Hamiltonian formalisms and symmetries of the Pais-Uhlenbeck oscillator

TL;DR

This paper explores the Hamiltonian formalism and symmetries of the Pais-Uhlenbeck oscillator, revealing extended algebraic structures and connecting different Hamiltonian approaches with the free higher derivatives theory.

Contribution

It derives symmetry generators in various Hamiltonian frameworks and establishes their algebraic relations, including the centrally extended l-conformal Newton-Hooke algebra for odd frequencies.

Findings

Symmetry generators are derived in both original and Ostrogradski Hamiltonian formalisms.

The algebra of generators is a central extension of the Lagrangian symmetry algebra.

A canonical transformation relates different Hamiltonian formalisms and aligns with the orbit method.

Abstract

The study of the symmetry of Pais-Uhlenbeck oscillator initiated in [Nucl. Phys. B 885 (2014) 150] is continued with special emphasis put on the Hamiltonian formalism. The symmetry generators within the original Pais and Uhlenbeck Hamiltonian approach as well as the canonical transformation to the Ostrogradski Hamiltonian framework are derived. The resulting algebra of generators appears to be the central extension of the one obtained on the Lagrangian level; in particular, in the case of odd frequencies one obtains the centrally extended l-conformal Newton-Hooke algebra. In this important case the canonical transformation to an alternative Hamiltonian formalism (related to the free higher derivatives theory) is constructed. It is shown that all generators can be expressed in terms of the ones for the free theory and the result agrees with that obtained by the orbit method.