Bohm & Vigier: Ideas as A Basis for A Fractal Universe

Abstract

Bohm and Vigier introduced the notion of random fluctuations occurring from interaction with a subquantum medium. Fényes-Nelson’s stochastic mechanics generalises these ideas in terms of a Markov process and tries to reconcile the individual particle trajectory notion with the quantum (Schrödinger) theory. Bohm-Vigier deterministic trajectories are in fact the mean displacement paths of the underlying Nelson’s diffusion process. However, random paths of stochastic mechanics are quite akin to Feynman paths which are non-differentiable and thus have fractal properties in the Mandelbrot sense. How a random field makes particles to propagate? This is the question. Can we speak about a stochastic acceleration property of (vacuum) spacetime which has stochastic (and chaotic) features? Can this offer an explanation of the inertial properties of matter? What is the source of randomness? The present paper tries to find an answer to these questions in the framework of the universality of a fractal structure of spacetime and of stochastic acceleration. Some arguments in favour of a fractal structure of spacetime at small and large scales areas follows. (i) Fractal trajectories in space with Hausdorff dimension 2 (e.g. a Peano-Moore curve) exhibit both an uncertainty principle and a de Broglie relation. Quantum mechanical particles move statistically on such fractal (Feynman) paths. Thus, Schrödinger equation may be interpreted as a fractal signature of spacetime. (ii) The formal analytic continuation (t → it or D → iD) iD) which relates the Schrödinger and diffusion equations has a physical alternative: there exists a (classical or quantum) stochastic fluid which can be either a fluid of probability for a unique element or a real fluid composed of elements undergoing quasi-Brownian motion. A particle (corpuscule) may be one or a small cluster of stochastic elements. There is a sort of democracy (statistical self-similarity) between the stochastic elements constituting the particle. As regards the cause of the randomness, the parton model involves a fragmentation of the partons. (iii) Nature does not “fractalize” (and quantize); it is intrinsically fractal (and quantum). Wave function of the universe is a solution of the Wheeler-DeWitt equation of quantum cosmology and corresponds to a Schrödinger equation. This can be related to the fact that observations of galaxy-galaxy and clustercluster correlations as well as other large-scale structure can be fit with a fractal with D ≈ 1.2 which may have grown from two-dimensional sheetlike objects such as domain walls or string wakes. The fractal dimension D can serve as a constraint on the properties of the stochastic motion responsible for limiting the fractal structure. (iV) The nonlinear (soliton) equation corresponds to a (linear) Schrödinger equation coupled to a medium with a specific nonlocal response. Physically, this model is similar to a simple case of linear propagation of thin beams in a wave guide. Thus, a free photon in (fractal) space represents in fact a “bouncing ball” in a wave guide. In other words, spacetime is structured as a (fractal) web of optical fibers (channels) which represents the skeleton of spacetime. (V) The proper wave functions describing a hydrogen-like atom (ψ₁, ψ₂, ψ₃, ...) describing a hydrogen-like atom are similar to the electromagnetic modes (TEM ₀ , TEM ₁ , TEM ₂ , ...) in optical resonating cavities.