Scale Relativity

Laurent Nottale's concept of scale relativity is a theoretical framework that extends Einstein's theories of relativity to include the notion of scale. It proposes that the laws of physics should be invariant not only under changes in reference frames, as in special and general relativity, but also under changes in scale. The theory suggests that space-time has a fractal structure at very small scales, which could explain quantum phenomena and bridge the gap between quantum mechanics and classical physics.

Scale relativity introduces the idea of a "state of scale" in coordinate systems, similar to how we consider states of position or movement. It posits that there are minimum and maximum scales in physics - the Planck length and time scales at the smallest end, and a maximum scale related to the cosmological constant at the largest end. These scales are proposed to be invariant under dilations, analogous to how the speed of light is invariant in special relativity.

The theory is constructed in three levels: Galilean, special, and general scale relativity, mirroring the development of relativity theories. Galilean scale relativity involves linear transformations with a constant fractal dimension, while special scale relativity introduces more complex transformations and the concept of a maximum scale.

One of the key aspects of scale relativity is its approach to quantum mechanics. It suggests that the non-differentiability of space-time at very small scales leads to quantum-like behaviours, potentially offering a geometric interpretation of quantum phenomena. This could provide a new perspective on the quantum-classical transition and contribute to efforts to unify quantum mechanics and general relativity.

It should be noted that scale relativity continues to be an area of ongoing research and debate. However, for the purposes of demonstrating how dynamic symmetry can be integrated with other theories, scale relativity is a useful tool, prompting further investigation into the intricate relationships between symmetry, scale, and physical laws.