Projectively and conformally invariant star-products

We consider the Poisson algebra S(M) of smooth functions on T^*M which are fiberwise polynomial. In the case where M is locally projectively (resp. conformally) flat, we seek the star-products on S(M) which are SL(n+1,R) (resp. SO(p+1,q+1))-invariant. We prove the existence of such star-products using the projectively (resp. conformally) equivariant quantization, then prove their uniqueness, and study their main properties. We finally give an explicit formula for the canonical projectively invariant star-product.Comment: 37 pages, Latex; minor correction

Representations of the conformal Lie algebra in the space of tensor densities on the sphere

Let ${\mathcal F}_\lambda(\mathbb{S}^n)$ be the space of tensor densities on $\mathbb{S}^n$ of degree $\lambda$. We consider this space as an induced module of the nonunitary spherical series of the group $\mathrm{SO}_0(n+1,1)$ and classify $(\mathrm{so}(n+1,1),\mathrm{SO}(n+1))$-sim$unitary submodules of${\mathcal F}_\lambda(\mathbb{S}^n)$as a function of$\lambda$.Comment: Published by JNMP at http://www.sm.luth.se/math/JNMP

Inelastic light, neutron, and X-ray scatterings related to the heterogeneous elasticity of glasses

The effects of plasticization of poly(methyl methacrylate) glass on the boson peaks observed by Raman and neutron scattering are compared. In plasticized glass the cohesion heterogeneities are responsible for the neutron boson peak and partially for the Raman one, which is enhanced by the composition heterogeneities. Because the composition heterogeneities have a size similar to that of the cohesion ones and form quasiperiodic clusters, as observed by small angle X-ray scattering, it is inferred that the cohesion heterogeneities in a normal glass form nearly periodic arrangements too. Such structure at the nanometric scale explains the linear dispersion of the vibrational frequency versus the transfer momentum observed by inelastic X-ray scattering.Comment: 9 pages, 2 figures, to be published in J. Non-Cryst. Solids (Proceedings of the 4th IDMRCS

Non-Commutative Corrections to the MIC-Kepler Hamiltonian

Non-commutative corrections to the MIC-Kepler System (i.e. hydrogen atom in the presence of a magnetic monopole) are computed in Cartesian and parabolic coordinates. Despite the fact that there is no simple analytic expression for non-commutative perturbative corrections to the MIC-Kepler spectrum, there is a term that gives rise to the linear Stark effect which didn't exist in the standard hydrogen model.Comment: 5 page

Conformally equivariant quantization: Existence and uniqueness

We prove the existence and the uniqueness of a conformally equivariant symbol calculus and quantization on any conformally flat pseudo-Riemannian manifold (M,\rg). 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 \so(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.Comment: LaTeX document, 32 pages; improved versio

Transverse Shifts in Paraxial Spinoptics

The paraxial approximation of a classical spinning photon is shown to yield an "exotic particle" in the plane transverse to the propagation. The previously proposed and observed position shift between media with different refractive indices is modified when the interface is curved, and there also appears a novel, momentum [direction] shift. The laws of thin lenses are modified accordingly.Comment: 3 pages, no figures. One detail clarified, some misprints corrected and references adde

Decomposition of symmetric tensor fields in the presence of a flat contact projective structure

Let $M$ be an odd-dimensional Euclidean space endowed with a contact 1-form $\alpha$. We investigate the space of symmetric contravariant tensor fields on $M$ as a module over the Lie algebra of contact vector fields, i.e. over the Lie subalgebra made up by those vector fields that preserve the contact structure. If we consider symmetric tensor fields with coefficients in tensor densities, the vertical cotangent lift of contact form $\alpha$ is a contact invariant operator. We also extend the classical contact Hamiltonian to the space of symmetric density valued tensor fields. This generalized Hamiltonian operator on the symbol space is invariant with respect to the action of the projective contact algebra $sp(2n+2)$. The preceding invariant operators lead to a decomposition of the symbol space (expect for some critical density weights), which generalizes a splitting proposed by V. Ovsienko