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

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.

Figures (6)

• Figure 1: Phase-space trajectories and quantum-classical diagnostics in the rigid-transport regime. (a) Phase-space projections of three trajectory families: classical ensemble (blue dashed), Bohmian trajectories (red solid), and the Gaussian packet centre (black solid). The initial Gaussian wavepacket parameters are $A_{11}(0)=1/2\sigma_x^2$, $A_{22}(0)=1/2\sigma_y^2$, $A_{12}(0)=A_{21}(0)=0.2$, with $\sigma_x=1.2$, $\sigma_y=1.0$, $q_c(0)=(-3,2)$, $p_c(0)=(1,-0.75)$, and $B(0)=0$. The initial and final points, at $t=0$ and $t=115$, are marked by a filled circle and a triangle, respectively. (b) Time series of the internal deviation $u(t)=q_{\mathrm{Bohm}}(t)-q_c(t)$ (black) and the quantum–classical separation $\Delta(t)=q_{\mathrm{Bohm}}(t)-q_{\mathrm{classical}}(t)$ (magenta), averaged over the Bohmian ensemble. The potential is characterised by $\nu =0.200703$, $\Omega = -0.105$ and $g=-0.0305556$.
• Figure 2: Phase-space trajectories and quantum-classical diagnostics in the quasi-semiclassical regime (a) for the classical ensemble (blue dashed), Bohmian ensemble (red), and Gaussian centre (black) in a regime with $\Omega$ reduced by $0.4$ relative to figure \ref{['rigid']}. (b) Time series of the internal deviation $u(t)$ (black) and quantum–classical separation $\Delta(t)$ (magenta). (c) Time evolution of $\det\Lambda(t)$.
• Figure 3: Phase-space trajectories in the unstable spiral regime. Panel (a) shows the classical ensemble (blue dashed), Bohmian ensemble (red), and Gaussian centre (black) in a regime with $\nu$ reduced by $0.1$ relative to figure \ref{['rigid']}. (b) Time series of the internal deviation $u(t)$ (green) and quantum–classical separation $\Delta(t)$ (magenta). (c) Time evolution of $\det\Lambda(t)$.
• Figure 4: Phase-space trajectories and quantum-classical diagnostics at the critical point $\det C =0$ (a) for the classical ensemble (blue dashed), Bohmian ensemble with $B=0$ (red), $B_{xx}=B_{yy}=0$, $B_{xy}=B_{yx}=-0.45$ (cyan), and Gaussian centre (black). (b) Time series of the internal deviation $u(t)$ (black) and quantum–classical separation $\Delta(t)$ (magenta). (c) Time evolution of $\det\Lambda(t)$. The potential is characterised by $\nu =0.200703$, $\Omega =0.00579446$ and $g=-0.0305556$.
• Figure 5: Phase-space trajectories and diagnostics in the rigid-transport regime for a non-normalisable Gaussian wavepacket. (a) Classical ensemble (blue dashed), Bohmian ensemble (red), and Gaussian centre (black). (b) Time series of internal deviation $u(t)$ (green) and quantum–classical separation $\Delta(t)$ (magenta). The initial Gaussian wavepacket parameters are $A_{11}(0)=-1/2\sigma_x^2$, $A_{22}(0)=-1/2\sigma_y^2$, $A_{12}(0)=A_{21}(0)=c$ with the remaining ones identical to those in figure \ref{['rigid']}. The potential is characterised by $\nu =0.200703$, $\Omega = -0.105$ and $g=0.0305556$.