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

Abstract

In two recent papers (Aizawa et al., 2013 [15]) and (Aizawa et al., 2015 [16]), representation theory ofthe centrally extended l-conformal Galilei algebra with half-integer l has been applied so as to constructsecond order differential equations exhibiting the corresponding group as kinematical symmetry. It wassuggested to treat them as the Schrodinger equations which involve Hamiltonians describing dynamicalsystems without higher derivatives. The Hamiltonians possess two unusual features, however. First, theyinvolve the standard kinetic term only for one degree of freedom, while the remaining variables providecontributions linear in momenta. This is typical for Ostrogradsky’s canonical approach to the description ofhigher derivative systems. Second, the Hamiltonian in the second paper is not Hermitian in the conventionalsense. In this work, we study the classical limit of the quantum Hamiltonians and demonstrate that the firstof them is equivalent to the Hamiltonian describing free higher derivative nonrelativistic particles, whilethe second can be linked to the Pais–Uhlenbeck oscillator whose frequencies form the arithmetic sequence?k = (2k ? 1), k = 1,..., n. We also confront the higher derivative models with a genuine second ordersystem constructed in our recent work (Galajinsky and Masterov, 2013 [5]) which is discussed in detailfor l = 32

Ricci-flat spacetimes with l -conformal Galilei symmetry

“…where α(y) is a function to be fixed below and ǫ = ±1. It is assumed that y remains intact under the l-conformal Galilei transformations so that the metric maintains the symmetry of its predecessor (18). One could promote k n in (19) to become real functions k n (y) as well.…”

Remark on higher-derivative mechanics with l-conformal Galilei symmetry

“…The remarkable property of the particles, which described by (19), (20), is that their motion is bounded. According to (17), one and the same eigenvalues of the matrix (15) appear for different values of parameter l. Therefore, one can realize different l-conformal Galilei algebras in one and the same system of equations. For example, the system (1a), (6) may accommodate l-conformal Galilei symmetry for any integer l > 2.…”

Higher-derivative generalization of conformal mechanics

“…Recently, symmetries of these higher-derivative models have been extensively studied 13,14,[16][17][18][19][20][21][22][23] . In particular it has been shown that the l-conformal Galilei group is the maximal symmetry group of the free (2l + 1)-order particle 14 .…”