User:Maschen/Fractal spacetime

In applied mathematics and mathematical physics, fractal spacetime is the generalization of spacetime to non-integer fractal dimensions and fractal properties, i.e. scale-dependent, self-similar, and nondifferentiable. Rather than taking spacetime to be a integer dimensions at all scales, as in Euclidean space, Minkowski space, or curved spacetimes, exploited in established mainstream physics from general relativity to quantum field theory, fractal spacetime has an interesting and peculiar property that the properties of the spacetime continuum depends on scale.

It is a modern area of research in theoretical physics; some proponents suggest it could be a new route to unifying quantum mechanics and relativity. It collects and organizes theoretical developments and applications in many fields, including physics, mathematics, astrophysics, cosmology and life sciences. In this new formulation of geometry; position, orientation, motion and now scale transformations, the fundamental laws of physics may be given a general form that unifies and thus goes beyond the classical and quantum regimes taken separately. Active work is in progess to attempt at developing a relativistic version of stochastic quantum mechanics. In this approach, one introduces a four-dimensional Wiener or Bernstein process in terms of a fifth ("proper time") variable. As a consequence some parts of the trajectories are running backward in time, and as in modern particle physics; time-reversed paths in spacetime correspond to antiparticles, so this behaviour is interpreted in terms of particle-antiparticle pairs.

Introduction
Fractal geometry, the term coined by and field pioneered by Mandelbrot in the 1970s, is a relatively new area of pure and applied mathematics offering a revolutionized interpretation of irregular shapes naturally found in science, from ferns to galaxies. When the quantum and relativistic theories were developed, spacetime was of integer dimension and scale-independent.

The concept that the quantum space-time of microphysics is fractal, instead of flat Minkowskian space has been suggested before, based on earlier results [3-6], obtained at first by Feynman who pioneered the path integral formulation of 1948, considering the geometrical structure of quantum paths. This concept is also key in fractional quantum mechanics by Nick Laskin. The typical trajectories of quantum mechanical particles are continuous but nondifferentiable, and can be characterized by a fractal dimension which jumps from D = 1 at large length-scales to D = 2 at small length-scales, the transition occurring about the de Broglie scale... The case D = 2 is the fractal dimension of Brownian motion of a Markov-Wiener process.

Scale relativity
In scale relativity, Einstein’s principles of relativity are modifed with scale transformations.

In the principle of scale relativity, the Planck length-scale is invariant under dilations for scale laws, in the same way that the speed of light is invariant for motion laws.

Introducing fractal spacetime in quantum mechanics
The extension of the principle of relativity can be seen as follows.

Firstly, that quantum paths are nondifferentiable, while the principle of relativity, in its “general” form, requires the equations of physics to be covariant under continuous and at least two times differentiable transformations of curvilinear coordinate systems [17]: such a general covariance leads, with the principle of equivalence, to Einstein’s field equations.

The general form of equations which would be invariant under continuous but nondifferentiable transformations...

Secondly, the Heisenberg uncertainty relations state there is a universal limit on the resolution of measurement apparatus. Basing on this universality and on the relative character of all scales in nature, including resolutions into the definition of coordinate systems have been proposed, defined as their "state of scale". In this form, Einstein’s formulation of the principle of relativity, stating that the laws of physics must apply to any coordinate frame whatever its state, can incorporate not only the effect of motion transformations via kinematical quantities which characterize the motion of the reference system, such as velocity and acceleration, but also of scale transformations.

A new version of stochastic quantum mechanics is described by fractal and renormalization group approaches.

The classical time derivative is replaced by a "quantum-covariant" derivative. The Schrödinger equation fulfills the the correspondence principle in terms of this derivative.

Theoretical/experimental justifications
An application of the new framework is to the problem of the mass spectra of elementary particles.

Some consequences of fractal spacetime in physics include:


 * Scale of Grand Unification: Because of the new relation between length-scale and mass-scale, the theory yields a new fundamental scale, given by the length-scale corresponding to the Planck energy.
 * Unification of the fundamental interactions: as a consequence'', the four fundamental couplings, U(1), SU(2), SU(3) and gravitational converge in the new framework towards about the same scale, which now corresponds to the Planck mass scale.
 * Fine structure constant: The problem of the divergence of charges (coupling constants) and self-energy is solved.