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Projectively equivariant quantizations over the superspace R^{p|q} Radoux, Fabian Conference (2010, November 03) Detailed reference viewed: 6 (4 ULg)Equivariant quantization of orbifolds ; Radoux, Fabian ; in Journal of Geometry & Physics (2010), 60(9), 1103-1111 Equivariant quantization is a new theory that highlights the role of symmetries in the relationship between classical and quantum dynamical systems. These symmetries are also one of the reasons for the ... [more ▼] Equivariant quantization is a new theory that highlights the role of symmetries in the relationship between classical and quantum dynamical systems. These symmetries are also one of the reasons for the recent interest in quantization of singular spaces, orbifolds, stratified spaces, etc. In this work, we prove the existence of an equivariant quantization for orbifolds. Our construction combines an appropriate desingularization of any Riemannian orbifold by a foliated smooth manifold, with the foliated equivariant quantization that we built in Poncin et al. (2009) [19]. Further, we suggest definitions of the common geometric objects on orbifolds, which capture the nature of these spaces and guarantee, together with the properties of the mentioned foliated resolution, the needed correspondences between singular objects of the orbifold and the respective foliated objects of its desingularization. [less ▲] Detailed reference viewed: 62 (23 ULg)Existence of natural and conformally invariant quantizations of arbitrary symbols Mathonet, Pierre ; Radoux, Fabian in Journal of Nonlinear Mathematical Physics (2010), 17 A quantization can be seen as a way to construct a diﬀerential operator with prescribed principal symbol. The map from the space of symbols to the space of diﬀerential operators is moreover required to be ... [more ▼] A quantization can be seen as a way to construct a diﬀerential operator with prescribed principal symbol. The map from the space of symbols to the space of diﬀerential operators is moreover required to be a linear bijection. In general, there is no natural quantization procedure, that is, spaces of symbols and of differential operators are not equivalent, if the action of local diﬀeomorphisms is taken into account. However, considering manifolds endowed with additional structures, one can seek for quantizations that depend on this additional structure and that are natural if the dependence with respect to the structure is taken into account. The existence of such a quantization was proved recently in a series of papers in the context of projective geometry. Here, we show that the construction of the quantization based on Cartan connections can be adapted from projective to pseudo-conformal geometry to yield the natural and conformally invariant quantization for arbitrary symbols, outside some critical situations. [less ▲] Detailed reference viewed: 42 (5 ULg)A first approximation for quantization of singular spaces ; Radoux, Fabian ; in Journal of Geometry & Physics (2009), 59(4), 503-518 Many mathematical models of physical phenomena that have been proposed in recent years require more general spaces than manifolds. When taking into account the symmetry group of the model, we get a ... [more ▼] Many mathematical models of physical phenomena that have been proposed in recent years require more general spaces than manifolds. When taking into account the symmetry group of the model, we get a reduced model on the (singular) orbit space of the symmetry group action. We investigate quantization of singular spaces obtained as leaf closure spaces of regular Riemannian foliations on compact manifolds. These contain the orbit spaces of compact group actions and orbifolds. Our method uses foliation theory as a desingularization technique for such singular spaces. A quantization procedure on the orbit space of the symmetry group–that commutes with reduction–can be obtained from constructions which combine different geometries associated with foliations and new techniques originated in Equivariant Quantization. The present paper contains the first of two steps needed to achieve these just detailed goals. [less ▲] Detailed reference viewed: 33 (9 ULg)On natural and conformally invariant quantizations Mathonet, Pierre ; Radoux, Fabian in Journal of the London Mathematical Society (2009), 80(1), 256-272 The concept of conformally equivariant quantization was introduced by Duval, Lecomte and Ovsienko for manifolds endowed with flat conformal structures. They obtained results of existence and uniqueness ... [more ▼] The concept of conformally equivariant quantization was introduced by Duval, Lecomte and Ovsienko for manifolds endowed with flat conformal structures. They obtained results of existence and uniqueness (up to normalization) of such a quantization procedure. A natural generalization of this concept is to seek for a quantization procedure, over a manifold M, that depends on a pseudo-Riemannian metric, is natural, and is invariant with respect to a conformal change of the metric. The existence of such a procedure was conjectured by P. Lecomte and proved by C. Duval and V. Ovsienko for symbols of degree at most 2 and by S. Loubon Djounga for symbols of degree 3. In two recent papers, we investigated the question of existence of projectively equivariant quantizations using the framework of Cartan connections. Here we will show how the formalism developed in these works adapts in order to deal with the conformally equivariant quantization for symbols of degree at most 3. This will allow us to easily recover earlier results on the subject. We will then show how it can be modified in order to prove the existence of conformally equivariant quantizations for symbols of degree 4. [less ▲] Detailed reference viewed: 14 (4 ULg)An Explicit Formula for the Natural and Conformally Invariant Quantization Radoux, Fabian in Letters in Mathematical Physics (2009), 89 Lecomte (Prog Theor Phys Suppl 144:125–132, 2001) conjectured the existence of a natural and conformally invariant quantization. In Mathonet and Radoux (Existence of natural and conformally invariant ... [more ▼] Lecomte (Prog Theor Phys Suppl 144:125–132, 2001) conjectured the existence of a natural and conformally invariant quantization. In Mathonet and Radoux (Existence of natural and conformally invariant quantizations of arbitrary symbols, math.DG 0811.3710), we gave a proof of this theorem thanks to the theory of Cartan connections. In this paper, we give an explicit formula for the natural and conformally invariant quantization of trace-free symbols thanks to the method used in Mathonet and Radoux and to tools already used in Radoux [Lett Math Phys 78(2):173–188, 2006] in the projective setting. This formula is extremely similar to the one giving the natural and projectively invariant quantization in Radoux. [less ▲] Detailed reference viewed: 14 (3 ULg)Natural equivariant quantizations Radoux, Fabian Conference (2008, November 12) Detailed reference viewed: 8 (4 ULg)Quantifications naturelles projectivement équivariantes Radoux, Fabian Scientific conference (2008, March 21) Detailed reference viewed: 12 (5 ULg)Non-uniqueness of the natural and projectively equivariant quantization Radoux, Fabian in Journal of Geometry & Physics (2008), 58 In [C. Duval, V. Ovsienko, Projectively equivariant quantization and symbol calculus: Noncommutative hypergeometric functions, Lett. Math. Phys. 57 (1) (2001) 61–67], the authors showed the existence and ... [more ▼] In [C. Duval, V. Ovsienko, Projectively equivariant quantization and symbol calculus: Noncommutative hypergeometric functions, Lett. Math. Phys. 57 (1) (2001) 61–67], the authors showed the existence and the uniqueness of a sl(m+1,R)-equivariant quantization in non-critical situations. The curved generalization of the sl(m+1,R)-equivariant quantization is the natural and projectively equivariant quantization. In [M. Bordemann, Sur l’existence d’une prescription d’ordre naturelle projectivement invariante (submitted for publication). math.DG/0208171] and [Pierre Mathonet, Fabian Radoux, Natural and projectively equivariant quantizations by means of Cartan connections, Lett. Math. Phys. 72 (3) (2005) 183–196], the existence of such a quantization was proved in two different ways. In this paper, we show that this quantization is not unique. [less ▲] Detailed reference viewed: 15 (3 ULg)Natural and projectively equivariant quantizations Radoux, Fabian Conference (2007, October 21) Detailed reference viewed: 20 (2 ULg)Natural and projectively equivariant quantizations Radoux, Fabian Conference (2007, October 13) Detailed reference viewed: 20 (2 ULg)Natural and projectively equivariant quantizations Radoux, Fabian Conference (2007, May 01) Detailed reference viewed: 18 (2 ULg)Quantifications naturelles projectivement équivariantes Radoux, Fabian Scientific conference (2007, February 13) Detailed reference viewed: 11 (2 ULg)Cartan connections and natural and projectively equivariant quantizations Mathonet, Pierre ; Radoux, Fabian in Journal of the London Mathematical Society (2007), 76 In this paper, the question of existence of a natural and projectively equivariant symbol calculus is analysed using the theory of projective Cartan connections. A close relationship is established ... [more ▼] In this paper, the question of existence of a natural and projectively equivariant symbol calculus is analysed using the theory of projective Cartan connections. A close relationship is established between the existence of such a natural symbol calculus and the existence of an sl(m+1,R)-equivariant calculus over R^m . Moreover, it is shown that the formulae that hold in the non-critical situations over R^m for the sl(m+1,R)-equivariant calculus can be directly generalized to an arbitrary manifold by simply replacing the partial derivatives by invariant diﬀerentiations with respect to a Cartan connection. [less ▲] Detailed reference viewed: 14 (7 ULg)Natural and projectively equivariant quantizations Radoux, Fabian Conference (2006, November 25) Detailed reference viewed: 6 (3 ULg)Quantifications naturelles projectivement équivariantes Radoux, Fabian Doctoral thesis (2006) Detailed reference viewed: 40 (13 ULg)Quantifications naturelles projectivement équivariantes Radoux, Fabian Scientific conference (2006, September 28) Detailed reference viewed: 9 (4 ULg)Explicit formula for the natural and projectively equivariant quantization Radoux, Fabian in Letters in Mathematical Physics (2006), 78 In [Prog Theor Phys Suppl 49(3):173–196, 1999], Lecomte conjectured the existence of a natural and projectively equivariant quantization. In [math.DG/0208171, Submitted], Bordemann proved this existence ... [more ▼] In [Prog Theor Phys Suppl 49(3):173–196, 1999], Lecomte conjectured the existence of a natural and projectively equivariant quantization. In [math.DG/0208171, Submitted], Bordemann proved this existence using the framework of Thomas–Whitehead connections. In [Lett Math Phys 72(3):183–196, 2005], we gave a new proof of the same theorem thanks to the Cartan connections. After these works, there was no explicit formula for the quantization. In this paper, we give this formula using the formula in terms of Cartan connections given in [Lett Math Phys 72(3):183–196, 2005]. This explicit formula constitutes the generalization to any order of the formulae at second and third orders soon published by Bouarroudj in [Lett Math Phys 51(4):265–274, 2000] and [C R Acad Sci Paris Sér I Math 333(4):343–346, 2001]. [less ▲] Detailed reference viewed: 5 (2 ULg)Natural and projectively equivariant quantizations Radoux, Fabian Conference (2005, September 02) Detailed reference viewed: 5 (3 ULg)Natural and projectively equivariant quantizations by means of Cartan connections Mathonet, Pierre ; Radoux, Fabian in Letters in Mathematical Physics (2005), 72 The existence of a natural and projectively equivariant quantization in the sense of Lecomte was proved recently by M. Bordemann, using the framework of Thomas-Whitehead connections. We give a new proof ... [more ▼] The existence of a natural and projectively equivariant quantization in the sense of Lecomte was proved recently by M. Bordemann, using the framework of Thomas-Whitehead connections. We give a new proof of existence using the notion of Cartan projective connections and we obtain an explicit formula in terms of these connections. Our method yields the existence of a projectively equivariant quantization if and only if an sl(m+1,R)-equivariant quantization exists in the flat situation, thus solving one of the problems left open by M. Bordemann. [less ▲] Detailed reference viewed: 16 (6 ULg) |
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