A Consistent Path Integral Approach to Higher Derivative Oscillators

TL;DR

The authors study the quantum field theory of the Pais–Uhlenbeck higher-derivative oscillator, establishing a consistent canonical and path-integral formulation. By comparing Ostrogradsky and Hawking–Hertog coordinates, they show that introducing a Lagrange multiplier in the path integral is essential for a proper treatment of canonical coordinates, yet does not affect physical Green functions. Their analysis demonstrates that the PU theory can be quantized consistently, with a Fock space separating ghost and standard modes, and that adding non-derivative interactions yields improved UV convergence, making a wide class of φ-operators renormalizable. They explicitly compute one-loop renormalization for φ^4 and φ^6 interactions and argue that any even-power φ^{2n} interaction is renormalizable within this framework. The results have implications for higher-derivative theories, including variants of quadratic gravity, by providing a solid basis for their perturbative treatment while highlighting remaining challenges such as the unbounded energy spectrum and unitarity issues.

Abstract

In this work, we study the Quantum Field Theory version of the higher derivative Pais-Uhlenbeck oscillator. We quantize canonically this system and construct its Fock space, as well as study its path integral. We demonstrate that the inclusion of canonical coordinates in the path integral necessarily introduces a new field, a Lagrange multiplier, which is essential for the consistent application of these coordinates in the canonical quantization framework. Finally, we analyze the improved ultraviolet convergence of the Green functions that this theory exhibits in the presence of an interaction.