231 research outputs found
On dynamical realizations of l-conformal Galilei and Newton-Hooke algebras
In two recent papers [N. Aizawa, Y. Kimura, J. Segar, J. Phys. A 46 (2013)
405204] and [N. Aizawa, Z. Kuznetsova, F. Toppan, J. Math. Phys. 56 (2015)
031701], 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 Schrodinger 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 omega_k=(2k-1), k=1,...,n. We also confront the
higher derivative models with a genuine second order system constructed in our
recent work [A. Galajinsky, I. Masterov, Nucl. Phys. B 866 (2013) 212] which is
discussed in detail for l=3/2.Comment: V2:12 pages,clarifying remarks included into the Introduction and
Conclusion, the version to appear in NP
A variant of Schwarzian mechanics
The Schwarzian derivative is invariant under SL(2,R)-transformations and, as
thus, any function of it can be used to determine the equation of motion or the
Lagrangian density of a higher derivative SL(2,R)-invariant 1d mechanics or the
Schwarzian mechanics for short. In this note, we consider the simplest variant
which results from setting the Schwarzian derivative to be equal to a
dimensionful coupling constant. It is shown that the corresponding dynamical
system in general undergoes stable evolution but for one fixed point solution
which is only locally stable. Conserved charges associated with the
SL(2,R)-symmetry transformations are constructed and a Hamiltonian formulation
reproducing them is proposed. An embedding of the Schwarzian mechanics into a
larger dynamical system associated with the geodesics of a Brinkmann-like
metric obeying the Einstein equations is constructed.Comment: V2: 8 pages, typos fixed. The version to appear in NP
Remark on integrable deformations of the Euler top
The Euler top describes a free rotation of a rigid body about its center of
mass and provides an important example of a completely integrable system. A
salient feature of its first integrals is that, up to a reparametrization of
time, they uniquely determine the dynamical equations themselves. In this note,
this property is used to construct integrable deformations of the Euler top.Comment: V2:the version published in JMA
N=2 superparticle near horizon of extreme Kerr-Newman-AdS-dS black hole
Conformal mechanics related to the near horizon extreme Kerr-Newman-AdS-dS
black hole is studied. A unique N=2 supersymmetric extension of the conformal
mechanics is constructed.Comment: V2: the version to appear in NP
Geometry of the isotropic oscillator driven by the conformal mode
Geometrization of a Lagrangian conservative system typically amounts to
reformulating its equations of motion as the geodesic equations in a properly
chosen curved spacetime. The conventional methods include the Jacobi metric and
the Eisenhart lift. In this work, a modification of the Eisenhart lift is
proposed which describes the isotropic oscillator in arbitrary dimension driven
by the one-dimensional conformal mode.Comment: V3: 10 pages, presentation improved, the version to appear in Eur.
Phys. J.
Remarks on l-conformal extension of the Newton-Hooke algebra
The l-conformal extension of the Newton-Hooke algebra proposed in [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.Comment: V3: terminology improved, one reference added; the version to appear
in PL
N=4 l-conformal Galilei superalgebras inspired by D(2,1;a) supermultiplets
N=4 supersymmetric extensions of the l-conformal Galilei algebra are
constructed by properly extending the Lie superalgebra associated with the most
general N=4 superconformal group in one dimension D(2,1;a). If the acceleration
generators in the superalgebra form analogues of the irreducible (1,4,3)-,
(2,4,2)-, (3,4,1)-, and (4,4,0)-supermultiplets of D(2,1;a), the parameter a
turns out to be constrained by the Jacobi identities. In contrast, if the tower
of the acceleration generators resembles a component decomposition of a generic
real superfield, which is a reducible representation of D(2,1;a), a remains
arbitrary. An N=4 l-conformal Galilei superalgebra recently proposed in [Phys.
Lett. B 771 (2017) 401] is shown to be a particular instance of a more general
construction in this work.Comment: V2: 9 pages. Introductory part extended, two references added. The
version to appear in JHE
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