is there another place where I can find it?

Application: The commutation relations of quantum mechanics arise in constructing quantum theory in classical geometry. Now suppose that we construct quantum mechanics in a geometry that is already noncommutative? In particular, if we have a first-order in some parameter noncommutative algebra of functions, can writing quantum mechanics give information on a second order deformation?

1 Answer

Indeed neither Fedosov's book nor his original paper (A Simple Geometrical Construction of Deformation Quantization, J. Diff. Geom. 40 (1993) 213-238) have an explicit formula for the second order term of his star product. To my knowledge, the first place where the recursive formulas for the terms in Fedosov's star product to all orders are written down explicitly is the paper by M. Bordemann and S. Waldmann, A Fedosov Star Product of the Wick Type for Kähler Manifolds, Lett. Math. Phys. 41 (1997) 243-253, arXiv:q-alg/9605012, particularly Theorems 3.1, 3.2 and 3.4 therein. Since these results are all due to Fedosov, only the proof of Theorem 3.4 is briefly sketched. However, if you want more details on how to get these formulae and are more or less comfortable reading German, I recommend the excellent book by S. Waldmann, Poisson-Geometrie und Deformationsquantisierung (Springer-Verlag, 2007), specially Section 6.4, pp. 444-473 and Exercise 6.12, pp. 484.

As for the applications you have in mind, I believe the answer is known if your noncommutative geometry comes from a strict deformation quantization à la Rieffel (e.g. the Moyal plane). In that case, one can use the action of translations to deform both the geometry and the (already noncommutative) C*-algebra of observables.

• Thanks! That is a useful reference. My German may not be up to the second one... – Edwin Beggs Jun 5 '16 at 8:43