A semi-heretical attempt to unify quantum theory and relativity, proposed by Laurent Nottale, an astrophysicist working at the Observatoire Paris-Meudon.

Scale relativity is essentially the idea that instead of using four coordinates to specify a location in the manifold, x, y, z and t (three space coordinates and a time one) we should rather use a 5-tuple: (x, y, z, t, D), where D stands for the local Hausdorff dimension at the level of scale that we are concerned with. The theory posits that the Hausdorff dimension of spacetime (in this theory, spacetime is non-differentiable, discontinuous, and fractal, though not self-similar at all scales) varies according to the scale at which you are located.

The theory aims to extend Einstein's principle that the laws of physics should work the same for any observer, no matter their frame of reference (which way they are moving and accelerating - this is the famous 'Principle of Equivalence') to state that the laws of physics should be the same for any observer in any state of motion at any scale.

The Planck length occupies a similar role in scale relativity to that of the speed of light in special relativity. It constitutes an absolute infimum for length, which though finite in measure, has all the properties one would expect from the infinitely small. In the theory, the Planck length is invariant under relativistic dilations and scale-relativistic transformations of resolutions, in the same way that the speed of light is invariant in Einstein's relativity. It represents a bottom limit for scale just as the speed of light represents an upper limit for velocity.

As you might expect, for a theory which explicitly deals with resolution in measurement and the very small, there are consequences for quantum mechanics. There are connections between stochastic quantum mechanics, which deals with the propagation of particles considered as a fractal random walk, and the posited fractal aspect of spacetime. Summarizing chapter 4 of his 1993 book Fractal Space-time and Microphysics, Nottale says:

As first demonstrated by Feynman, typical quantum mechanical trajectories are characterized by their non-differentiability and their fractal structure. It is demonstrated that the Heisenberg relations can be translated in terms of a fractal dimension of all four space-time coordinates jumping from D = 2 in the quantum and quantum relativistic domain to D = 1 (i.e. non-fractal) in the classical domain, the transition being identified as the de Broglie scale Lm = h/pm.

Nottale's theory also makes predictions in the macro world of planets and solar systems. If planetary formation occurs through a gradual accretion of smaller bodies into planet-sized chunks, under scale relativity the probability distribution of the orbitals of the resulting planets is given by the "solutions of a generalised Schroedinger equation." In 1996, when very few extrasolar planetary systems were known, Nottale made predictions based on his theory about the average size of planetary orbits which have (it is claimed) been strongly in accordance with the data emerging from the glut of extrasolar planet discoveries that have occurred since then. This aspect of the theory may provide an indirect theoretical justification of Bode's Law, which is otherwise an entirely puzzling regularity in the structure of the Solar System.

The theory has not met with universal approval, to say the least, in the conventional physics world, and Nottale himself has admitted that the formalism to adequately handle the theory is still 'under development.' But of course this is just as true for other attempts to unify quantum mechanics and relativity, such as loop quantum gravity and M-Theory.

Scale relativity is one of the ingredients in Carlos Castro's 'New Relativity.' 1

1. see, for example, Castro's The status and programs of the New Relativity Theory at

http://arxiv.org/format/physics/0011040
(Castro, it must be said, writes in the exuberant manner one associates with crackpottery, or with unacnowledged geniuses who have discovered the authentic new direction for physics, but no-one will believe them...)