Treating Ostrogradski instability for Gallilean invariant Chern Simon's model via \mathcal{PT} symmetry

TL;DR

This paper addresses the Ostrogradski instability in higher derivative theories by applying PT symmetry, demonstrating a method to obtain a stable, bounded Hamiltonian in a Galilean invariant Chern-Simons model.

Contribution

It introduces a PT symmetry approach to resolve Ostrogradski ghosts in second class constrained systems, exemplified by the Galilean invariant Chern-Simons model.

Findings

Hamiltonian becomes free from linear momenta

Hamiltonian is bounded from below

Method applicable to second class constrained systems

Abstract

The Ostrogradski ghost problem that appears in higher derivative theories containing constraints has been considered here. Specifically we have considered systems where only the second class constraints appear. For these kind of systems, it is not possible to gauge away the linear momenta that cause the instability. To solve this issue, we have considered the PT symmetric aspects of the theory. As an example we have considered the Galilean invariant Chern Simons model in $2+1$ D which is a purely second class system. By solving the constraints, in the reduced phasespace, we have derived the \p similarity transformed Hamiltonian and putting conditions on we found that the final form of the Hamiltonian is free from any linear momenta and bounded from below.