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Journal of Physics A Mathematical and Theoretical

Bohmian trajectories in the relativistic Hamiltonian dynamics

K. Kowalski

Published Oct 8, 2026• 0 citations
Publisher

Abstract

Abstract The Bohmian trajectories for a free spinless relativistic particle that is initially in a coherent state are studied in the Hamiltonian dynamics based on the Salpeter equation as an alternative to the Klein–Gordon Equation. The guidance equation is introduced utilizing the probability density and relativistic probability current obtained from the Salpeter equation.

DOI
10.1088/1751-8121/aeb20e

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Abstract The Bohmian trajectories for a free spinless relativistic particle that is initially in a coherent state are studied in the Hamiltonian dynamics based on the Salpeter equation as an alternative to the Klein–Gordon Equation. The guidance equation is introduced utilizing the probability density and relativistic probability current obtained from the Salpeter equation.
Conventional relativistic quantum mechanics, based on the Klein-Gordon equation, does not possess a natural probabilistic interpretation in configuration space. The Bohmian interpretation, in which probabilities play a secondary role, provides a viable interpretation of relativistic quantum mechanics. We formulate the Bohmian interpretation of many-particle wave functions in a Lorentz-covariant way. In contrast with the nonrelativistic case, the relativistic Bohmian interpretation may lead to measurable predictions on particle positions even when the conventional interpretation does not lead to such predictions. Key words: relativistic quantum mechanics, Klein-Gordon equation, Bohmian interpretation. 1
In this paper we intend to extend some ideas of a recently proposed Lorentz-invariant Bohmian model, obeying Klein-Gordon equations, but considering particles with a spin different than zero. First we build a Bohmian model for a single particle, closely related to the Bohm-Dirac model, but using Nikolic's ideas on a space-time probability density and the introduction of a synchronization parameter σfor the particle trajectory. We prove that, in the case of a spinor whose component phases become equal when \plank \to 0, it is possible to use the usual Bohmian Mechanics procedures to obtain the classical limit. We extend these ideas for a two-particle system, based on two non-local but covariant Dirac-like equations.
Abstract We investigate potential scattering and tunneling dynamics of a particle wavepacket evolving according to the relativistic Schrödinger equation (also known as the Salpeter equation). The tunneling properties of the Salpeter equation differ from those of the standard relativistic wave equations (such as the Klein–Gordon or Dirac equations). In particular, the tunneling solutions must be found by working in momentum space, given that the equation in configuration space contains a pseudo-differential operator. The resulting integral equations are derived and solved numerically for wavepackets scattering on model potential barriers. The solutions are characterized by the absence of Klein tunneling and an effect of the potential on the fraction of the transmitted wavepacket that propagates outside the light cone, a feature that has in the past been well-studied only for free propagation.
The Klein-Gordon equation describes the wave-like behavior of spinless particles since it is Lorentz invariant. While it seemed initially ripe for explaining the electronic structure of the hydrogen atom, the lack of a unconditional positive probability density really limited its applications. Yet, it is intimately connected with fermions. Any solution to the Dirac equation is automatically a solution to the Klein-Gordon equation. What is even more surprising, the Klein-Gordon equation for a free particle turns into the Schrödinger equation in the non-relativistic limit. In this work we show that these problems disappear when we use the relativistic mass instead of the rest mass. While the Klein-Gordon equation losses its Lorentz invariance because of this transformation, it gains most of the features present the Schrödinger equation, including the unconditional positivity of probability density, while keeping most of its relativistic characteristics intact, including the matter-wave dispersion relation. What is even more surprising, the non-relativistic, quasi-static limit of the Klein-Gordon equation with relativistic mass is simply the Schrödinger equation under all possible conditions. So, it can be argued that this Klein-Gordon equation is a sort of relativistic Schrödinger equation.
A new Bohmian quantum-relativistic model, in which from the Klein-Gordon equation a generalization of the standard Zitterbewegung arises, is explored. It is obtained by introducing a new independent time parameter, whose relative motions are not directly observable but cause the quantum uncertainties of the observables. Unlike Bohm's original theory, the quantum potential does not affect the observable motion, as for a normal external potential, but it only determines that one relative to the new time variable, of which the Zitterbewegung of a free particle is an example. The model also involves a relativistic revision of the uncertainty principle.