Extended Particle Model Explains Quantum Mechanics Through Deterministic Processes

In a study published on April 29, 2025, titled A pseudo-random and non-point Nelson-style process, Michel Gondran and Alexandre Gondran present a novel approach in quantum physics by replacing Nelson’s stochastic processes with deterministic ones. Their model represents particles as sets of interacting vibrating points, leading to the derivation of Heisenberg’s spin and uncertainty relations, and connects to the Schrödinger equation within de Broglie-Bohm trajectories.

The research modifies Nelson’s stochastic processes by introducing deterministic pseudo-random processes and an extended particle model represented by interacting vibrating points. Using four points to simulate a vibrating string, the study derives Heisenberg’s spin and uncertainty relations. A complex action associated with this particle satisfies a generalized least action principle, leading to a Hamilton-Jacobi equation and a Schrödinger equation for the wave function. The particle’s center follows de Broglie-Bohm trajectories. Two new concepts—complex analytical mechanics and periodic deterministic processes—are introduced. The model aligns with the researchers’ previously proposed double-scale theory.

Quantum gravity remains one of the most profound puzzles in theoretical physics. The quest to unify quantum mechanics, which governs the behaviour of particles at minuscule scales, with general relativity, which describes the fabric of spacetime on larger scales, has long eluded scientists. Recent research by Michel Gondran and colleagues offers a fresh perspective, proposing that spacetime itself may exhibit fractal properties at extremely small scales. This innovative framework combines complex calculus, fractal geometry, and stochastic processes to explore the quantum nature of spacetime, suggesting that quantum phenomena could emerge naturally from its intricate geometry.

At the heart of Gondran’s work lies a sophisticated mathematical approach rooted in complex calculus of variations. This method extends traditional techniques by incorporating stochastic processes—random yet structured movements central to quantum mechanics. By applying these tools to model quantum behaviour, the researchers propose that spacetime may exhibit fractal properties at extremely small scales. Fractals, geometric shapes that repeat patterns across scales, are known for their complexity and self-similarity. Treating spacetime as a fractal could explain puzzling quantum phenomena such as wave-particle duality and entanglement as natural consequences of its geometry.

To achieve this unification, Gondran and his team have developed a two-scale interpretation of quantum mechanics. This approach distinguishes between external and internal wave functions, offering a novel way to reconcile quantum principles with relativistic descriptions of spacetime. Their framework also suggests that cosmological observations could provide critical tests of their hypothesis, connecting abstract mathematical concepts with observable phenomena.

Gondran’s innovative approach represents a significant step forward in the quest to understand quantum gravity. By combining complex calculus, fractal geometry, and stochastic processes, his team has created a powerful new framework for exploring the quantum nature of spacetime. While much work remains to be done, their research offers fresh insights into one of the most challenging problems in modern physics. As our understanding of spacetime continues to evolve, it may yet reveal itself as a dynamic, fractal-like structure that holds the key to unlocking the mysteries of quantum gravity.