International audienceChiral fermions can be embedded into Souriau's massless spinning particle model by "enslaving" the spin, viewed as a gauge constraint. The latter is not invariant under Lorentz boosts; spin enslavement can be restored, however, by a Wigner-Souriau (WS) translation, analogous to a compensating gauge transformation. The combined transformation is precisely the recently uncovered twisted boost, which we now extend to finite transformations. WS-translations are identified with the stability group of a motion acting on the right on the Poincare group, whereas the natural Poincare action corresponds to action on the left

The isometries of an exact plane gravitational wave are symmetries for both massive and massless particles. Their conformal extensions which are in fact Chrono-Projective transformations introduced by Duval et al are symmetries for massless particles and play an important role for the Memory Effect. The generically 5-parameter isometry group extends under special circumstances to a 6 parameter group, further extended by homotheties, which are universal Chrono-Projective symmetries for any profile.

The recently discovered conserved quantity associated with Kepler rescaling is generalised by an extension of Noether's theorem which involves the classical action integral as an additional term.For a free particle the familiar Schroedinger-dilations are recovered. A general pattern arises for homogeneous potentials. The associated conserved quantity allows us to derive the virial theorem.The relation to the Bargmann framework is explained and illustrated by exact plane gravitational waves.

For a plane gravitational wave whose profile is given, in Brinkmann coordinates, by a 2 × 2 symmetric traceless matrix K(U ), the matrix Sturm-Liouville equationP = KP plays a multiple and central rôle: (i) it determines the isometries; (ii) it appears as the key tool for switching from Brinkmann to BJR coordinates and vice versa; (iii) it determines the trajectories of particles initially at rest. All trajectories can be obtained from trivial "Carrollian" ones by a suitable action of the (broken) Carrollian isometry group. PACS numbers: 04.20.-q Classical general relativity; 02.20.Sv Lie algebras of Lie groups; 04.30.-w Gravitational waves General Relativity and Gravitation (2018) 50:107 https://doi.

Kepler's rescaling becomes, when "Eisenhart-Duval lifted" to 5-dimensional "Bargmann" gravitational wave spacetime, an ordinary spacetime symmetry for motion along null geodesics, which are the lifts of Keplerian trajectories. The lifted rescaling generates a well-behaved conserved Noether charge upstairs, which takes an unconventional form when expressed in conventional terms. This conserved quantity seems to have escaped attention so far. Applications include the Virial Theorem and also Kepler's Third Law. The lifted Kepler rescaling is a Chrono-Projective transformation. The results extend to celestial mechanics and Newtonian Cosmology.

The Eisenhart lift of a Paul Trap used to store ions in molecular physics is a linearly polarized periodic gravitational wave. A modified version of Dehmelt's Penning Trap is in turn related to circularly polarized periodic gravitational waves, sought for in inflationary models. Similar equations rule also the Lagrange points in Celestial Mechanics. The explanation is provided by anisotropic oscillators.Phys. Rev. D 98 (2018) no.4, 044037

The integrals of the motion associated with conformal Killing vectors of a curved space-time with an additional electromagnetic background are studied for massive particles. They involve a new term which might be non-local. The difficulty disappears for pp-waves, for which explicit, local conserved charges are found. Alternatively, the mass can be taken into account by "distorting" the conformal Killing vectors. The relation of these non-point symmetries to the charges is analysed both in the Lagrangian and Hamiltonian approaches, as well as in the framework of Eisenhart-Duval lift.

Employing the Foldy-Wouthuysen transformation it is demonstrated straightforwardly that the first and second Chern numbers are equal to the coefficients of the 2+1 and 4+1 dimensional Chern-Simons actions which are generated by the massive Dirac fermions coupled to the Abelian gauge fields. A topological insulator model in 2 + 1 dimensions is discussed and by means of a dimensional reduction approach the 1 + 1 dimensional descendant of the 2 + 1 dimensional Chern-Simons theory is presented. Field strength of the Berry gauge field corresponding to the 4+1 dimensional Dirac theory is explicitly derived through the Foldy-Wouthuysen transformation. Acquainted with it the second Chern numbers are calculated for specific choices of the integration domain. A method is proposed to obtain 3 + 1 and 2 + 1 dimensional descendants of the effective field theory of the 4 + 1 dimensional time reversal invariant topological insulator theory. Inspired by the spin Hall effect in graphene, a hypothetical model of the time reversal invariant spin Hall insulator in 3 + 1 dimensions is proposed.

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