Irreducible Representations of the Poincaré Group Including Inversions

We classify completely all irreducible representations of the Poincaré Group (including inversions) corresponding to positive or zero mass systems, using a generalized version of the method of VARADARAJAN [9] which seems to be more convenient for practical computation than the method of PARTHASARATHY [5]. The case of zero mass — infinite helicity systems which was not considered previously in the literature is treated in a complete fashion.     

Applications of the Poincaré sphere in polarization metrology

The Poincaré sphere representation of polarization states is used to derive two auxiliary equations for phase retardance measurements. These equations, in addition to another two previously derived equations, allow for extending the range of validity of a model for calibrating phase plates in pairs. In another application, the sphere is used to explain a new method for identifying the principal axes of two birefringent phase plates during their calibration.     

Massless Fields as Unitary Representations of the Poincaré Group

Relativistic zero-mass fields are described as manifestly covariant unitary irreducible representations of the Poincaré group. The wave-equations, which are a necessary condition for unitarity, are constructed for spinor fields of arbitrary spin and for tensor fields of integer spin. Poincaré covariance together with causality and positive energy are used to determine the commutators of quantized fields up to a positive multiple and to prove the spin-statistics theorem. The use of potentials for boson fields is discussed and it is shown that, at the expense of manifest covariance, potentials may be introduced as zero-mass limits of the massive Wigner representations.     

Hamiltonian structure and gauge symmetries of Poincaré gauge theory

This is a review of the constrained dynamical structure of Poincaré gauge theory which concentrates on the basic canonical and gauge properties of the theory, including the identification of constraints, gauge symmetries and conservation laws. As an interesting example of the general approach, we discuss the teleparallel formulation of general relativity.     

Automorphic Functions,Poincaré Series,and Conformal Fields on Riemann Surfaces

Poincaré series and automorphic functions for SU(1, 1) and a discrete subgroup Γ are studied with harmonic analysis. We consider automorphic functions on the open unit circle with general “spin label” m and their decomposition into irreducible automorphic functions by means of the Plancherel formula. These automorphic functions are bijectively mapped onto automorphic distributions on the boundary of the unit circle by meam of the Poisson kernel. The exponent of convergence of Poincaré series is expressed in representation theory language. The results are applied to two-point functions of conformal fields.     

The Representations of the Poincaré Group in the Framework of Free Quantum Fields

This review article arose while the author tried to get acquainted with the recent developments in the domain of the representations of the Poincaré group in the quantum field theory. Special emphasis was put in this article on the case of massless free fields. Some ideas concerning general restrictions imposed upon the spinorial fields in order to get irreducible representations seem to be new and original.     

The method of the Poincaré map for compressional and torsional waves in composite rods

A generalized Poincaré map is introduced and used to find the compressional and torsional normal-mode frequencies and wave amplitudes of a one-dimensional rod composed by N coaxial cylinders of different length and radius. The method is approximately 100 times faster and more stable than the transfer matrix method. The Poincaré map method is used in a rod composed of two locally periodic systems, each one with a different length of the unit cell. This system can be used as an almost perfect mechanical noise filter.     

Hamiltonian Structure of Poincaré Gauge Theory and Separation of Non-Dynamical Variables in Exact Torsion Solutions

The canonical Hamiltonian of the Poincaré gauge theory of gravity is reanalyzed for generic Lagrangians. It is shown that the time components e₀^α and Γ₀^αβ of the tetrad and the linear connection fields of a Riemann-Cartan space-time U₄ constitute gauge degrees of freedom which remain non-dynamical during the time evolution of the system. Whereas the e₀^α are to be identified with the lapse and shift functions N^α known from the ADM formalism in Einstein's theory, the additional Lorentz degrces of freedom Γ₀^αβ are pertinent to Poincaré gauge models. These non-dynamical variables are instrumental in the derivation of exact torsion solutions obeying modified double duality conditions for the U₄-curvature. Thereby, in the case of spherical symmetry and for the charged Taub-NUT metric, we obtain the most general torsion configuration for a large class of quadratic Lagrangians. Previously found solutions are contained therein and can be recovered after fixing special “gauge”.     

Poincaré gauge theory with coupled even and odd parity spin‐0 modes: Cosmological normal modes

We are investigating the dynamics of a new Poincaré gauge theory of gravity model, which has cross coupling between the spin‐0⁺ and spin‐0^‐ modes. To this end we are considering a very appropriate situation – homogeneous‐isotropic cosmologies – which is relatively simple, and yet all the modes have non‐trivial dynamics which reveals physically interesting and possibly observable results. More specifically we consider manifestly isotropic Bianchi class A cosmologies. Here the first order equations obtained from an effective Lagrangian are linearized and the normal modes are found. These turn out to control the asymptotic late time cosmological normal modes. Numerical evolution confirms the late time asymptotic approximation and shows the expected effects of the cross parity pseudoscalar coupling.     

Étude géométrique du monopole magnétique

Le mouvement d'une particule chargée soumise au champ d'un monopole magnétique est étudié dans un cadre géométrique.

Le formalisme sans corde de Wu et de Yang permet d'interprêter géométriquement la symmétrie de rotation mais s'avére insuffisant pour traiter les symétries cachées découvertes récemment par Jackiw. Cette tache est accomplie par la quantification géométrique de Souriau et de Kostant. La relation des deux constructions est expliquée en détail.     

Les matériaux magnétiques hyperfréquences à l'ère des métamatériaux

This article investigates the advances brought by metamaterials in order to obtain attractive magnetic microwave properties. Can magnetic metamaterials outperform conventional soft magnetic materials? In the case where permeability levels and bandwidth are the key figures of merit, it is acknowledged that copper-based metamaterials can exceed the performance of soft magnetic materials, but only at operating frequencies above 10 GHz. As for low frequency operation, magnetic metamaterials may also be preferred to conventional magnetic materials when requirements include excellent temperature stability or immunity to external magnetic fields. However, in many cases, metamaterials need to include certain conventional magnetic constituents in order to compete with conventional magnetic materials. Several types of metamaterials containing conventional magnetic inclusions and well suited for industrial production are presented. Last but not least, it is underlined that the investigations on metamaterials are beneficial to scientific exchanges between scientists from different areas. To cite this article: O. Acher, C. R. Physique 10 (2009).     

Le théorème de réduction de Marsden-Weinstein en géométrie cosymplectique et de contact

We show that the classical Marsden-Weinstein Reduction theorem for Hamiltonian systems with symmetries is still true for contact manifolds and cosympletic manifolds (i.e. canonical manifolds in the sense of A. Lichnerowicz).

In fact, we precise the notion of transitive almost contact structure, which enables us to consider the cosymplectic geometry as a limit of the contact geometry when a certain parameter goes to zero. This point of view unifies both theories.

However, we have to give two distinct proofs for the contact Reduction theorem and the cosymplectic one.     

Courbure scalaire et géométrie conforme

We present a study of the scalar curvature problems in conformal geometry. We deal with the Nirenberg and Yamabe problems, the prescribed scalar curvature problem, the Yamabe equivariant problem, and the questions connected with the multiplicity of metrics having the same scalar curvature. The problems of the best constants in the Sobolev imbedding theorem and the positive mass theorem (which is highly physical and connected with general relativity) are first treated. The mathematics of these two subjects will appear throughout the text.

Résumé

On présente une étude des problèmes de courbure scalaire en géométrie conforme. Sont ainsi traités: les problèmes de Nirenberg et Yamabe, le problème de la courbure scalaire prescrite, le problème de Yamabe équivariant et les questions relatives à la multiplicité des métriques ayant une même courbure scalaire. Les problèmes de meilleures constantes dans les inclusions de Sobolev et le théorème de la masse positive (qui est hautement physique et lié à la relativité générale) sont d'abord présentés. Les mathématiques de ces deux sujets apparaitront tout au long du texte.     

Décomposition du fibré des spineurs d''une variété spin Kähler-quanternionenienne sous l''action de la 4-forme fondamentale

On a spin quaternionic-Kähler manifold M^4m, we give the spectral decomposition of the spin bundle the action of the fundamental 4-form Ω. Moreover, we compute the eigenvalues of Ω which, in the compact case, play an essential role in the problem of estimating the eigenvalues of the Dirac operator. The proof is based on the decomposition of the spin representation into irreducible components under the action of the group Sp(1) × Sp(m). We show that this algebraic results induces a Whitney decomposition of the spin bundle which, when restricted to the fibres, is the spectral decomposition under the action of Ω.     

Les nouvelles générations de détecteurs infrarouge à base de HgCdTeNew generations of infrared detectors based on HgCdTe

The technology of very high performance cooled infrared detectors made with HgCdTe has progressed continuously for ten years and reached today an industrial maturity that allows the production of large size arrays at a more and more reasonable cost. At the same time, new prototypes of more complex sensors have appeared (megapixel arrays, multicolour arrays, high definition long linear arrays, …) that show the strong potentialities of this very high performance technology. This paper presents the technology developed in France and gives the state of the art of products available in industry; it then focuses on some very recent realizations of advanced prototypes made at LETI (dualband arrays, megapixel arrays, …). To cite this article: G. Destefanis, C. R. Physique 4 (2003).     

Moiré fringes

This review paper clarifies the characteristics of the incoherent-and-multiplicative type moiré fringe phenomenon, presents examples of the application of these characteristics to measurement and explains the profile prediction method and the sharpening conditions of moiré fringes formed by two binary gratings.     

Geometric micron-moiré

A high spatial resolution and high-sensitivity geometric micron-moiré method is presented. The method is based upon classical geometric moiré method and microscopic principle, the frequency of the specimen and reference gratings used in this method can be higher than 1000 lines/mm. By a microscopic moiré system, the high-frequency gratings could be magnified to a low-frequency level, so geometric moiré patterns can be formed. Experiments show that both the spatial resolution and the sensitivity can reach or exceed micron-level.