Quantum-classical diagnostics and Bohmian inequivalence for higher time-derivative Hamiltonians

TL;DR

This paper applies Bohmian mechanics to analyze higher-derivative quantum systems, revealing that classically equivalent Hamiltonians can exhibit distinct quantum trajectories and potentials, highlighting a quantum ambiguity.

Contribution

It introduces a Bohmian framework for higher-derivative Hamiltonians, demonstrating quantum inequivalence despite classical similarity, and provides detailed dynamical diagnostics.

Findings

Bohmian trajectories differ for classically equivalent Hamiltonians.

Quantum potentials reveal discrepancies in quantum dynamics.

The analysis characterizes different dynamical regimes such as bounded and runaway.

Abstract

We develop a Bohmian analysis of a two-dimensional ghost Hamiltonian and its mapping to the degenerate Pais-Uhlenbeck model. Using Gaussian wavepackets, we derive the corresponding guidance equations, the centre and width evolution, and the quantum potential. We use these quantities to characterise bounded, quasi-semiclassical, spiral, and runaway regimes. The Bohmian trajectories provide a direct dynamical diagnostic of coherence, packet deformation, and quantum-classical separation. We then compare a bi-Hamiltonian pair consisting of the ghost Hamiltonian and a classically equivalent alternative formulation. While the two descriptions produce identical classical trajectories, they lead to different Bohmian trajectories and different quantum potentials evaluated along those trajectories. This demonstrates that classical equivalence need not extend to Bohmian quantum dynamics and identifies a concrete quantum ambiguity in the degenerate higher-derivative system.