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Physical Review

Model of the Causal Interpretation of Quantum Theory in Terms of a Fluid with Irregular Fluctuations

David Böhm, Jean Pierre Vigier

Published Oct 1, 1954• 608 citations
Publisher

Abstract

In this paper, we propose a physical model leading to the causal interpretation of the quantum theory. In this model, a set of fields which are equivalent in many ways to a conserved fluid, with density ${|\ensuremath{\psi}|}^{2}$, and local stream velocity, $\frac{d\ensuremath{\xi}}{\mathrm{dt}}=\frac{\ensuremath{\nabla}S}{m}$, act on a particle-like inhomogeneity which moves with the local stream velocity of the equivalent fluid. By introducing the hypothesis of a very irregular and effectively random fluctuation in the motions of the fluid, we are able to prove that an arbitrary probability density ultimately decays into ${|\ensuremath{\psi}|}^{2}$. Thus, we answer an important objection to the causal interpretation, made by Pauli and others. This result is extended to the Dirac equation and to the many-particle problem.

DOI
10.1103/physrev.96.208

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In this paper, we propose a physical model leading to the causal interpretation of the quantum theory. In this model, a set of fields which are equivalent in many ways to a conserved fluid, with density ∣\ensuremathψ∣2{|\ensuremath{\psi}|}^{2}∣\ensuremathψ∣2, and local stream velocity, d\ensuremathξdt=\ensuremath∇Sm\frac{d\ensuremath{\xi}}{\mathrm{dt}}=\frac{\ensuremath{\nabla}S}{m}dtd\ensuremathξ​=m\ensuremath∇S​, act on a particle-like inhomogeneity which moves with the local stream velocity of the equivalent fluid. By introducing the hypothesis of a very irregular and effectively random fluctuation in the motions of the fluid, we are able to prove that an arbitrary probability density ultimately decays into ∣\ensuremathψ∣2{|\ensuremath{\psi}|}^{2}∣\ensuremathψ∣2. Thus, we answer an important objection to the causal interpretation, made by Pauli and others. This result is extended to the Dirac equation and to the many-particle problem.
In two previous papers a causal interpretation of the quantum theory was developed which involved the hypothesis that a quantum-mechanical system contains a precisely defined particle variable x but that, at present, we are restricted to calculating the probability density P(x,t)P(\mathrm{x}, t)P(x,t) that the particle is at the position x. It was shown that the assumption that $P(\mathrm{x}, t)={|\ensuremath{\psi}(\mathrm{x}, t)|}^{2}$ is consistent, in the sense that if it holds initially, the equations of motion of the particles will cause this relation to be maintained for all time. In this paper, we extend the theory by showing that as a result of random collisions, an arbitrary probability density will ultimately decay into one with a density of ${|\ensuremath{\psi}(\mathrm{x}, t)|}^{2}$. Since all quantum-mechanical experiments to date have been concerned with statistical ensembles of systems that have been colliding with other systems for a very long time, it is therefore inevitable that as we draw samples from such ensembles, the probability density of systems with particles at the point x will be equal to ${|\ensuremath{\psi}(\mathrm{x}, t)|}^{2}$.In the previous papers we also pointed out that, within the conceptual framework of the causal interpretation, it was possible to suggest mathematical theories more general than are permitted by the usual interpretation and that these more general theories might be needed in the domain of ${10}^{\ensuremath{-}13}$ cm, where present theories seem to fail. However, if these more general theories should apply at the level of ${10}^{\ensuremath{-}13}$ cm, then there would be a tendency to create discrepancies between PPP and ${|\ensuremath{\psi}|}^{2}$, a tendency whose cumulative effects should be felt even at the atomic level, where the more general theory ought to approach the usual theory. However, because those discrepancies have been shown to die out as a result of collisions, we can expect that under normal conditions the difference between PPP and ${|\ensuremath{\psi}|}^{2}$ would be negligible. Conditions are suggested, however, in which this difference might be appreciable, and experiments are indicated which might be able to test for the existence of such discrepancies.
In our previous works, we suggest that quantum particles are composite physical objects endowed with the geometric and topological structures of their corresponding differentiable manifolds that would allow them to imitate and adapt to physical environments. In this work, we show that Dirac equation in fact describes quantum particles as composite structures that are in a fluid state in which the components of the wavefunction can be identified with the stream function and the velocity potential of a potential flow formulated in the theory of classical fluids. We also show that Dirac quantum particles can manifest as standing waves which are the result of the superposition of two fluid flows moving in opposite directions. However, for a steady motion a Dirac quantum particle does not exhibit a wave motion even though it has the potential to establish a wave within its physical structure, therefore, without an external disturbance a Dirac quantum particle may be considered as a classical particle defined in classical physics. And furthermore, from the fact that there are two identical fluid flows in opposite directions within their physical structures, the fluid state model of Dirac quantum particles can be used to explain why fermions are spin-half particles.
The authors explore certain interconnections between density-functional theory and quantum fluid dynamics of many-electron systems, in the relativistic domain following the hydrodynamical approach adopted by Takabayashi for the one-particle Dirac equation. In order to build a 'classical' hydrodynamical interpretation, the spinor formulation is transformed into a tensor formulation by defining a number of density functions (local observables). These lead to six 'classical' fluid dynamic equations, together with two subsidiary conditions, for a complete specification of the system. The various density functions and the hydrodynamical equations are physically interpreted. The relativistic hydrodynamics discussed here correspond to a 'spinning' fluid. The net many-electron fluid consists of components each of which is characterised by fluid dynamic quantities corresponding to each spinor. The net hydrodynamical quantities are obtained by summing over the occupied spinors. Thus, the earlier nonrelativistic 'classical' picture of the many-electron fluid as a collection of individual fluid components is also valid in the relativistic domain.
An interpretation of non-relativistic quantum mechanics is presented in the spirit of Erwin Madelung's hydrodynamic formulation of QM and Louis de Broglie's and David Bohm's pilot wave models. The aims of the approach are as follows: 1) to have a clear ontology for QM, 2) to describe QM in a causal way, 3) to get rid of the wave-particle dualism in pilot wave theories, 4) to provide a theoretical framework for describing creation and annihilation of particles, and 5) to provide a possible connection between particle QM and virtual particles in QFT. These goals are achieved, if the wave function is replaced by a fluid of so called virtual particles. It is also assumed that in this fluid of virtual particles exist a few real particles and that only these real particles can be directly observed. This has relevance for the measurement problem in QM and it is found that quantum probabilities arise in a very natural way from the structure of the theory. The model presented here is very similar to a recent computational model of quantum physics and recent Bohmian models of QFT.
The reinterpretation of quantum mechanical formalism in terms of a classical model with a continuous material "ΨΨΨ-field" acting upon a point-like particle which is subjected to large friction and random forces is proposed. This model gives a mechanism for sudden "quantum jumps" and provides a simple explanation of "Schrödinger Cat" phenomena.