Galilean Isometries

We introduce three nested Lie algebras of infinitesimal `isometries' of a Galilei space-time structure which play the r\^ole of the algebra of Killing vector fields of a relativistic Lorentz space-time. Non trivial extensions of these Lie algebras arise naturally from the consideration of Newton-Cartan-Bargmann automorphisms.Comment: Plain TeX, 8 page

Polarized Spinoptics and Symplectic Physics

We recall the groundwork of spinoptics based on the coadjoint orbits, of given color and spin, of the group of isometries of Euclidean three-space; this model has originally been put forward by Souriau in his treatise "Structure des Syst\'emes Dynamiques", whose manuscript was initially entitled "Physique symplectique". We then set up a model of polarized spinoptics, namely an extension of geometrical optics accounting for elliptically polarized light rays in terms of a certain fibre bundle associated with the bundle of Euclidean frames of a given Riemannian three-manifold. The characteristic foliation of a natural presymplectic two-form introduced on this bundle via the Ansatz of minimal coupling is determined, yielding a set of differential equations governing the trajectory of light, as well as the evolution of polarization in this Riemannian manifold. Those equations, when specialized to the Fermat metric (for a slowly varying refractive index), enable us to recover, and justify, a set of differential equations earlier proposed in the literature, in another context, namely in terms of a semi-classical limit of wave optics. They feature a specific anomalous velocity responsible for the recently observed Spin Hall Effect of Light, namely a tiny spatial deflection of polarized light rays, transversally to the gradient of the refractive index. Our model, constructed from the start on purely geometric grounds, turns out to encode automatically the Berry as well as the Pancharatnam connections that usually appear in the framework of wave optics.Comment: 27 pages, 3 figure

A recollection of Souriau's derivation of the Weyl equation via geometric quantization

These notes merely intend to memorialize Souriau's overlooked achievements regarding geo\-metric quantization of Poincar\'e-elementary symplectic systems. Restricting attention to his model of massless, spin-\half, particles, we faithfully rephrase and expound here Sections (18.82)--(18.96) & (19.122)--(19.134) of his book \cite{SSD} edited in 1969. Missing details about the use of a preferred Poincar\'e-invariant polarizer are provided for completeness

Non-relativistic conformal symmetries and Newton-Cartan structures

This article provides us with a unifying classification of the conformal infinitesimal symmetries of non-relativistic Newton-Cartan spacetime. The Lie algebras of non-relativistic conformal transformations are introduced via the Galilei structure. They form a family of infinite-dimensional Lie algebras labeled by a rational "dynamical exponent", $z$. The Schr\"odinger-Virasoro algebra of Henkel et al. corresponds to $z=2$. Viewed as projective Newton-Cartan symmetries, they yield, for timelike geodesics, the usual Schr\"odinger Lie algebra, for which z=2. For lightlike geodesics, they yield, in turn, the Conformal Galilean Algebra (CGA) and Lukierski, Stichel and Zakrzewski [alias "\alt" of Henkel], with $z=1$. Physical systems realizing these symmetries include, e.g., classical systems of massive, and massless non-relativistic particles, and also hydrodynamics, as well as Galilean electromagnetism.Comment: LaTeX, 47 pages. Bibliographical improvements. To appear in J. Phys.

A new integrable system on the sphere and conformally equivariant quantization

Taking full advantage of two independent projectively equivalent metrics on the ellipsoid leading to Liouville integrability of the geodesic flow via the well-known Jacobi-Moser system, we disclose a novel integrable system on the sphere $S^n$, namely the "dual Moser" system. The latter falls, along with the Jacobi-Moser and Neumann-Uhlenbeck systems, into the category of (locally) St\"ackel systems. Moreover, it is proved that quantum integrability of both Neumann-Uhlenbeck and dual Moser systems is insured by means of the conformally equivariant quantization procedure.Comment: LaTeX, 33 pages. Minor corrections. Published versio

Conformal Galilei groups, Veronese curves, and Newton-Hooke spacetimes

Finite-dimensional nonrelativistic conformal Lie algebras spanned by polynomial vector fields of Galilei spacetime arise if the dynamical exponent is z=2/N with N=1,2,.... Their underlying group structure and matrix representation are constructed (up to a covering) by means of the Veronese map of degree N. Suitable quotients of the conformal Galilei groups provide us with Newton-Hooke nonrelativistic spacetimes with a quantized reduced negative cosmological constant \lambda=-N.Comment: LaTeX, 31 pages. Sections 3 and 5 reorganized. Conclusion expanded. New references adde

Schwarzian derivative and Numata Finsler structures

LaTeX, 4 pages. Reference added. To appear in Advances in Pure and Applied MathematicsInternational audienceThe flag curvature of the Numata Finsler structures is shown to admit a non\-trivial prolongation to the one-dimensional case, revealing an unexpected link with the Schwarzian derivative of the diffeomorphisms associated with these Finsler structures

Geometrical Spinoptics and the Optical Hall Effect

33 pages. Two subsections and new references added. To appear in the Journal of Geometry and PhysicsGeometrical optics is extended so as to provide a model for spinning light rays via the coadjoint orbits of the Euclidean group characterized by color and spin. This leads to a theory of ``geometrical spinoptics'' in refractive media. Symplectic scattering yields generalized Snell-Descartes laws that include the recently discovered optical Hall effect

An overview of signal processing issues in chemical sensing

International audienceThis tutorial paper aims at summarizing some problems, ranging from analytical chemistry to novel chemical sensors, that can be addressed with classical or advanced methods of signal and image processing. We gather them under the denomination of "chemical sensing". It is meant to introduce the special session "Signal Processing for Chemical Sensing" with a large overview of issues which have been and remain to be addressed in this application domain, including chemical analysis leading to PARAFAC/tensor methods, hyper spectral imaging, ion-sensitive sensors, artificial nose, chromatography, mass spectrometry, etc. For enlarging and illustrating the points of view of this tutorial, the invited papers of the session consider other applications (NMR, Raman spectroscopy, recognition of explosive compounds, etc.) addressed by various methods, e.g. source separation, Bayesian, and exploiting typical chemical signal priors like positivity, linearity, unit-concentration or sparsity