Model of the Causal Interpretation of Quantum Theory in Terms of a Fluid with Irregular Fluctuations
D. Bohm · J. P. Vigier
Summary and citation · read the original at the source
In one page
David Bohm and Jean-Pierre Vigier take the causal interpretation of quantum mechanics — the one in which a particle really does have a position and is guided by a wave — and ask what physical thing could be doing the guiding. Their answer is a fluid. Rewrite Schrödinger’s equation in terms of the wave’s amplitude and phase and it turns into the equations of a conserved fluid with a density and a flow, plus one extra term, the quantum potential, which they read as an internal stress in that fluid. The particle is then a small persistent lump — a vortex, or a stable pulse — carried along at the local stream velocity. The new move in this paper is to insist that any real fluid jitters, and to show the jitter does work: whatever probability distribution you start with decays in time to the Born rule, so that rule becomes a statistical equilibrium rather than a postulate. They extend the result to the Dirac equation and to many particles.
Why it matters hereChapter 5 treats the vacuum as a quantum fluid, and this is the paper that first put that picture down as equations rather than metaphor — density, flow, internal stress, turbulence, and a particle as a stable structure inside the medium. Chapter 2 gets the deeper claim underneath it: that quantum randomness is the visible surface of real fluctuation in a real substructure. Chapter 13 gets a lineage — Vigier carried this programme forward for four decades, and the modern stochastic-electrodynamics work on the same site is its direct descendant.
What it claims
01Write the wave function as an amplitude times a phase, and Schrödinger’s equation becomes two equations for a fluid: one expressing conservation of the fluid, whose density is the squared amplitude and whose stream velocity is the gradient of the phase divided by the mass; and one determining the changes of the velocity potential in terms of the classical potential plus the quantum potential. Following Takabayasi and Schönberg, Bohm and Vigier take the quantum potential to arise from an internal stress in the fluid — a stress that depends on derivatives of the fluid density, and so is not quite like an ordinary pressure.Section 2, The Hydrodynamic Model, page 208, equations 1 to 3
Published and peer-reviewed02The model is completed by postulating a particle in the form of a highly localized inhomogeneity that moves with the local fluid velocity. Its precise nature is deliberately left open: it could be a foreign body of nearly the same density carried along as a small floating object is carried on water, or a stable dynamic structure existing in the fluid — a small stable vortex, or a pulse-like inhomogeneity — stabilised by some nonlinearity present in a more accurate approximation than the linear equations give.Section 2, page 209
Published and peer-reviewed03The physical argument for fluctuation is simply that real media always fluctuate. Bohm and Vigier list three sources: irregular disturbance arriving from outside the fluid at its boundaries; instability of the nonlinear equations of motion producing turbulence within the fluid itself; and a residual Brownian movement arising from a granular substructure underlying the fluid, analogous to the molecular structure underlying ordinary fluids. All they require is that the density equals the squared amplitude and the velocity equals the phase gradient over the mass as averages.Section 3, Fluctuations of the Madelung Fluid, page 209
Published and peer-reviewed04The result the paper is written for: with those random fluctuations assumed, an arbitrary probability density ultimately decays into the squared amplitude of the wave function. That answers the objection made by Pauli and others — that the causal interpretation should allow any initial distribution — by making the Born rule a statistical equilibrium the fluid relaxes into, in the way Boltzmann’s H theorem works in classical mechanics, rather than a separate postulate. The proof is then extended to the Dirac equation and to the many-particle problem.Abstract, page 208; Sections 4 and 5, pages 211 to 214; Conclusion, page 215
Published and peer-reviewed05The model has to survive measurement, and the authors show it does. Once a measurement splits the wave function into packets separated by a classical distance, the particle must stay in the packet it entered, or results would not be definite. Because the mean density between packets is very small, a large probability of diffusing across the gap would require enormous fluctuation velocities there; merely assuming that fluctuation velocities do not differ by large orders of magnitude from place to place is enough to make that diffusion negligible.Section 6, pages 214 to 215
Published and peer-reviewed06What the authors put forward to watch: because the causal interpretation permits an unlimited number of new physical models of types not consistent with the usual interpretation, models that reproduce ordinary quantum theory only as an approximation, they may lead to appreciably different results at a new level of very small distances. Bohm and Vigier name three openings — vortex motion in the fluid as a natural model for a particle with spin, nonlinear field equations with a spectrum of stable pulse-like localized solutions that follow the local stream velocity, and transitions between such structures as a description of changes from one type of particle to another.Section 1, Introduction, page 208; Section 7, Conclusion, pages 215 to 216
What to watch
The way in
https://doi.org/10.1103/PhysRev.96.208Published as Physical Review volume 96, pages 208 to 216, issue of 1 October 1954, received 14 June 1954. Bohm was then at the Universidade de São Paulo and Vigier at the Institut Henri Poincaré, Paris. The record is closed access under the APS default licence and Unpaywall and OpenAlex report no open deposit. The full text was read for this sheet on 2026-09-08 through the APS harvest endpoint carried in the Crossref record; that endpoint is provided for similarity checking, not for redistribution, so no text of the paper is reproduced here. The scanned original carries optical-recognition damage in places — one distance scale in the introduction is unreadable and is therefore not quoted — and every claim below is drawn from passages that read cleanly.
How to cite it
D. Bohm, J. P. Vigier (1954) Model of the Causal Interpretation of Quantum Theory in Terms of a Fluid with Irregular Fluctuations. doi:10.1103/PhysRev.96.208
Where it sits in the curriculum
The vacuum as a quantum fluidWhat the vacuum isThe unified picture