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.

arXiv commentV2:12 pages,clarifying remarks included into the Introduction and Conclusion, the version to appear in NPB

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• Hamiltonian formulation for perfect fluid equations with the l-conformal Galilei symmetry
• Meta-Schrödinger invariance
• Higher-derivative generalization of conformal mechanics
• On Casimir Operators of Conformal Galilei Algebras
• Generalized Niederer's transformation for quantum Pais-Uhlenbeck oscillator
• Invariant PDEs of Conformal Galilei Algebra as deformations: cryptohermiticity and contractions
• Superfield approach to higher derivative N=1 superconformal mechanics
• Perfect fluid dynamics with conformal Newton-Hooke symmetries
• Nonlocal dynamics and infinite non-relativistic conformal symmetries
• N=2 supersymmetric odd-order Pais-Uhlenbeck oscillator
• Remarks on Galilean electromagnetism
• Fractional Galilean Symmetries