De Broglie-Bohm Guidance Equations for Arbitrary Hamiltonians

TL;DR

This paper derives guidance equations for de Broglie-Bohm theory applicable to any Hamiltonian expressed as a differential operator, linking quantum currents to system evolution.

Contribution

It provides a general derivation of guidance equations for arbitrary Hamiltonians, extending the applicability of pilot-wave theory.

Findings

Derived guidance equations from Schrödinger equation for all differential operator Hamiltonians

Connected guidance equations to Noether currents from phase symmetry

Unified framework for guidance equations in pilot-wave theory

Abstract

In a pilot-wave theory, an individual closed system is described by a wavefunction $\psi(q)$ and configuration $q$. The evolution of the wavefunction and configuration are respectively determined by the Schr\"odinger and guidance equations. The guidance equation states that the velocity field for the configuration is given by the quantum current divided by the density $|\psi(q)|^2$. We present the currents and associated guidance equations for any Hamiltonian given by a differential operator. These are derived directly from the Schr\"odinger equation, and also as Noether currents arising from a global phase symmetry associated with the wavefunction in configuration space.