Advances in Geometry by Alexander Astashkevich (auth.), Jean-Luc Brylinski, Ranee

X (and hence a polynomial function on the fibers) is an order d symbol. The order d symbol map sends Vd(X) into Rd(T* X) where Vd(X) is the space of differential operators of order at most d and Rd(T* X) is the space of order d symbols. We say that ¢ E Rd(T* X) quantizes into D if D E Vd(X) has order d and the principal symbol of D is ¢. Unless X is smooth and affine, there is no guarantee that a given symbol will quantize.

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