Laurent Nottale CNRS LUTH, Paris-Meudon Observatory http://www.luth.obspm.fr/~luthier/nottale/ Scales in nature 1 10 -33 cm Planck scale 10 -28 cm Grand Unification 10 10 accelerators: today's limit 10 -16 cm electroweak unification 20 10 3 10 -13 cm quarks 4 10 -11 cm electron Compton length 1 Angstrom Bohr radius atoms 30 virus 10 40 microns bacteries 1 m human scale 40 10 6000 km Earth radius 700000 km Sun radius 1 millard km Solar System 50 10 1 parsec distances to Stars 10 kpc Milky Way radius 1 Mpc Clusters of galaxies 60 100 Mpc very large structures 10 28 10 cm Cosmological scale Scales of living systems Relations between length-scales and mass- scales l/lP = m/mP (GR) l/lP = mP / m (QM) FIRST PRINCIPLES RELATIVITY COVARIANCE EQUIVALENCE weak / strong Action Geodesical CONSERVATION Noether SCALE RELATIVITY Continuity + Giving up the hypothesis Generalize relativity of differentiability of of motion ? space-time Transformations of non- differentiable coordinates ? …. Theorem Explicit dependence of coordinates in terms of scale variables + divergence FRACTAL SPACE-TIME Complete laws of physics by fundamental scale laws Constrain the new scale laws… Principle of scale relativity Generalized principle Scale covariance of equivalence Linear scale-laws: “Galilean” Linear scale-laws : “Lorentzian” Non-linear scale-laws: self-similarity, varying fractal dimension, general scale-relativity, constant fractal dimension, scale covariance, scale dynamics, scale invariance invariant limiting scales gauge fields Continuity + Non-differentiability implies Fractality X A 1 A0 t 0 0.01 0.1 0.11 1 1. Continuity + nondifferentiability Scale dependence when Continuity + Non-differentiability implies Fractality 1.5 0 1.5 5 1.25 1.25 1 1 0.75 0.75 0.5 0.5 0.25 0.25 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1.5 1 1.5 6 1.25 1.25 1 1 0.75 0.75 0.5 0.5 0.25 0.25 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1.5 2 1.5 7 1.25 1.25 1 1 0.75 0.75 0.5 0.5 0.25 0.25 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1.5 3 1.5 8 1.25 1.25 1 1 0.75 0.75 0.5 0.5 0.25 0.25 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1.5 4 1.5 9 1.25 1.25 1 1 0.75 0.75 0.5 0.5 0.25 0.25 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Construction by successive bisections Continuity + Non-differentiability implies Fractality Continuity + Non-differentiability implies Fractality 2. Continuity + nondifferentiability divergence Lebesgue theorem (1903): « a curve of finite length is almost everywhere differentiable » Since F is continuous and no where or almost no where differentiable when i.e., F is a fractal curve Fractal function: construction by bisection DF = 1.5 Fractal function: construction by bisection. 2 DF = 2 Typical paths of a quantum mechanical particle Feynman (1948), Feynman & Hibbs (1965): « It appears that quantum-mechanical paths are very irregular. However these irregularities average out over a reasonable length of time to produce a reasonable drift, or « average » velocity, although for short intervals of time the « average » value of the velocity is very high. » « Typical paths of a quantum-mechanical particle are highly irregular on a fine scale. Thus, although a mean velocity can be defined, no mean-square velocity exists at any point. In other words, the paths are nondifferentiable. » Typical paths of a quantum mechanical particle t x Feynman & Hibbs (1965) p. 177: Fractals: Abbott & Wise 1981 Typical paths of a quantum mechanical particle Einstein, letter to Pauli (1948): «[…] this complete description could not be content with the fundamental concepts used in point mechanics. I have told you more than once that I am an inveterate supporter, not of differential equations, but quite of the principle of general relativity whose heuristic force is indispensable to us. However, despite much research, I have not succeeded satisfying the principle of general relativity in another way than thanks to differential equations; maybe someone will find out another possibility, provided he searches with enough perseverance.» Cf also G. Bachelard 1927, 1940, A. Buhl 1934…: non-analycity, see Alunni 2001 Principle of relativity of scales *Re-definition of space-time resolution intervals as characterizing the state of scale of the coordinate system *Relative character of the « resolutions » (scale-variables):only scale ratios do have a physical meaning, never an absolute scale *Principle of scale relativity: « the fundamental laws of nature are valid in any coordinate system, whatever its state of scale » *Principle of scale covariance: the equations of physics keep their form (the simplest possible)* in the scale transformations of the coordinate system Weak: same form under generalized transformations Strong: Galilean form (vacuum, inertial motion) Coordinate system Motion t Velocity Acceleration Scale δ t Resolution Origin δ x Orientation x Scale Relativity : structure of the theory Laws of scale Induced dynamics Scale-motion transformations (in (motion laws) in coupling scale space) standard space-(time) « Galilean » D =2 T Standard quantum Special scale relativity mechanics Abelian and non- Abelian Generalized gauge fields scale relativity Generalized quantum mechanics Quantum laws in scale space.