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

Abstract

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.

V2:12 pages,clarifying remarks included into the Introduction and Conclusion, the version to appear in NPB

References in corpus (5)

• No-ghost theorem for the fourth-order derivative Pais-Uhlenbeck oscillator model
• Acceleration-Extended Galilean Symmetries with Central Charges and their Dynamical Realizations
• Remarks on l-conformal extension of the Newton-Hooke algebra
• $\ell$-oscillators from second-order invariant PDEs of the centrally extended Conformal Galilei Algebras
• Higher-derivative mechanics with N=2 l-conformal Galilei supersymmetry

Cited by in corpus (11)

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• Remark on higher-derivative mechanics with l-conformal Galilei symmetry
• Hamiltonian formulation for perfect fluid equations with the l-conformal Galilei symmetry
• Meta-Schrödinger invariance
• Perfect fluid dynamics with conformal Newton-Hooke symmetries
• Superfield approach to higher derivative N=1 superconformal mechanics
• N=2 supersymmetric odd-order Pais-Uhlenbeck oscillator
• Remarks on Galilean electromagnetism