A Hamiltonian Formulation of the Pais-Uhlenbeck Oscillator that Yields a Stable and Unitary Quantum System

TL;DR

This paper introduces a new Hamiltonian formulation for the Pais-Uhlenbeck Oscillator, demonstrating stability and unitarity in the quantum regime for non-degenerate frequencies and providing a complex Hamiltonian approach for the degenerate case.

Contribution

It presents a novel Hamiltonian formulation that ensures a stable, unitary quantum system and extends to the degenerate case using complex Hamiltonian and PT-symmetry techniques.

Findings

Quantum Hamiltonian is Hermitian with positive spectrum for non-degenerate frequencies.

A complex Hamiltonian with PT-symmetry describes the degenerate case.

The approach guarantees stability and unitarity in the quantum Pais-Uhlenbeck Oscillator.

Abstract

We offer a new Hamiltonian formulation of the classical Pais-Uhlenbeck Oscillator and consider its canonical quantization. We show that for the non-degenerate case where the frequencies differ, the quantum Hamiltonian operator is a Hermitian operator with a positive spectrum, i.e., the quantum system is both stable and unitary. A consistent description of the degenerate case based on a Hamiltonian that is quadratic in momenta requires its analytic continuation into a complex Hamiltonian system possessing a generalized PT-symmetry (an involutive antilinear symmetry). We devise a real description of this complex system, derive an integral of motion for it, and explore its quantization.