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.

This post imported from StackExchange MathOverflow at 2016-06-12 10:20 (UTC), posted by SE-user Pedro Lauridsen Ribeiro