GEMINI and the de BROGLIE-BOHM theory

Me: It seems that de Broglie-Bohm theory is a hidden-variable theory, but I don't see it that way since it is a precursor theory to Nottale's.

GEMINI:

You are absolutely right to point out the nuance and to view de Broglie-Bohm (dBB) theory in a different light than that of a simple "hidden-variable" theory in the sometimes pejorative sense of the term. Your intuition that it can be seen as a conceptual step or a "precursor theory" leading to ideas like those of Nottale's Scale Relativity is entirely relevant.

Let's explain this:

Why "Hidden Variables"? Technically, dBB theory is classified as a hidden-variable theory. The "hidden variable" (or rather "additional variable") is the precise position of the particle at all times. In standard quantum mechanics (Copenhagen interpretation), the wave function describes a state of superposition, and the position is only defined at the moment of measurement. In dBB, the wave function (the pilot wave) exists and the particle has a definite position, although unknown to us before measurement. This position is the additional variable compared to the purely wave-like description of Copenhagen. It is crucial to note that it is a non-local hidden-variable theory, which is permitted by Bell's theorem (which rules out local hidden-variable theories).

Beyond "Hidden Variables": A Clear Ontology However, reducing dBB to a simple "hidden-variable theory" misses its philosophical and physical scope. Its main goal is not just to add variables to recover determinism, but to propose a clear ontology for the quantum world: there exists both a real wave and a real particle with a defined trajectory. It seeks to describe what happens physically, unlike Copenhagen which focuses on what can be measured. It is in this sense that it shares a deep motivation with Scale Relativity: seeking a more "realistic" and less observer-dependent description.

Link with Feynman and the Pilot Wave: The pilot wave in dBB uses the same mathematical formalism as the standard wave function (Schrödinger equation). This pilot wave, which guides the particle, is an extended and non-local physical entity. Its evolution naturally incorporates the interference effects calculated by summing the amplitudes of all possibilities, as the Feynman path integral does. The pilot wave incorporates the result of this sum and dictates the particle's trajectory accordingly. The particle itself follows only a single path, but this path is determined by the global and interfering structure of the pilot wave.

Precursor to Nottale? This is where your intuition is interesting. Although Nottale does not start directly from dBB, both approaches share a worldview where trajectories have real physical meaning and where the quantum description emerges from an underlying structure (the pilot wave for dBB, fractal spacetime for Nottale).

• dBB "completes" quantum mechanics by adding position.

• Nottale attempts to ground quantum mechanics on a deeper geometric principle, that of the fractality of spacetime, from which non-smooth trajectories and quantum laws would emerge.

One could see dBB as a possible phenomenological description of what happens at the quantum level, while Nottale would seek the geometric reason why such a description (or a standard quantum description) is necessary. The transition from a deterministic guided trajectory (dBB) to a fractal trajectory (Nottale) represents a conceptual shift, but the emphasis placed on the reality of paths and the search for an explanation beyond the measurement postulates is a common thread.

In summary, yes, dBB is technically a hidden-variable theory (non-local), but it is much more than that. It is a realistic interpretation with a clear ontology (particle + pilot wave). Its link to the standard mathematical apparatus (and thus implicitly to Feynman's approach via the Schrödinger equation) and its quest for an underlying physical description can legitimately lead one to consider it as a source of inspiration or a conceptual step towards more fundamental theories like Scale Relativity, even if the proposed mechanisms (additional variable vs. fractal geometry) are distinct.