Acceleration-extended Newton–Hooke symmetry and its dynamical realization

Newton–Hooke group is the nonrelativistic limit of de Sitter (anti-de Sitter) group, which can be enlarged with transformations that describe constant acceleration. We consider a higher order Lagrangian that is quasi-invariant under the acceleration-extended Newton–Hooke symmetry, and obtain the Schrödinger equation quantizing the Hamiltonian corresponding to its first order form. We show that the Schrödinger equation is invariant under the acceleration-extended Newton–Hooke transformations. We also discuss briefly the exotic conformal Newton–Hooke symmetry in 2+12+1 dimensions.