Physica Scripta

Relativistic quantum mechanics as a Markovian stochastic process in a five-dimensional complex space-time

R Ramanathan

Published Apr 1, 1993• 1 citation

Abstract

A heuristic model of space time under the influence of an underlying stochastic background is presented. It is shown that it is possible to view the whole of relativistic quantum mechanics as a Markovian stochastic process under certain conditions in a complex five dimensional space which models the influence of the background. Compactification of the extra dimension leads to the Klein-Gordon equation with the probabilistic interpretation in-built in the theory. The question of positivity of the variance in the context of complex space-time is circumvented by suggesting that positivity can break down after compactification.

DOI: 10.1088/0031-8949/47/4/002

Related papers

On the stochastic mechanics of the free relativistic particle

Michele Pavon

Given a positive energy solution of the Klein–Gordon equation, the motion of the free, spinless, relativistic particle is described in a fixed Lorentz frame by a Markov diffusion process with nonconstant diffusion coefficient. Proper time is an increasing stochastic process and we derive a probabilistic generalization of the equation (dτ)2=−(1/c2)dXν dXν. A random time-change transformation provides the bridge between the t and the τ domain. In the τ domain, we obtain an M4-valued Markov process with singular and constant diffusion coefficient. The square modulus of the Klein–Gordon solution is an invariant, nonintegrable density for this Markov process. It satisfies a relativistically covariant continuity equation.

arXiv:quant-ph/0108139Journal of Mathematical PhysicsOct 1, 200115 citations

Quantum Mechanics, Stochasticity and Space-Time

R Ramanathan

An extended and more rigorous version of a recent proposal for an objective stochastic formulation of quantum mechanics along with its extension to the relativistic case without spin is presented. The relativistic Klein-Gordon equation is shown to be a particular form of the relativistic Kolmogorov-Fokker-Planck equation which is derived from a covariant formulation of the Chapman-Kolmogorov condition. Complexification of probability amplitudes is again achieved only through a conformal rotation of Minkowski space-time M 4 .

Physica ScriptaNov 1, 19865 citations

Relativistic quantum mechanics and the Bohmian interpretation

Hrvoje Nikolić

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

Jan 1, 200454 citations

Relativistic stochastic processes associated to Klein-Gordon equation

Maurizio Serva

A stochastic interpretation of Klein-Gordon equation is proposed. The starting point is the construction of relativistically covariant diffusions satisfying a property similar but not identical to the Markov property. It is then shown that the continuity equation associated with these processes together with the relativistic version of mean Newton equation are equivalent to the Klein-Gordon equation On propose une interpretation stochastique de l'equation de Klein Gordon. Le point de depart est la construction d'une diffusion relativiste covariante qui satisfait une propriete semblable (mais pas identique) a la propriete de Markov. On montre ensuite que l'equation de continuite associee a ce processus plus la version relativiste de l'equation de Newton en moyenne sont equivalentes a l'equation de Klein-Gordon

Annales De L Institut Henri Poincare-physique TheoriqueJan 1, 198812 citations

Quantum mechanics can be formulated as a non-Markovian stochastic process

German A. Skorobogatov, Sergey Igorevich Svertilov

Both quantum mechanics and stochastic-process theory deal with dynamical systems the behaviors of which can be predicted only probabilistically. It is shown here that the measurable behavior of an elementary quantum system can be modeled as a non-Markovian stochastic process. The conclusion that quantum behavior can generally be modeled as a real stochastic process eliminates some mysterious characteristics of quantum mechanics when viewed in Copenhagen-interpretation terms.

Physical Review ANov 1, 199816 citations

Quantum mechanics without quantum potentials

Adam Brownstein

The issue of non-locality in quantum mechanics can potentially be resolved by considering relativistically covariant diffusion in four-dimensional spacetime. Stochastic particles described by the Klein-Gordon equation are shown to undergo a classical diffusion process in spacetime coordinates, which is seen by transforming the quantum Cauchy-momentum equations to a Lagrangian frame of reference. Since the quantum potential term is removed under this transformation, the equations for momentum propagation along particle trajectories assume a classical form. A local stochastic de Broglie-Bohm interpretation for the Klein-Gordon system can subsequently be derived. We also introduce the concept of momentum equivariance to replace the second-order Bohm-Newton equations of motion, which break down due to non-linear terms of the stochastic Lagrangian derivative.

arXiv:2401.04091Jan 8, 20240 citations