# On the equivalence of the Pais-Uhlenbeck oscillator model and two non-Hermitian Harmonic Oscillators

## TL;DR

This paper demonstrates the equivalence between the Pais-Uhlenbeck oscillator and two non-Hermitian harmonic oscillators, providing insights into quantization issues and divergence problems in higher-order quantum systems.

## Contribution

It explicitly shows how to convert two harmonic oscillators into a Pais-Uhlenbeck model and addresses quantization challenges, including divergence issues in vacuum states and path integrals.

## Key findings

- Established the conversion procedure between harmonic oscillators and Pais-Uhlenbeck model
- Analyzed quantization to avoid divergence problems in vacuum states
- Discussed implications for PT-symmetric quantum systems and quantum field theory

## Abstract

A system of two independent Bosonic Harmonic Oscillators is converted into the respective fourth-order derivative Pais-Uhlenbeck oscillator model. The conversion procedure displays transparently how the quantization of the fourth-order derivative Pais-Uhlenbeck oscillator has to be performed in order not to suffer from the divergence problems of the vacuum state and path integrals as conjectured most recently by P. D. Mannheim in his article ``Determining the normalization of the quantum field theory vacuum, with implications for quantum gravity" [arXiv:2301.13029 [hep-th]]. In order to make the case we present the construction of the path integral, generating functionals and vacuum persistence amplitudes for PT-symmetry completed systems in Quantum Mechanics and Quantum Field Theory and discuss some implications to Quantum Field Theory and Particle Physics.