Ivan Masterov

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

Abstract 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.

Conformal Newton–Hooke symmetry of Pais–Uhlenbeck oscillator

K.A. and J.G. are grateful to Piotr Kosinski for helpful and illuminating discussions. We thank Peter Horvathy and Andrei Smilga for useful correspondence. This work was sup-ported by the NCN grant DEC-2013/09/B/ST2/02205 (K.A. and J.G.) and by the RFBR grants 13-02-90602-Arm (A.G.) and 14-02-31139-Mol (I.M.) as well as by the MSU program “Nauka” under the project 825 (A.G. and I.M.). I.M. gratefully acknowledges the support of the Dynasty Foundation. ©2014 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/3.0/). Funded by SCOAP3.

Dynamical realizations of l-conformal Newton–Hooke group

Abstract 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 = 3 2 -conformal Newton–Hooke symmetry for a particular choice of its frequencies.

Remark on quantum mechanics with N=2 Schrödinger supersymmetry

Abstract A unitary transformation which relates a many-body quantum mechanics with N = 2 Schrodinger supersymmetry to a set of decoupled superparticles is proposed. The simplification in dynamics is achieved at a price of a nonlocal realization of the full N = 2 Schrodinger superalgebra in a Hilbert space. The transformation is shown to be universal and applicable to conformal many-body models confined in a harmonic trap.

Dynamical realization of l-conformal Galilei algebra and oscillators

Abstract The method of nonlinear realizations is applied to the l -conformal Galilei algebra to construct a dynamical system without higher derivative terms in the equations of motion. A configuration space of the model involves coordinates, which parametrize particles in d spatial dimensions, and a conformal mode, which gives rise to an effective external field. It is shown that trajectories of the system can be mapped into those of a set of decoupled oscillators in d dimensions.

N=2 supersymmetric extension of l-conformal Galilei algebra

N=2 supersymmetric extension of the l-conformal Galilei algebra is constructed. A relation between its representations in flat spacetime and in Newton-Hooke spacetime is discussed. An infinite-dimensional generalization of the superalgebra is given.

An alternative Hamiltonian formulation for the Pais–Uhlenbeck oscillator

Ostrogradskys method allows one to construct Hamiltonian formulation for a higher derivative system. An application of this approach to the Pais–Uhlenbeck oscillator yields the Hamiltonian which is unbounded from below. This leads to the ghost problem in quantum theory. In order to avoid this nasty feature, the technique previously developed in [7] is used to construct an alternative Hamiltonian formulation for the multidimensional Pais–Uhlenbeck oscillator of arbitrary even order with distinct frequencies of oscillation. This construction is also generalized to the case of an N=2 supersymmetric Pais–Uhlenbeck oscillator.

Higher-derivative mechanics with N=2l-conformal Galilei supersymmetry

The analysis previously developed in [J. Math. Phys. 55 102901 (2014)] is used to construct systems which hold invariant under N=2l-conformal Galilei superalgebra. The models describe two different supersymmetric extensions of a free higher-derivative particle. Their Newton-Hooke counterparts are derived by applying appropriate coordinate transformations.

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 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 Ostrogradskys 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.

The odd-order Pais-Uhlenbeck oscillator

We consider a Hamiltonian formulation of the (2n + 1)-order generalization of the Pais–Uhlenbeck oscillator with distinct frequencies of oscillation. This system is invariant under time translations. However, the corresponding Noether integral of motion is unbounded from below and can be presented as a direct sum of 2n one-dimensional harmonic oscillators with an alternating sign. If this integral of motion plays a role of a Hamiltonian, a quantum theory of the Pais–Uhlenbeck oscillator faces a ghost problem. We construct an alternative canonical formulation for the system under study to avoid this nasty feature.