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 Schrödinger 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 ωk=(2k−1)ωk=(2k−1), k=1,…,nk=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.