Conformally equivariant quantization : existence and uniqueness

Christian Duval; Pierre Lecomte; Valentin Ovsienko

Annales de l'institut Fourier (1999)

  • Volume: 49, Issue: 6, page 1999-2029
  • ISSN: 0373-0956

Abstract

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We prove the existence and the uniqueness of a conformally equivariant symbol calculus and quantization on any conformally flat pseudo-riemannian manifold ( M , g ) . 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 o ( p + 1 , q + 1 ) where p + q = dim ( M ) . This quantization exists for generic values of the weights of the tensor densities and we 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.

How to cite

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Duval, Christian, Lecomte, Pierre, and Ovsienko, Valentin. "Conformally equivariant quantization : existence and uniqueness." Annales de l'institut Fourier 49.6 (1999): 1999-2029. <http://eudml.org/doc/75408>.

@article{Duval1999,
abstract = {We prove the existence and the uniqueness of a conformally equivariant symbol calculus and quantization on any conformally flat pseudo-riemannian manifold $(M,g)$. 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 $\{\rm o\}(p+1,q+1)$ where $p+q=\dim (M)$. This quantization exists for generic values of the weights of the tensor densities and we 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.},
author = {Duval, Christian, Lecomte, Pierre, Ovsienko, Valentin},
journal = {Annales de l'institut Fourier},
keywords = {quantisation; conformal structures; modules of differential operators; tensor densities; Casimir operators; star products; half-densities},
language = {eng},
number = {6},
pages = {1999-2029},
publisher = {Association des Annales de l'Institut Fourier},
title = {Conformally equivariant quantization : existence and uniqueness},
url = {http://eudml.org/doc/75408},
volume = {49},
year = {1999},
}

TY - JOUR
AU - Duval, Christian
AU - Lecomte, Pierre
AU - Ovsienko, Valentin
TI - Conformally equivariant quantization : existence and uniqueness
JO - Annales de l'institut Fourier
PY - 1999
PB - Association des Annales de l'Institut Fourier
VL - 49
IS - 6
SP - 1999
EP - 2029
AB - We prove the existence and the uniqueness of a conformally equivariant symbol calculus and quantization on any conformally flat pseudo-riemannian manifold $(M,g)$. 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 ${\rm o}(p+1,q+1)$ where $p+q=\dim (M)$. This quantization exists for generic values of the weights of the tensor densities and we 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.
LA - eng
KW - quantisation; conformal structures; modules of differential operators; tensor densities; Casimir operators; star products; half-densities
UR - http://eudml.org/doc/75408
ER -

References

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