The Evolution of the Path: From Newton’s Line to the Fractal Geodesic
The concept of a "path"—how a particle moves from A to B—is the central thread of physics. It is more than just a trajectory; it is the fundamental link that connects events in the universe. The history of physics is, in effect, the history of our understanding of this line.
This is the story of how the simple, singular line of Newton shattered into a chaotic web, only to be reconstructed as the geometry of the universe itself.
1. Classical Mechanics: The Path of Least Action
In the classical world, the path is singular and absolute. If you throw a stone, it follows one specific curve.
Mathematically, this is governed by the Principle of Least Action. Out of all the hypothetical curves the stone could take, nature selects the single one where the "Action" (the balance of kinetic and potential energy) is minimized.
The View: Determinism. One particle, one path.
2. Einstein and Brownian Motion: The Path Becomes Jagged
Before Quantum Mechanics fully took hold, Albert Einstein (1905) provided a critical clue while studying Brownian Motion—the jittery, chaotic movement of pollen grains suspended in water.
Einstein realized that at microscopic scales, a particle is bombarded incessantly. Its path is continuous (it never teleports), but it is non-differentiable (it has no defined velocity at any single point because it changes direction infinitely often).
The Insight: If you zoom in on a Brownian path, it doesn't smooth out into a straight line; it reveals ever more zig-zags. This was the first glimpse of fractal geometry in physics, hinting that the "smoothness" of Newton was merely an illusion of large scales. Einstein described the first "fractal path" in physics, bridging the gap between thermodynamics and the quantum world to come.
3. The Quantum Schism: Copenhagen vs. Bohm
When Quantum Mechanics arrived, the concept of the path faced a crisis.
The Copenhagen Interpretation (Bohr/Heisenberg): They argued that because we cannot track a particle without disturbing it, the path does not exist. Between A and B, the particle is a probability cloud. To speak of a trajectory is considered "metaphysical," not physical.
The Bohmian Interpretation (Pilot Wave): David Bohm attempted to salvage the classical path. He argued that the particle does have a specific trajectory, but it is guided by a "Quantum Potential"—a non-local field that probes the environment. The path is real, but hidden.
4. Feynman’s Synthesis: The Sum of Histories
Richard Feynman revolutionized the debate by fully embracing the "weirdness." He proposed the Path Integral.
To calculate how a particle gets from A to B, Feynman stated that we must sum every possible path:
The particle goes straight.
The particle curves left.
The particle goes around the planet and comes back.
The Selection Mechanism (Interference):
You correctly noted that we do not see particles flying around the planet to cross a room. Why? Destructive Interference.
The "extreme" paths (like looping around the planet) have phases that vary wildly; they cancel each other out mathematically. The only paths that add up constructively (finding the "Stationary Phase") are those clustered essentially near the classical path.
However, Feynman discovered that the paths contributing the most to the quantum result are the jagged, non-differentiable paths—exactly like Einstein’s Brownian motion (Fractal Dimension D= 2).
5. Nottale & Scale Relativity: The Geodesic Bundle
Laurent Nottale (introducing Scale Relativity) takes Feynman’s discovery and Einstein’s relativity and unifies them.
Nottale argues that this "jaggedness" is not the particle shaking; it is space-time itself that is fractal.
Just as a coastline has infinite length if you measure it with a zero-size ruler, the distance between A and B in a quantum space-time is infinite.
The Geodesic Bundle:
In General Relativity (Einstein), a particle follows a Geodesic—the shortest path in a curved space.
In Scale Relativity (Nottale), the space is fractal. In a fractal landscape, there is no single "shortest" line. There is an infinity of curves that are equally "short."
The "Quantum Path" is not a single line, nor is it a magical cloud.
It is a Bundle of Geodesics.
When a particle moves, it follows the valleys and ridges of this fractal space-time. The "fuzziness" of the electron is not a lack of precision; it is the particle traversing a tube of infinite fractal geodesics.
Conclusion: Paths as Movement
The history of the path is a history of "Zooming In":
Newton: We looked from afar and saw a smooth line.
Einstein: We looked at pollen and saw a jagged dance.
Feynman: We realized the "smooth line" is just the sum of infinite jagged paths.
Nottale: We realized the jagged paths are actually the geodesics of space-time itself.
Conclusion: Paths as Communication
This evolution changes how we view connection and entanglement.
Newton: Connection is contact.
Feynman: Connection is the sum of all possibilities.
Nottale: Connection is geometric.
In Scale Relativity, the "Path" is the structure of space-time. When two particles are entangled, their geodesic bundles remain intertwined in the fractal dimension, regardless of their separation in classical 3D space. The path is the communication channel; it has simply evolved from a 1D line into a multi-dimensional fractal structure that binds the quantum system together.
The "entanglement of paths" is therefore not a magic connection between abstract possibilities, but the geometric reality that two particles are sharing a single, complex fractal geodesic bundle through space-time.
