Recent results on the semiclassical dynamics of an electron in a solid are explained using techniques developed for "exotic" Galilean dynamics. The system is indeed Hamiltonian and Liouville's theorem holds for the symplectic volume form. Suitably defined quantities satisfy hydrodynamic equations.cond-mat/0506051. to appear in Mod. Phys. Lett. B.

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Mimicking the description of spinning particles in General Relativity, the Fermat Principle is extended to spinning photons. Linearization of the resulting Papapetrou-Souriau type equations yields the semiclassical model used recently to derive the "Optical Hall Effect" for polarized light (alias the "Optical Magnus Effect").PACS numbers: 42.25. Bs, 03.65.Sq, 03.65.Vf, Light is an electromagnetic wave, whose propagation is described by Maxwell's theory. It can also be viewed, however, as a particle (a "photon"). Here we adhere to the second approach: we describe light by a bona fide mechanical model in that we use a Lagrangian.In traditional geometrical optics the spin degree of freedom is neglected, and the light rays obey the Fermat Principle [1]. In the intermediate model advocated by Landau and Lifchitz [2], the photon is polarized, but the polarization is simply carried along by the light rays, and has no influence on the trajectory of light. Recent approaches [3, 4] go one step further : the feedback from the polarization deviates the trajectory from that given by the Fermat Principle. A dramatic consequence is that, for polarized light, the Snel(-Descartes) law of refraction requires correction : the plane of the refracted (or reflected) ray is shifted perpendicularly to that of the incident ray [3]. This "Hall Effect for light" is a manifestation of the Magnus-type interaction between the refractive medium and the photon's polarization [4]. It can be derived in a semiclassical framework, which also includes a Berry-type term [5,6,7].In this Rapid Communication, we argue that the deviation of polarized light from the trajectory predicted by ordinary geometrical optics is indeed analogous to the deviation of a spinning particle from geodesic motion in General Relativity. The resulting equations are reminiscent of those of Papapetrou and Souriau [8].In detail, the Fermat Principle of geometrical optics * UMR 6207 du CNRS associée aux Universités d'Aix-Marseille I et II et Université du Sud Toulon-Var; Laboratoire affiliéà la FRUMAM-FR2291.; Electronic address: duval@cpt.univ-mrs.fr † Electronic address: e-mail:zalanh@ludens.elte.hu ‡ Electronic address: horvathy@lmpt.univ-tours.fr says that light in an isotropic medium of refractive index n = n(r) propagates along curves that minimize the optical length. Light rays are hence geodesics of the "optical" metric g ij = n 2 (r)δ ij of 3-space. To extend this theory to spin we consider the bundle of positively oriented orthonormal frames over a 3-manifold endowed with a Riemannian metric g ij . At each point, such a "Dreibein" is given by three orthogonal vectors U i , V i , W i of unit length that span unit volume. We stress that the [6-dimensional] orthonormal frame bundle we are using here is a mere artifact that allows us to define a variational formalism. Eliminating unphysical degrees of freedom will leave us with 4 independent physical variables.Introducing the covariant exterior derivative associated with the Levi-Civita connection, DU k = dU k + Γ k ij d...

Quasi-breathers (QB) are time-periodic solutions with weak spatial localization introduced in G. Fodor et al. in Phys. Rev. D. 74, 124003 (2006). QB's provide a simple description of oscillons (very long-living spatially localized time dependent solutions). The small amplitude limit of QB's is worked out in a large class of scalar theories with a general self-interaction potential, in D spatial dimensions. It is shown that the problem of small amplitude QB's is reduced to a universal elliptic partial differential equation. It is also found that there is the critical dimension, Dcrit = 4, above which no small amplitude QB's exist. The QB's obtained this way are shown to provide very good initial data for oscillons. Thus these QB's provide the solution of the complicated, nonlinear time dependent problem of small amplitude oscillons in scalar theories.

The radiation loss of small amplitude oscillons (very long-living, spatially localized, time dependent solutions) in one dimensional scalar field theories is computed in the small-amplitude expansion analytically using matched asymptotic series expansions and Borel summation. The amplitude of the radiation is beyond all orders in perturbation theory and the method used has been developed by Segur and Kruskal in Phys. Rev. Lett. 58, 747 (1987). Our results are in good agreement with those of long time numerical simulations of oscillons.Comment: 22 pages, 9 figure