Newton Hooke Group Research Articles

(1 + 1) Newton–Hooke group for the simple and damped harmonic oscillator

It is demonstrated that, in the framework of the orbit method, a simple and damped harmonic oscillator is indistinguishable at the level of an abstract Lie algebra. This opens a possibility for treating the dissipative systems within the orbit method. An in-depth analysis of the coadjoint orbits of the (1 + 1) dimensional Newton-Hooke group is presented. Furthermore, it is argued that the physical interpretation is carried by a specific realisation of the Lie algebra of smooth functions on a phase space rather than by an abstract Lie algebra.

Dynamical realizations of l-conformal Newton–Hooke group

The method of nonlinear realizations and the technique previously developed in [A. Galajinsky, I. Masterov, Nucl. Phys. B 866 (2013) 212, arXiv:1208.1403] are used to construct a dynamical system without higher derivative terms, which holds invariant under the l-conformal Newton–Hooke group. A configuration space of the model involves coordinates, which parametrize a particle moving in d spatial dimensions and a conformal mode, which gives rise to an effective external field. The dynamical system describes a generalized multi-dimensional oscillator, which undergoes accelerated/decelerated motion in an ellipse in accord with evolution of the conformal mode. Higher derivative formulations are discussed as well. It is demonstrated that the multi-dimensional Pais–Uhlenbeck oscillator enjoys the l=32-conformal Newton–Hooke symmetry for a particular choice of its frequencies.

Goryachev-Chaplygin, Kovalevskaya, and Brdička-Eardley-Nappi-Witten pp-waves spacetimes with higher rank Stäckel-Killing tensors

Hidden symmetries of the Goryachev-Chaplygin and Kovalevskaya gyrostats spacetimes, as well as the Brdička-Eardley-Nappi-Witten pp-waves are studied. We find out that these spacetimes possess higher rank Stäckel-Killing tensors and that in the case of the pp-wave spacetimes, the symmetry group of the Stäckel-Killing tensors is the well-known Newton-Hooke group.

Kohn condition and exotic Newton–Hooke symmetry in the non-commutative Landau problem

N “exotic” [alias non-commutative] particles with masses ma, charges ea and non-commutative parameters θa, moving in a uniform magnetic field B, separate into center-of-mass and internal motions if Kohnʼs condition ea/ma=const is supplemented with eaθa=const. Then the center-of-mass behaves as a single exotic particle carrying the total mass and charge of the system, M and e, and a suitably defined non-commutative parameter Θ. For vanishing electric field off the critical case eΘB≠1, the particles perform the usual cyclotronic motion with modified but equal frequency. The system is symmetric under suitable time-dependent translations which span a (4+2)-parameter centrally-extended subgroup of the “exotic” [i.e., two-parameter centrally-extended] Newton–Hooke group. In the critical case B=Bc=(eΘ)−1 the system is frozen into a static “crystal” configuration. Adding a constant electric field, all particles perform, collectively, a cyclotronic motion combined with a drift perpendicular to the electric field when eΘB≠1. For B=Bc the cyclotronic motion is eliminated and all particles move, collectively, following the Hall law. Our time-dependent symmetries are reduced to the (2+1)-parameter Heisenberg group of centrally-extended translations.

Kohn’s theorem, Larmor’s equivalence principle and the Newton–Hooke group

We consider non-relativistic electrons, each of the same charge to mass ratio, moving in an external magnetic field with an interaction potential depending only on the mutual separations, possibly confined by a harmonic trapping potential. We show that the system admits a “relativity group” which is a one-parameter family of deformations of the standard Galilei group to the Newton–Hooke group which is a Wigner–İnönü contraction of the de Sitter group. This allows a group-theoretic interpretation of Kohn’s theorem and related results. Larmor’s theorem is used to show that the one-parameter family of deformations are all isomorphic. We study the “Eisenhart” or “lightlike” lift of the system, exhibiting it as a pp-wave. In the planar case, the Eisenhart lift is the Brdička–Eardley–Nappi–Witten pp-wave solution of Einstein–Maxwell theory, which may also be regarded as a bi-invariant metric on the Cangemi–Jackiw group.

Galilean quantum gravity with cosmological constant and the extended q-Heisenberg algebra

We define a theory of Galilean gravity in 2+1 dimensions with cosmological constant as a Chern-Simons gauge theory of the doubly-extended Newton-Hooke group, extending our previous study of classical and quantum gravity in 2+1 dimensions in the Galilean limit. We exhibit an r-matrix which is compatible with our Chern-Simons action (in a sense to be defined) and show that the associated bi-algebra structure of the Newton-Hooke Lie algebra is that of the classical double of the extended Heisenberg algebra. We deduce that, in the quantisation of the theory according to the combinatorial quantisation programme, much of the quantum theory is determined by the quantum double of the extended q-deformed Heisenberg algebra.

Acceleration-extended Newton–Hooke symmetry and its dynamical realization

Newton–Hooke group is the nonrelativistic limit of de Sitter (anti-de Sitter) group, which can be enlarged with transformations that describe constant acceleration. We consider a higher order Lagrangian that is quasi-invariant under the acceleration-extended Newton–Hooke symmetry, and obtain the Schrödinger equation quantizing the Hamiltonian corresponding to its first order form. We show that the Schrödinger equation is invariant under the acceleration-extended Newton–Hooke transformations. We also discuss briefly the exotic conformal Newton–Hooke symmetry in 2 + 1 dimensions.

On a curvature-statistics theorem

The spin-statistics theorem in quantum field theory relates the spin of a particle to the statistics obeyed by that particle. Here we investigate an interesting correspondence or connection between curvature (κ = ±1) and quantum statistics (Fermi-Dirac and Bose-Einstein, respectively). The interrelation between both concepts is established through vacuum coherent configurations of zero modes in quantum field theory on the compact O(3) and noncompact O(2; 1) (spatial) isometry subgroups of de Sitter and Anti de Sitter spaces, respectively. The high frequency limit, is retrieved as a (zero curvature) group contraction to the Newton-Hooke (harmonic oscillator) group. We also make some comments on the physical significance of the vacuum energy density and the cosmological constant problem.

NEWTON–HOOKE LIMIT OF BELTRAMI–DE SITTER SPACETIME, PRINCIPLE OF GALILEI–HOOKE'S RELATIVITY AND POSTULATE ON NEWTON–HOOKE UNIVERSAL TIME

Based on the Beltrami–de Sitter spacetime, we present the Newton–Hooke model under the Newton–Hooke contraction of the BdS spacetime with respect to the transformation group, algebra and geometry. It is shown that in Newton–Hooke space–time, there are inertial-type coordinate systems and inertial-type observers, which move along straight lines with uniform velocity. And they are invariant under the Newton–Hooke group. In order to determine uniquely the Newton–Hooke limit, we propose the Galilei–Hooke's relativity principle as well as the postulate on Newton–Hooke universal time. All results are readily extended to the Newton–Hooke model as a contraction of Beltrami–anti-de Sitter spacetime with negative cosmological constant.

Locally operating realizations of transformation Lie groups

Using the Mackey theory of induced representations, a systematic study of the locally operating multiplier realizations of a connected Lie group G that acts transitively on a space-time manifold is presented. We obtain a mathematical characterization of the locally operating multiplier realizations and a reduction of the problem of multiplier locally operating realizations to linear ones via a splitting group G‘;m for G. In this way the locally operating multiplier realizations are obtained by induction from finite-dimensional linear representations of a well-determined subgroup of Ḡ. Some examples, such as the two-dimensional Euclidean group, the Galilei group, and the one-dimensional Newton–Hooke group, are given.

Gauge equivalence of representations of symmetry groups in quantum mechanics

The equivalence of representations of symmetry groups operating upon wavefunctions in configuration space is studied with regard to the (intuitive) notion of physical equivalence. A refinement of the usual projective equivalence relation is introduced, called gauge equivalence, for which the allowed unitary equivalence transformations are gauge transformations. For a Euclidean as well as for a Newton-Hooke symmetry group the gauge equivalence classes of unitary multiplier representations are determined. These examples support the assertion that equivalence from a physical viewpoint corresponds better to this new gauge equivalence concept than to the usual notion of projective equivalence.