Newton-Hooke Algebra Research Articles

Symmetry harmonization: exploring deformed oscillators and dissipative dynamics through the glass of Newton–Hooke algebra

Motivated by the symmetry in the non-relativistic limit of anti-de Sitter geometry, we employ planar dynamical models featuring exotic (deformed) harmonic oscillators, presented through direct and indirect Lagrangian representations. The latter introduces Bateman dissipative oscillator system. Analyzing these dynamic systems with a first-order Lagrangian scheme, our phase-space-based approach utilizes the moment map components to reveal the underlying symmetry algebra. This obtained algebra, interpreted as an extended version of Newton–Hooke (NH) cosmological symmetry algebras, has the potential to cast an augmented non-relativistic shadow over the expanding universe, offering an insightful perspective on extended NH spacetime in 2+1 dimensions through our dynamical realizations.

Equivalence principle, de-Sitter space, and cosmological twistors

In this paper, I discuss the impact of the positive cosmological constant on the interplay between the equivalence principle in general relativity, and the rules of quantum mechanics. At the nonrelativistic level, there is an ambiguity in the definition of a phase of a wave function measured by inertial and accelerating observes. This is the cosmological analogue of the Penrose effect, which can also be seen as a nonrelativistic limit of the Unruh effect. The symmetries of the associated Schrödinger equation are generated by the Newton–Hooke algebra, which arises from a nonrelativistic limit of a cosmological twistor space.

Non-relativistic gravity theories in four spacetime dimensions

In this work we present a non-relativistic gravity theory defined in four spacetime dimensions using the MacDowell-Mansouri geometrical formulation. We obtain a Newtonian gravity action which is constructed from the curvature of a Newton-Hooke version of the so-called Newtonian algebra. We show that the non-relativistic gravity theory presented here contains the Poisson equation in presence of a cosmological constant. Moreover we make contact with the Modified Newtonian Dynamics (MOND) approach for gravity by considering a particular ansatz for a given gauge field. We extend our results to a generalized non-relativistic MacDowell-Mansouri gravity theory by considering a generalized Newton-Hooke algebra.

Carrollian and non-relativistic Jackiw–Teitelboim supergravity

We present non- and ultra-relativistic Jackiw–Teitelboim (JT) supergravity as metric BF theories based on the extended Newton–Hooke and extended AdS Carroll superalgebras in two spacetime dimensions, respectively. The extended Newton–Hooke structure, and, in particular, the invariant metric necessary for the BF construction of non-relativistic JT supergravity, is obtained by performing an expansion of the mathcal {N}=2 AdS_2 superalgebra. Subsequently, we introduce the extended AdS_2 Carroll superalgebra, and the associated invariant metric, as a suitable redefinition of the extended Newton–Hooke superalgebra. The mapping involved can be seen as the supersymmetric extension of the duality existing at the purely bosonic level between the extended Newton–Hooke algebra with (positive) negative cosmological constant and the extended (A)dS Carroll algebra in two dimensions. Finally, we provide the Carrollian JT supergravity action in the BF formalism. Moreover, we show that both the non-relativistic and the ultra-relativistic theories presented can also be obtained by direct expansion of mathcal {N}=2 JT supergravity.

Coset space actions for nonrelativistic strings

We formulate the stringy nonrelativistic limits of the flat space and AdS5×S5 string as coset models, based on the string Bargmann and extended string Newton-Hooke algebras respectively. Our construction mimics the typical relativistic one, but differs in several interesting ways. Using our coset formulation we give a Lax representation of the equations of motion of both models.

Beyond Lorentzian symmetry

This thesis presents a framework in which to explore kinematical symmetries beyond the standard Lorentzian case. This framework consists of an algebraic classification, a geometric classification, and a derivation of the geometric properties required to define physical theories on the classified spacetime geometries. The work completed in substantiating this framework for kinematical, super-kinematical, and super-Bargmann symmetries constitutes the body of this thesis. To this end, the classification of kinematical Lie algebras in spatial dimension $D = 3$, as presented in [3, 4], is reviewed; as is the classification of spatially-isotropic homogeneous spacetimes of [5]. The derivation of geometric properties such as the non-compactness of boosts, soldering forms and vielbeins, and the space of invariant affine connections is then presented. We move on to classify the $\mathcal{N}=1$ kinematical Lie superalgebras in three spatial dimensions, finding 43 isomorphism classes of Lie superalgebras. Once these algebras are determined, we classify the corresponding simply-connected homogeneous (4|4)-dimensional superspaces and show how the resulting 27 homogeneous superspaces may be related to one another via geometric limits. Finally, we turn our attention to generalised Bargmann superalgebras. In the present work, these will be the $\mathcal{N}=1$ and $\mathcal{N}=2$ super-extensions of the Bargmann and Newton-Hooke algebras, as well as the centrally-extended static kinematical Lie algebra, of which the former three all arise as deformations. Focussing solely on three spatial dimensions, we find $9$ isomorphism classes in the $\mathcal{N}=1$ case, and we identify $22$ branches of superalgebras in the $\mathcal{N}=2$ case.

Three-dimensional exotic Newtonian gravity with cosmological constant

In this work we introduce a cosmological constant in the extended Newtonian gravity theory. To this end, we extend the exotic Newton-Hooke algebra by introducing new generators and central charges. The new algebra obtained here has been denoted as exotic Newtonian algebra and reproduces the extended Newtonian one in the flat limit ℓ→∞. A three-dimensional Chern-Simons action for the exotic Newtonian algebra is presented. We show that the non-relativistic gravity theory proposed here reproduces the most general extended Newtonian gravity theory in the flat limit.

Non-relativistic symmetries in three space-time dimensions and the Nappi-Witten algebra

We show that the Extended Bargmann and Newton-Hooke algebras in 2+1 dimensions can be obtained as expansions of the Nappi-Witten algebra. The procedure can be generalized to obtain two infinite families of non-relativistic symmetries, which include the Maxwellian Exotic Bargmann symmetry, its generalized Newton-Hooke counterpart, and its Hietarinta dual. In each case, the invariant bilinear form on the Nappi-Witten algebra leads to the invariant tensor on the expanded algebra, allowing one to construct the corresponding Chern-Simons gravity theory.

Stringy (Galilei) Newton-Hooke Chern-Simons gravities

We construct Chern-Simons gravities in (2 + 1)-dimensional space-time considering the Stringy Galilei algebra both with and without non-central extensions. In the first case, there is an invariant and non-degenerate bilinear form, however the field equations do not allow to express the spin connections in terms of the dreibeins. In the second case there is no invariant non-degenerate bilinear form. Therefore, in both cases we do not have an ordinary gravity theory. Instead, if we consider the stringy Newton-Hooke algebra with extensions as gauge group we have an invariant non-degenerate metric and from the field equations we express the spin connections in terms of the geometric fields.

Nonrelativistic limits of colored gravity in three dimensions

The three-dimensional non-relativistic isometry algebras, namely Galilei and Newton-Hooke algebras, are known to admit double central extensions, which allows for non-degenerate bilinear forms hence for action principles through Chern-Simons formulation. In three-dimensional colored gravity, the same central extension helps the theory evade the multi-graviton no-go theorems by enlarging the color-decorated isometry algebra. We investigate the non-relativistic limits of three-dimensional colored gravity in terms of generalized \.In\"on\"u-Wigner contractions.

Higher-derivative superparticle inAdS3space

Employing the coset approach we construct component actions for a superparticle moving in AdS$_3$ with $N{=}(2,0)$, $D{=}3$ supersymmetry partially broken to $N{=}2$, $d{=}1$. These actions may contain higher time-derivative terms, which are chosen to possess the same (super)symmetries as the free superparticle. In terms of the nonlinear-realization superfields, the component actions always take a simpler form when written in terms of covariant Cartan forms. We also consider in detail the reduction to the nonrelativistic case and construct the corresponding action a Newton-Hooke superparticle and its higher-derivative generalizations. The structure of these higher time-derivative generalizations is completely fixed by invariance under the supersymmetric Newton-Hooke algebra extended by two central charges.

On dynamical realizations of l-conformal Galilei and Newton–Hooke algebras

In two recent papers (Aizawa et al., 2013 [15]) and (Aizawa et al., 2015 [16]), representation theory of the centrally extended l-conformal Galilei algebra with half-integer l has been applied so as to construct second order differential equations exhibiting the corresponding group as kinematical symmetry. It was suggested to treat them as the Schrödinger equations which involve Hamiltonians describing dynamical systems without higher derivatives. The Hamiltonians possess two unusual features, however. First, they involve the standard kinetic term only for one degree of freedom, while the remaining variables provide contributions linear in momenta. This is typical for Ostrogradsky's canonical approach to the description of higher derivative systems. Second, the Hamiltonian in the second paper is not Hermitian in the conventional sense. In this work, we study the classical limit of the quantum Hamiltonians and demonstrate that the first of them is equivalent to the Hamiltonian describing free higher derivative nonrelativistic particles, while the second can be linked to the Pais–Uhlenbeck oscillator whose frequencies form the arithmetic sequence ωk=(2k−1), k=1,…,n. We also confront the higher derivative models with a genuine second order system constructed in our recent work (Galajinsky and Masterov, 2013 [5]) which is discussed in detail for l=32.

Hamiltonian formalisms and symmetries of the Pais–Uhlenbeck oscillator

The study of the symmetry of Pais–Uhlenbeck oscillator initiated in Andrzejewski et al. (2014) [24] is continued with special emphasis put on the Hamiltonian formalism. The symmetry generators within the original Pais and Uhlenbeck Hamiltonian approach as well as the canonical transformation to the Ostrogradski Hamiltonian framework are derived. The resulting algebra of generators appears to be the central extension of the one obtained on the Lagrangian level; in particular, in the case of odd frequencies one obtains the centrally extended l-conformal Newton–Hooke algebra. In this important case the canonical transformation to an alternative Hamiltonian formalism (related to the free higher derivatives theory) is constructed. It is shown that all generators can be expressed in terms of the ones for the free theory and the result agrees with that obtained by the orbit method.

Conformal Newton–Hooke algebras, Niederer's transformation and Pais–Uhlenbeck oscillator

Dynamical systems invariant under the action of the l-conformal Newton–Hooke algebras are constructed by the method of nonlinear realizations. The relevant first order Lagrangians together with the corresponding Hamiltonians are found. The relation to the Galajinsky and Masterov [24] approach as well as the higher derivatives formulation is discussed. The generalized Niederer's transformation is presented which relates the systems under consideration to those invariant under the action of the l-conformal Galilei algebra [25]. As a nice application of these results an analogue of Niederer's transformation, on the Hamiltonian level, for the Pais–Uhlenbeck oscillator is constructed.

(In)finite extensions of algebras from their İnönü–Wigner contractions

The method to obtain massive non-relativistic states from the Poincaré algebra is twofold. First, following İnönü and Wigner, the Poincaré algebra has to be contracted to the Galilean one. Second, the Galilean algebra has 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 İnönü–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 (ultrarelativistic) contraction of the Poincaré algebra, the exotic Newton–Hooke algebra and some others.This paper is dedicated to the memory of Laurent Houart (1967–2011).

Remarks on l-conformal extension of the Newton–Hooke algebra

The l-conformal extension of the Newton–Hooke algebra proposed in [J. Negro, M.A. del Olmo, A. Rodriguez-Marco, J. Math. Phys. 38 (1997) 3810] is formulated in the basis in which the flat space limit is unambiguous. Admissible central charges are specified. The infinite-dimensional Virasoro–Kac–Moody type extension is given.

Formula omitted] superconformal Newton–Hooke algebra and many-body mechanics

A representation of the conformal Newton–Hooke algebra on a phase space of n particles in arbitrary dimension which interact with one another via a generic conformal potential and experience a universal cosmological repulsion or attraction is constructed. The minimal N=2 superconformal extension of the Newton–Hooke algebra and its dynamical realization in many-body mechanics are studied.

(Anti)de Sitter/Poincaré symmetries and representations from Poincaré/Galilei through a classical deformation approach

A classical deformation procedure, based on universal enveloping algebras, Casimirs and curvatures of symmetrical homogeneous spaces, is applied to several cases of physical relevance. Starting from the (3 + 1)D Galilei algebra, we describe at the level of representations the process leading to its two physically meaningful deformed neighbours. The Poincaré algebra is obtained by introducing a negative curvature in the flat Galilean phase space (or space of worldlines), while keeping a flat spacetime. To be precise, starting from a representation of the Galilei algebra with both Casimirs different from zero, we obtain a representation of the Poincaré algebra with both Casimirs necessarily equal to zero. The Poincaré angular momentum, Pauli–Lubanski components, position and velocity operators, etc are expressed in terms of ‘Galilean’ operators through some expressions deforming the proper Galilean ones. Similarly, the Newton–Hooke algebras appear by endowing spacetime with a nonzero curvature, while keeping a flat phase space. The same approach, starting from the (3 + 1)D Poincaré algebra provides representations of the (anti)de Sitter as Poincaré deformations.

Newton-Hooke algebras, nonrelativistic branes, and generalized pp-wave metrics

The Newton-Hooke algebras in d dimensions are constructed as contractions of dS(AdS) algebras. Nonrelativistic brane actions are WZ terms of these Newton-Hooke algebras. The NH algebras appear also ! ~

Quantum (2+1) kinematical algebras: a global approach

In this paper we give an approach to quantum deformations of the (2+1) kinematical Lie algebras within a scheme that simultaneously describes all groups of motions of classical geometries in N=3 dimensions. We cover at once all the kinematical geometries including the quantum versions of Inonu-Wigner contractions, which are defined in a natural way and relate q-deformations as expected. We thus obtain some q-deformations previously known for the three-dimensional Euclidean and (2+1)-Poincare algebras and also some new q-deformations for these and other kinematical algebras, such as the (2+1)-de Sitter, Galilei and Newton-Hooke algebras.