arXiv subjects
Publications and source records attributed to Ivan Masterov.
We show that the In\"{o}n\"{u}-Wigner contraction of $so(\ell+1,\ell+d)$ with the integer $\ell>1$ may lead to algebra which contains a variety of conformal extensions of the Galilei algebra as subalgebras. These extensions involve the $\ell$-conformal Galilei algebra in $d$ spatial dimensions as well as $l$-conformal Galilei algebras in one spatial dimension with $l=3$, $5$, ..., $(2\ell-1)$.
Two Lagrangian formulations for describing of the damped harmonic oscillator have been introduced by Bateman. For these models we construct higher derivative generalization which enjoys the l-conformal Newton-Hooke symmetry. The dynamics of generalized systems corresponds to the damped Pais-Uhlenbeck oscillator for a particular choice of its frequencies.
The issue of constructing N=1,2,3 supersymmetric extensions of the l-conformal Galilei algebra is reconsidered following the approach in [JHEP 1709 (2017) 131]. Drawing a parallel between acceleration generators entering the superalgebra and irreducible supermultiplets of d=1, N-extended superconformal group, a new N=1 l-conformal Galilei superalgebra, two new N=2 variants, and two new N=3 versions are built. Realisations in terms of differential operators in superspace are given.
It was recently demonstrated that N=1,2,3,4 super-Schwarzian derivatives can be constructed by applying the method of nonlinear realisations to finite-dimensional superconformal groups OSp(1|2), SU(1,1|1), OSp(3|2), SU(1,1|2), respectively, thus avoiding the use of superconformal field theory techniques. In this work, a similar construction is applied to the exceptional supergroup D(2,1;a), which describes the most general N=4 supersymmetric extension of SL(2,R), with the aim to study possible candidates for a D(2,1;a) super-Schwarzian derivative.
It is shown that, if the energy in the Schwarzian mechanics (SM) is equal to the coupling constant in the de Alfaro-Fubini-Furlan (DAFF) model, there exists a link between these two systems. In particular, the equation of motion, SL(2,R)-symmetry transformations and the corresponding conserved charges of SM can be derived from those of the DAFF model by applying a coordinate transformation of a special type, while the general solution of the DAFF system maps to the velocity function of SM. A way to reproduce this link via the method of nonlinear realizations is presented. Schwarzian counterparts of the DAFF mechanics in Newton-Hooke (NH) spacetime as well as a higher derivative generalization of the DAFF model are discussed.
We formulate the equations which determine a potential function in an $\mathcal{N}=1$ higher derivative supersymmetric mechanics compatible with the $osp(2|1)\oplus so(d)$ symmetry and provide a few explicit examples.
The full set of Casimir elements of the centrally extended l-conformal Galilei algebra is found in simple and tractable form.
An N=4 supersymmetric extension of the l-conformal Galilei algebra is constructed. This is achieved by combining generators of spatial symmetries from the l-conformal Galilei algebra and those underlying the most general superconformal group in one dimension D(2,1;a). The value of the group parameter a is fixed from the requirement that the resulting superalgebra is finite-dimensional. The analysis reveals a=-1/2 thus reducing D(2,1;a) to OSp(4|2).
The Eisenhart lift provides an elegant geometric description of a dynamical system of second order in terms of null geodesics of the Brinkmann-type metric. In this work, we attempt to generalize the Eisenhart method so as to encompass higher derivative models. The analysis relies upon Ostrogradsky's Hamiltonian. A consistent geometric description seems feasible only for a particular class of potentials. The scheme is exemplified by the Pais-Uhlenbeck oscillator.
Based on the results in [Nucl. Phys. B 866 (2013) 212], we consider a way to construct a higher-derivative mechanical model which possesses the $l$-conformal Galilei symmetry. The dynamical system describes generalized Pais-Uhlenbeck oscillator in the presence of an effective external field.
We consider an N=2 supersymmetric odd-order Pais-Uhlenbeck oscillator with distinct frequencies of oscillation. The technique previously developed in [Acta Phys. Polon. B 36 (2005) 2115; Nucl. Phys. B 902 (2016) 95] is used to construct a family of Hamiltonian structures for this system.
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.
Ostrogradsky's 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 [Acta Phys. Polon. B 36 (2005) 2115] 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.
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.
We construct an N=2 supersymmetric extension of the Pais-Uhlenbeck oscillator for distinct frequencies of oscillation. A link to a set of decoupled N=2 supersymmetric harmonic oscillators with alternating sign in the Hamiltonian is introduced. Symmetries of the model are discussed in detail. The investigation of a quantum counterpart of the constructed model shows that the corresponding Fock space contains negative norm states and the energy spectrum of the system is unbounded from below.
By applying Niederer--like transformation, we construct a representation of the N=2 l-conformal Newton-Hooke superalgebra for the case of a negative cosmological constant in terms of linear differential operators as well as its dynamical realization. Another variant of N=2 supersymmetric Pais-Uhlenbeck oscillator for a particular choice of its frequencies is proposed. The advantages of such realizations as compared to their analogues introduced in [J. Math. Phys. 53 (2012) 072904], [J. Math. Phys. 56 (2015) 022902] are discussed.
The analysis previously developed in [J. Math. Phys. 55 (2014) 102901] is used to construct systems which hold invariant under N=2 l-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.
Dynamical systems which are invariant under N=1 supersymmetric extension of the l-conformal Galilei algebra are constructed. These include a free N=1 superparticle which is governed by higher derivative equations of motion and an N=1 supersymmetric Pais-Uhlenbeck oscillator for a particular choice of its frequencies. A Niederer-like transformation which links the models is proposed.