## Conformally equivariant quantization: Existence and uniqueness

Citations: | 42 - 5 self |

### BibTeX

@TECHREPORT{Duval_conformallyequivariant,

author = {C. Duval and P. Lecomte and V. Ovsienko and In Memory and Moshe Flato and André Lichnerowicz},

title = {Conformally equivariant quantization: Existence and uniqueness},

institution = {},

year = {}

}

### OpenURL

### 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 o(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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Citation Context ...[G, [D, C]] + ) , (4.4) C = E 2 − 1 2 [R, T]+ (4.5) is the Casimir element of sl(2, R), (ii) if n ≥ 3, one has Z = {0}. (4.6) 15This theorem is a generalization of the celebrated Brauer-Weyl Theorem =-=[27]-=- (see also [16, 9]). Let us mention that we will, actually, need considering invariant operators with respect to homotheties generated by X0 (see (2.14) and (2.15)) inside e(p, q) ! . We readily have ... |

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Citation Context ...rsal. One guiding principle for the search of a quantization procedure is to impose further coherence with some natural symmetry of phase space. This constitutes the foundations of the “orbit method” =-=[14]-=-, geometric quantization [17, 26, 15] in the presence of symmetries, Moyal-Weyl quantization (see, e.g., [10]) defined by requiring invariance with respect to the linear symplectic group Sp(2n, R) of ... |

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Citation Context ...linear operation (depending on λ) ∗λ;� : S ⊗ S → S[[i�]] (3.4) 10such that Qλ;�(P ∗λ;� Q) = Qλ;�(P) ◦ Qλ;�(Q). (3.5) Recall that an associative operation ∗� : S ⊗ S → S[[i�]] is called a starproduct =-=[1, 7, 10]-=- if it is of the form P ∗� Q = PQ + i� 2 {P, Q} + O(�2 ) (3.6) where {·, ·} stands for the Poisson bracket on T ∗ M, and is given by bi-differential operators at each order in �. Theorem 3.4. The asso... |

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(Show Context)
Citation Context ...or the search of a quantization procedure is to impose further coherence with some natural symmetry of phase space. This constitutes the foundations of the “orbit method” [14], geometric quantization =-=[17, 26, 15]-=- in the presence of symmetries, Moyal-Weyl quantization (see, e.g., [10]) defined by requiring invariance with respect to the linear symplectic group Sp(2n, R) of R 2n . 1.1 Equivariant quantization p... |

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Citation Context ... In our framework, the Weyl quantization map, QWeyl, retains the very elegant form QWeyl ( i� = exp 2 D ) = Id + i� �2 D − 2 8 D2 + O(� 3 ) where the divergence operator D is as in (4.2). (See, e.g., =-=[10]-=- p. 87.) (6.5) 6.2 Study of the resonant modules For the sake of completeness, let us study in some more detail the particular modules of differential operators corresponding to the resonances (6.1). ... |

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Citation Context ...ular case n = 1, both conformal and projective structures coincide. We refer to [6] for a thorough study of sl(2, R)-equivariant quantization and of the corresponding invariant star-product. See also =-=[28]-=- for a classic monography on the structures of sl(2, R)-module on the space of differential operators on the real line. There exist various approaches to the quantization problem, however, our viewpoi... |

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Citation Context ...ce condition with respect to a (maximal) group, G, of symmetry in the context of deformation quantization. Now, G-equivariance is the root of geometric quantization [26, 17, 15], Berezin quantization =-=[2, 3]-=-, etc., but it seems to constitute a fairly new approach in the framework of symbol calculus, deformation theory and semi-classical approximations dealt with in this work. 1.2 Quantizing equivariantly... |

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Citation Context ... first step towards their equivariant quantization was taken in [9] in the case of second order operators. In the particular case n = 1, both conformal and projective structures coincide. We refer to =-=[6]-=- for a thorough study of sl(2, R)-equivariant quantization and of the corresponding invariant star-product. See also [28] for a classic monography on the structures of sl(2, R)-module on the space of ... |

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Citation Context ...al Laplacian), see [4], arose from the quantization of the same geodesic flow in a resonant case (recall that this operator is of special importance in field theory in a curved space-time, see, e.g., =-=[24]-=-). Let us finally mention that this work opens up a number of original questions under current investigation, viz the determination of the o(p + 1, q + 1)-invariant star-product and multi-dimensional ... |

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Citation Context ...C. The space Dλ,µ of linear differential operators A : Fλ → Fµ (2.1) from λ-densities to µ-densities on M is naturally a Diff(M)- and Vect(M)-module. These modules have been studied and classified in =-=[8, 21, 22, 12, 11, 23, 9, 19]-=-. ⊂ · · ·, where the module of There is a filtration D0 λ,µ ⊂ D1 λ,µ ⊂ · · · ⊂ Dk λ,µ zero-order operators D0 λ,µ ∼ = Fµ−λ consists of multiplication by (µ − λ)-densities. The higher-order modules are... |

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Citation Context ...C. The space Dλ,µ of linear differential operators A : Fλ → Fµ (2.1) from λ-densities to µ-densities on M is naturally a Diff(M)- and Vect(M)-module. These modules have been studied and classified in =-=[8, 21, 22, 12, 11, 23, 9, 19]-=-. ⊂ · · ·, where the module of There is a filtration D0 λ,µ ⊂ D1 λ,µ ⊂ · · · ⊂ Dk λ,µ zero-order operators D0 λ,µ ∼ = Fµ−λ consists of multiplication by (µ − λ)-densities. The higher-order modules are... |

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Citation Context ...or the search of a quantization procedure is to impose further coherence with some natural symmetry of phase space. This constitutes the foundations of the “orbit method” [14], geometric quantization =-=[17, 26, 15]-=- in the presence of symmetries, Moyal-Weyl quantization (see, e.g., [10]) defined by requiring invariance with respect to the linear symplectic group Sp(2n, R) of R 2n . 1.1 Equivariant quantization p... |

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Citation Context ... denoted by Sδ(M). Let us emphasize that equivariant quantization of G-structures has already been carried out in the case of projective structures, i.e. SL(n + 1, R)-structures, in the recent papers =-=[22, 20]-=-. As for conformal structures, a first step towards their equivariant quantization was taken in [9] in the case of second order operators. In the particular case n = 1, both conformal and projective s... |

8 |
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Citation Context ... denoted by Sδ(M). Let us emphasize that equivariant quantization of G-structures has already been carried out in the case of projective structures, i.e. SL(n + 1, R)-structures, in the recent papers =-=[22, 20]-=-. As for conformal structures, a first step towards their equivariant quantization was taken in [9] in the case of second order operators. In the particular case n = 1, both conformal and projective s... |

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Citation Context |

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Citation Context ...ra o(p + 1, q + 1) is maximal in the Lie algebra VectPol(R n ) of polynomial vector fields in the following sense: any bigger subalgebra of VectPol(R n ) necessarily coincides with VectPol(R n ). See =-=[5]-=- for a simple proof. The uniqueness and the canonical character of our quantization procedure definitely originates from this maximality property of o(p+1, q+1). See also [25] for a classification of ... |

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Citation Context ...ed out in the case of projective structures, i.e. SL(n + 1, R)-structures, in the recent papers [22, 20]. As for conformal structures, a first step towards their equivariant quantization was taken in =-=[9]-=- in the case of second order operators. In the particular case n = 1, both conformal and projective structures coincide. We refer to [6] for a thorough study of sl(2, R)-equivariant quantization and o... |

5 |
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Citation Context |

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5 |
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Citation Context ...s with VectPol(R n ). See [5] for a simple proof. The uniqueness and the canonical character of our quantization procedure definitely originates from this maximality property of o(p+1, q+1). See also =-=[25]-=- for a classification of a class of maximal Lie subalgebras of VectPol(R n ). Remark 2.3. From now on, we will use local coordinate systems adapted to the flat conformal structure on M in which the ge... |

2 |
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Citation Context ...f Enddiff(C[x 1 , . . ., x n , ξ1, . . .,ξn]) of those operators that commute with g. This classical notion of commutant has first been considered in the context of differential operators by Kirillov =-=[16]-=-. 4.1 Algebra of Euclidean invariants To work out a conformally equivariant quantization map, we need to study first equivariance with respect to the Euclidean subalgebra e(p, q). To this end, we will... |

1 |
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(Show Context)
Citation Context ...or the search of a quantization procedure is to impose further coherence with some natural symmetry of phase space. This constitutes the foundations of the “orbit method” [14], geometric quantization =-=[17, 26, 15]-=- in the presence of symmetries, Moyal-Weyl quantization (see, e.g., [10]) defined by requiring invariance with respect to the linear symplectic group Sp(2n, R) of R 2n . 1.1 Equivariant quantization p... |