In a pilot-wave theory, an individual closed system is described by a wavefunction $ψ(q)$ and configuration $q$. The evolution of the wavefunction and configuration are respectively determined by the Schrödinger and guidance equations. The guidance equation states that the velocity field for the configuration is given by the quantum current divided by the density $|ψ(q)|^2$. We present the currents and associated guidance equations for any Hamiltonian given by a differential operator. These are derived directly from the Schrödinger equation, and also as Noether currents arising from a global phase symmetry associated with the wavefunction in configuration space.
In a pilot-wave theory, an individual closed system is described by a wavefunction ψ(q) and configuration q. The evolution of the wavefunction and configuration are respectively determined by the Schrödinger and guidance equations. The guidance equation states that the velocity field for the configuration is given by the quantum current divided by the density ∣ψ(q)∣2. We present the currents and associated guidance equations for any Hamiltonian given by a differential operator. These are derived directly from the Schrödinger equation, and also as Noether currents arising from a global phase symmetry associated with the wavefunction in configuration space.
In De Broglie-Bohm Pilot-Wave Theory unique equations of motion and scalar fields for a particle can be formulated. This is done by finding a solution for a divergence free probability density current J(r,t) and then dividing by the Born Rule ∣Ψ(r,t)∣2 for velocity arising from Hamilton-Jacobi mechanics. This addition of divergence free probability current would still satisfy the Schrodinger continuity equation. It was found that for a 2D polar system a divergence free probability density is the gradient rotated by the matrix (S) applied to a scalar field as such S∇φ(r). A variety of equations of motion were used such that the dimensions of the equations are equivalent to a velocity, different scalar fields were then derived for a 2D quantum harmonic oscillator. It was found that for each unique velocity there was a unique scalar field associated with it, as such the equation of motion for a particle in De Broglie-Bohm pilot wave theory could be theoretically dependent on any arbitrary number of scalar fields. This ultimately means the equations of motion could be ambiguous as there are multiple mathematically valid solutions.
In the pilot-wave theory of quantum mechanics particles have definite positions and velocities and the system evolves deterministically. The velocity of a particle is determined by the wave function of the system (the guidance equation) and the wave function evolves according to Schrödinger’s equation. In this paper I first construct a Hamiltonian that gives Schrödinger’s equation and the guidance equation for the particle. I then find the Hamiltonian for a relativistic particle in Dirac’s theory and for a quantum scalar field.
de Broglie-Bohm theory (dBBT), treating quantum particles as point objects moving along well defined (Bohmian) trajectories, offers an appealing solution of the measurement problem in quantum mechanics; it has, however, problems relating to spin, relativity and lack of proper integration with the Hilbert space based framework. In this work, we present a consistent formalism which has the traditional state-observable framework integrated with the desirable features of dBBT. We adopt ensemble interpretation for the Schrodinger wave function ψ. Given a Schrodinger wave function ψ, we use its value ψ0 at some fixed time (say, t=0) to define the probability measure ∣ψ0∣2dx on the system configuration space M (=Rn). On the resulting probability space M0, we introduce a stochastic process ξ(t) corresponding to the Heisenberg position operator XH(t) such that, in the Heisenberg state ∣ψh⟩ corresponding to ψ0, the expectation value of XH(t) equals that of ξ(t) in M0. This condition leads to the de Broglie-Bohm guidance equation for the sample paths of the process ξ(t) which are, therefore, Bohmian trajectories supposedly representing time-evolutions of individual members of the ψ0-ensemble. Stochastic processes and Bohmian trajectories corresponding to observables with discrete eigenvalues (in particular spin) are treated by extending the configuration space to the spectral space of the commutative algebra obtained by adding appropriate discrete observables to the position observables. Pauli's equation is treated as an example. A straightforward derivation of von Neumann's projection rule employing the Schrodinger-Bohm evolution of individual systems along their Bohmian trajectories is given. Some comments on the potential application of the formalism developed here to quantum mechanics of the universe are included.
This paper and a following one develop a Hamiltonian formulation of the de Broglie-Bohm (deBB) interpretation of quantum mechanics. Drawing upon a unified Hamiltonian treatment of a classical particle and its associated ensemble, it is shown how the interaction of a particle and a spin-0 quantum wave can be consistently expressed in a canonical framework. A formulation is found in which the particle is acted upon by a force determined by the quantum potential but the particle does not react on the Schrodinger field. This requires the introduction of an additional field which the particle does act upon, and allows the deBB theory to be expressed in Hamiltonian terms. An ensemble theory for the wave-particle system based on Liouville's equation is developed, and it is shown that for a pure quantum state the particle obeys its own Liouville equation. The general particle phase space distribution that is compatible with ‖ψ‖ 2 is derived. A special case of this corresponds to the relation p = ∂S/∂q (and thus the guidance law of the deBB theory) but more general distributions, and hence non-deBB motions, are possible. This relation is interpreted as a constraint on the phase space of the composite system. Using the theory of Hamiltonian constraints it is shown that the first-order guidance law and Hamilton's second-order particle equations are dynamically equivalent.
We present a novel covariant relativistic guidance formula à la de Broglie-Bohm derived from conserved quantities associated with the Lagrangian of the Dirac equation in a 1+1-dimensional spacetime. In line with the standard guidance formula of Bohmian mechanics, this generalized guidance formula incorporates space-time derivatives of the Dirac spinor phases and consistently reproduces relativistic features such as mass-dependent trajectories and zitterbewegung. We formally show that this formulation explicitly reveals how the relativistic quantum potential affects the trajectories. Using numerical simulations, we compare these trajectories with those presented in the usual approaches. Finally, we show how the guidance formula naturally transitions to the standard de Broglie-Bohm theory in the non-relativistic limit and gives additional contributions in the expansions giving new insight into the zitterbewegung in pilot-wave theories.
Following the de Broglie's idea of pilot wave, this paper treats quantum mechanics as a problem of stochastic optimal guidance law design. The guidance scenario considered in the quantum world is that an electron is the flight vehicle to be guided and its accompanying pilot wave is the guidance law to be designed so as to guide the electron to a random target driven by Wiener process, while minimizing a cost-to-go function. After solving the stochastic optimal guidance problem by differential dynamic programming, we point out that the optimal pilot wave guiding the particle's motion is just the wavefunction ψ(t,x), a solution to Schrödinger equation. Quantum trajectories under the action of the optimal guidance law are solved and their statistical distribution is shown to coincide with the prediction of the probability density function ψ*ψ.