Properties of the geometric phase of a de Broglie-Bohm causal quantum mechanical trajectory

In a manner analogous to that used to define the geometric phase of a wavefunction, one can define the geometric phase of a periodic trajectory associated with the de Broglie-Bohm causal interpretation of quantum mechanics. This phase is proportional to the action integral over the trajectory. Numerical evidence indicates that this is an adiabatic invariant, despite the fact that the equations of motion are not separable and that the effective classical Hamiltonian depends explicitly on time, two conditions excluded in the standard proof of adiabatic invariance. The associated concept of geometric frequency, proportional to the time average of the kinetic energy, can be defined for both periodic and aperiodic trajectories. Numerical evidence indicates that this also has remarkable properties.