Conformally equivariant quantization: Existence and uniqueness

Details Conformally equivariant quantization: Existence and uniqueness Conformally equivariant quantization: Existence and uniqueness

1999-02-04

arxiv.org/abs/math/9902032v2

Mathematics

Differential Geometry

LaTeX document, 32 pages; improved version

Scientific paper

We prove the existence and the uniqueness of a conformally equivariant symbol calculus and quantization on any conformally flat pseudo-Riemannian manifold $(M,\rg)$. In other words, we establish a canonical isomorphism between the spaces of polynomials on $T^*M$ and of differential operators on tensor densities over $M$, both viewed as modules over the Lie algebra $\so(p+1,q+1)$ where $p+q=\dim(M)$. This quantization exists for generic values of the weights of the tensor densities and compute the critical values of the weights yielding obstructions to the existence of such an isomorphism. In the particular case of half-densities, we obtain a conformally invariant star-product.

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