∗Speaker. †We (S.D. and P.U.) thank the Galileo Galilei Institute for Theoretical Physics in Florence for the hospitality and the INFN for partial support during the completion of this work. P.U. is supported as Heisenberg Fellow of the Deutsche Forschungsgemeinschaft DFG. This work is supported in part by the European Community’s Marie-Curie Research Training Network HEPTOOLS under contract MR...

All indecomposable finite-dimensional representations of the homogeneous Galilei group which when restricted to the rotation subgroup are decomposed to spin 0, 1/2 and 1 representations are constructed and classified. These representations are also obtained via contractions of the corresponding representations of the Lorentz group. Finally the obtained representations are used to derive a gener...

We review both the kinematics and dynamics of non-lorentzian theories their associated geometries. First, we introduce kinematical spacetimes symmetry algebras. Next, construct actions describing particle in some these spaces using method nonlinear realisations. explain relation with coadjoint orbit method. continue discussing three types gravity theories: Galilei gravity, Newton-Cartan Carroll...

The method of nonlinear realizations is a convenient tool for building dynamical Lie group, which relies solely upon structure relations the corresponding algebra. goal this work to discuss advantages and limitations method, here applied construct perfect fluid equations with conformal symmetry. Four cases are studied in detail, include Schr\"odinger $\ensuremath{\ell}$-conformal Galilei Lifshi...

Abstract A physically relevant unitary irreducible non-projective representation of the Galilei group is possible in Koopman–von Neumann formulation classical mechanics. This characterized by vanishing central charge algebra. contrast to quantum case where mass plays role charge. Here we show, direct construction, that mechanics also allows for a projective We extend result certain kind quantum...

Philosophy [nature] is written in that great book which ever lies before our eyes—I mean the universe—but we cannot understand it if we do not first learn the language.in which it is written. This book is written in the mathematical language.without which one wanders in vain through a dark labyrinth. —Galileo Galilei, The Assayer, 1623 (quoted in Burtt EA, The Metaphysical Foundations of Modern...