Quantifications équivariantes

Scientific conference (2012, March 15)

Equivariant quantization in supergeometry

Scientific conference (2011, December 08)

Natural and invariant quantizations

Conference (2011, July 12)

Quantifications naturelles projectivement équivariantes

Book published by Editions Universitaires Européennes (2011)

Quantifications équivariantes

Scientific conference (2011, March 28)

Projectively equivariant quantizations over the superspace R^{p|q}

in Letters in Mathematical Physics (2011), 98(3), 311-331

We investigate the concept of projectively equivariant quantization in the framework of super projective geometry. When the projective superalgebra pgl(p+1|q) is simple, our result is similar to the ... [more ▼]

We investigate the concept of projectively equivariant quantization in the framework of super projective geometry. When the projective superalgebra pgl(p+1|q) is simple, our result is similar to the classical one in the purely even case: we prove the existence and uniqueness of the quantization except in some critical situations. When the projective superalgebra is not simple (i.e. in the case of pgl(n|n)\not\cong sl(n|n)), we show the existence of a one-parameter family of equivariant quantizations. We also provide explicit formulas in terms of a generalized divergence operator acting on supersymmetric tensor fields. [less ▲]

Equivariant quantization of orbifolds

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 ▲]

A first approximation for quantization of singular spaces

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 ▲]

An Explicit Formula for the Natural and Conformally Invariant Quantization

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 ▲]

Quantifications naturelles projectivement équivariantes

Scientific conference (2008, March 21)