Natural and projectively equivariant quantizations

Conference (2007, October 21)

Natural and projectively equivariant quantizations

Conference (2007, May 01)

Cartan connections and natural and projectively equivariant quantizations

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 ▲]

Quantifications naturelles projectivement équivariantes

Doctoral thesis (2006)

Explicit formula for the natural and projectively equivariant quantization

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 ▲]

Natural and projectively equivariant quantizations by means of Cartan connections

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 ▲]