On the Quantisation of Complex Higher Derivative Theories and Avoiding the Ostrogradsky Ghost

TL;DR

This paper demonstrates that complex classical mechanics enables the construction of higher derivative theories free from Ostrogradsky ghosts, which can be canonically quantized to produce stable, real-energy quantum theories.

Contribution

It introduces a novel approach using complex classical mechanics to avoid Ostrogradsky ghosts in higher derivative theories, allowing for stable, quantized models with real spectra.

Findings

Constructed complex higher derivative theories with bounded real spectra

Achieved canonical quantization of complex theories without ghosts

Provided a framework to analyze and compare previous ghost-avoidance methods

Abstract

Generic higher derivative theories are believed to be fundamentally unphysical because they contain Ostrogradsky ghosts. We show that within complex classical mechanics it is possible to construct higher derivative theories that circumvent the Ostrogradsky theorem and have a real energy spectrum that is bounded from below. The complex theory can be canonically quantised. The resulting quantum theory does not suffer from the kinetic instability and maintains the usual probabilistic interpretation without violating the correspondence principle. As a proof of concept, we construct a class of stable interacting complex higher derivative theories. This consistent and canonical framework allows us to analyse previous attempts to avoid the ghosts that use non-canonical quantisation schemes, such as the Lee-Wick theories, Dirac-Pauli quantisation or PT-symmetric quantum mechanics. The key to understand the would-be ghosts in any kinetically stable higher derivative theory is to accept the complex system behind it.