Second central extension in Galilean covariant field theory

The second central extension of the planar Galilei group has been alleged to have its origin in the spin variable. This idea is explored here by considering local Galilean covariant field theory for free fields of arbitrary spin. It is shown that such systems generally display only a trivial realization of the second central extension. While it is possible to realize any desired value of the extension parameter by suitable redefinition of the boost operator, such an approach has no necessary connection to the spin of the basic underlying field.Comment: 6 pgs., late

Anyons, group theory and planar physics

Relativistic and nonrelativistic anyons are described in a unified formalism by means of the coadjoint orbits of the symmetry groups in the free case as well as when there is an interaction with a constant electromagnetic field. To deal with interactions we introduce the extended Poincar\'e and Galilei Maxwell groups.Comment: 22 pages, journal reference added, bibliography update

Galilean Lee Model of the Delta Function Potential

The scattering cross section associated with a two dimensional delta function has recently been the object of considerable study. It is shown here that this problem can be put into a field theoretical framework by the construction of an appropriate Galilean covariant theory. The Lee model with a standard Yukawa interaction is shown to provide such a realization. The usual results for delta function scattering are then obtained in the case that a stable particle exists in the scattering channel provided that a certain limit is taken in the relevant parameter space. In the more general case in which no such limit is taken finite corrections to the cross section are obtained which (unlike the pure delta function case) depend on the coupling constant of the model.Comment: 7 pages, latex, no figure

Hopf instantons, Chern-Simons vortices, and Heisenberg ferromagnets

The dimensional reduction of the three-dimensional fermion-Chern-Simons model (related to Hopf maps) of Adam et el. is shown to be equivalent to (i) either the static, fixed--chirality sector of our non-relativistic spinor-Chern-Simons model in 2+1 dimensions, (ii) or a particular Heisenberg ferromagnet in the plane.Comment: 4 pages, Plain Tex, no figure

(In)finite extensions of algebras from their Inonu-Wigner contractions

The way to obtain massive non-relativistic states from the Poincare algebra is twofold. First, following Inonu and Wigner the Poincare algebra has to be contracted to the Galilean one. Second, the Galilean algebra is to be extended to include the central mass operator. We show that the central extension might be properly encoded in the non-relativistic contraction. In fact, any Inonu-Wigner contraction of one algebra to another, corresponds to an infinite tower of abelian extensions of the latter. The proposed method is straightforward and holds for both central and non-central extensions. Apart from the Bargmann (non-zero mass) extension of the Galilean algebra, our list of examples includes the Weyl algebra obtained from an extension of the contracted SO(3) algebra, the Carrollian (ultra-relativistic) contraction of the Poincare algebra, the exotic Newton-Hooke algebra and some others. The paper is dedicated to the memory of Laurent Houart (1967-2011).Comment: 7 pages, revtex style; v2: Minor corrections, references added; v3: Typos correcte

Galilée, de l’Enfer de Dante au purgatoire de la science

En 1587, le jeune Galilée est invité à donner Due lezioni all’Accademia Fiorentina circa la figura, sito e grandezza dell’Inferno di Dante (ci-après Leçons sur l’Enfer) [Galilei 1587] afin d’éclairer une vive controverse sur l’interprétation de la géographie de l’Enfer dantesque. Ce travail d’exégèse littéraire permet à Galilée de faire reconnaître ses talents mathématiques comme ses qualités pédagogiques. Mais la portée de ces leçons va bien au-delà, car on peut y voir apparaître plusieurs thèmes majeurs de l’œuvre ultérieure de Galilée : au plan mathématique, l’importance de la géométrie d’inspiration archimédienne, au plan physique, l’étude des questions de similitude que pose la résistance des matériaux – sans oublier l’intérêt constant du scientifique pour la langue et pour la culture littéraire.In 1587 the young Galileo was invited to give Due lezioni all’Accademia Fiorentina circa la figura, sito e grandezza dell’Inferno di Dante (hereafter Leçons sur l’Enfer) [Galilei 1587] aimed at settling an intense controversy regarding the geography of the Dantean Hell. This study in exegetics enabled Galileo to bring his mathematical talents and didactic qualities to the knowledge of the Tuscan scholars. But these lessons have a much greater importance, in that they reveal several of the major themes of Galileo’s further work from both mathematical and physical standpoints, such as the question of scale linked to the strength of materials, as well as the scientist’s unremitting interest in language and commitment to literary culture

Galilée, de l’Enfer de Dante au purgatoire de la science

En 1587, le jeune Galilée est invité à donner Due lezioni all’Accademia Fiorentina circa la figura, sito e grandezza dell’Inferno di Dante (ci-après Leçons sur l’Enfer) [Galilei 1587] afin d’éclairer une vive controverse sur l’interprétation de la géographie de l’Enfer dantesque. Ce travail d’exégèse littéraire permet à Galilée de faire reconnaître ses talents mathématiques comme ses qualités pédagogiques. Mais la portée de ces leçons va bien au-delà, car on peut y voir apparaître plusieurs thèmes majeurs de l’œuvre ultérieure de Galilée : au plan mathématique, l’importance de la géométrie d’inspiration archimédienne, au plan physique, l’étude des questions de similitude que pose la résistance des matériaux – sans oublier l’intérêt constant du scientifique pour la langue et pour la culture littéraire.In 1587 the young Galileo was invited to give Due lezioni all’Accademia Fiorentina circa la figura, sito e grandezza dell’Inferno di Dante (hereafter Leçons sur l’Enfer) [Galilei 1587] aimed at settling an intense controversy regarding the geography of the Dantean Hell. This study in exegetics enabled Galileo to bring his mathematical talents and didactic qualities to the knowledge of the Tuscan scholars. But these lessons have a much greater importance, in that they reveal several of the major themes of Galileo’s further work from both mathematical and physical standpoints, such as the question of scale linked to the strength of materials, as well as the scientist’s unremitting interest in language and commitment to literary culture

Moving vortices in noncommutative gauge theory

Exact time-dependent solutions of nonrelativistic noncommutative Chern - Simons gauge theory are presented in closed analytic form. They are different from (indeed orthogonal to) those discussed recently by Hadasz, Lindstrom, Rocek and von Unge. Unlike theirs, our solutions can move with an arbitrary constant velocity, and can be obtained from the previously known static solutions by the recently found ``exotic'' boost symmetry.Comment: Latex, 6 pages, no figures. A result similar to ours was obtained, independently, by Hadasz et al. in the revised version of their pape