Scale Relativity and Fractal Space-Time: Theory and Applications

In the first part of this contribution, we review the development of the theory of scale relativity and its geometric framework constructed in terms of a fractal and nondifferentiable continuous space-time. This theory leads (i) to a generalization of possible physically relevant fractal laws, written as partial differential equation acting in the space of scales, and (ii) to a new geometric foundation of quantum mechanics and gauge field theories and their possible generalisations. In the second part, we discuss some examples of application of the theory to various sciences, in particular in cases when the theoretical predictions have been validated by new or updated observational and experimental data. This includes predictions in physics and cosmology (value of the QCD coupling and of the cosmological constant), to astrophysics and gravitational structure formation (distances of extrasolar planets to their stars, of Kuiper belt objects, value of solar and solar-like star cycles), to sciences of life (log-periodic law for species punctuated evolution, human development and society evolution), to Earth sciences (log-periodic deceleration of the rate of California earthquakes and of Sichuan earthquake replicas, critical law for the arctic sea ice extent) and tentative applications to systems biology.

[1]  J. Cresson.  Scale relativity theory for one-dimensional non-differentiable manifolds , 2002 .

[2]  L. Nottale.  The Theory of Scale Relativity: Non‐Differentiable Geometry and Fractal Space‐Time , 2004 .

[3]  G. Jumarie.  Fractionalization of the complex-valued Brownian motion of order n using Riemann-Liouville derivative. Applications to mathematical finance and stochastic mechanics , 2006 .

[4]  L'arbre de la vie a-t-il une structure fractale ? , 1999 .

[5]  D. Sornette,et al.  Stock Market Crashes, Precursors and Replicas , 1995, cond-mat/9510036.

[6]  L. Nottale.  Relativité d'échelle et morphogenèse , 2001 .

[7]  T. Aushev,et al.  Radiative B meson decays into Kπγ and Kππγ final states , 2002 .

[8]  Gravitational structure formation in scale relativity , 2003, astro-ph/0310036.

[9]  Jacky Cresson,et al.  Fractional differential equations and the Schrödinger equation , 2005, Appl. Math. Comput..

[10]  Hayes,et al.  Review of Particle Physics. , 1996, Physical review. D, Particles and fields.

[11]  Laurent Nottale,et al.  Fractal Space-Time And Microphysics: Towards A Theory Of Scale Relativity , 1993 .

[12]  L. Nottale.  Scale-relativity and quantization of planet obliquities , 1998 .

[13]  Incompleteness of trajectory-based interpretations of quantum mechanics , 2004, quant-ph/0406054.

[14]  Didier Sornette,et al.  Scale Invariance and Beyond , 1997 .

[15]  J. Lévy-Leblond.  One more derivation of the Lorentz transformation , 1976 .

[16]  Divergence d'échelle et différentiabilité , 2000 .

[17]  Jacky Cresson,et al.  Quantum derivatives and the Schrödinger equation , 2004 .

[18]  M. S. E. Nasche.  Multi-dimensional Cantor Sets in Classical and Quantum Mechanics , 1992 .

[19]  G. Jumarie.  SCHRÖDINGER EQUATION FOR QUANTUM FRACTAL SPACE–TIME OF ORDER n VIA THE COMPLEX-VALUED FRACTIONAL BROWNIAN MOTION , 2001 .

[20]  L. Nottale.  Scale relativity and quantization of the planetary system around the pulsar PSR B1257 + 12 , 1998 .

[21]  R. Cash,et al.  Développement humain et loi log-périodique , 2002 .

[22]  L. Nottale.  Scale relativity and non-differentiable fractal space-time , 2001 .

[23]  R. Dalitz.  Elementary particle theory , 1980, Nature.

[24]  Benoit B. Mandelbrot,et al.  Fractal Geometry of Nature , 1984 .

[25]  Quantum–classical transition in scale relativity , 2004, quant-ph/0609161.

[26]  L. Nottale.  RELATIVITÉ, ÊTRE ET NE PAS ÊTRE , 2002 .

[27]  Alain Connes,et al.  Noncommutative Geometry and Matrix Theory: Compactification on Tori , 1997, hep-th/9711162.

[28]  L. Nottale.  ASTROPHYSICAL APPLICATIONS OF THE THEORY OF SCALE RELATIVITY , 2006 .

[29]  Wang,et al.  Comment on "Repeated measurements in stochastic mechanics" , 1993, Physical review. D, Particles and fields.

[30]  Scale Relativity : First Steps Toward a Field Theory , 1994 .

[31]  L. Nottale,et al.  SCALE-RELATIVITY, FRACTAL SPACE-TIME AND GRAVITATIONAL STRUCTURES , 1998 .

[32]  M. Berry.  Quantum fractals in boxes , 1996 .

[33]  G. Schumacher,et al.  Scale relativity and quantization of the solar system Orbit quantization of the planet's satellites , 1997 .

[34]  Carlos Castro,et al.  How the New Scale Relativity Theory resolves some quantum paradoxes , 2000 .

[35]  L. Nottale.  New formulation of stochastic Mechanics. Application to chaos , 1995 .

[36]  I. Khan.  The music of life , 1983 .

[37]  Peter Hänggi,et al.  Is quantum mechanics equivalent to a classical stochastic process , 1979 .

[38]  R. Feynman,et al.  Quantum Mechanics and Path Integrals , 1965 .

[39]  S. Saar,et al.  Time Evolution of the Magnetic Activity Cycle Period. II. Results for an Expanded Stellar Sample , 1999 .

[40]  D. Sornette.  Discrete scale invariance and complex dimensions , 1997, cond-mat/9707012.

[41]  Természettudományok.  Extrasolar Planets Encyclopaedia , 2010 .

[42]  L. Nottale.  Scale Relativity: A Fractal Matrix for Organization in Nature , 2007 .

[43]  Ke Chen,et al.  Applied Mathematics and Computation , 2022 .

[44]  C. Auffray,et al.  Scale relativity theory and integrative systems biology: 1. Founding principles and scale laws. , 2008, Progress in biophysics and molecular biology.

[45]  Laurent Nottale,et al.  THE THEORY OF SCALE RELATIVITY , 1992 .

[46]  The Pauli equation in scale relativity , 2006, quant-ph/0609107.

[47]  L. Nottale.  Origin of Complex and Quaternionic Wavefunctions in Quantum Mechanics: the Scale-Relativistic View , 2008 .

[48]  Laurent Nottale,et al.  The scale-relativity program , 1999 .

[49]  G. Jumarie,et al.  Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results , 2006, Comput. Math. Appl..

[50]  L. Nottale,et al.  Non-Abelian gauge field theory in scale relativity , 2006, hep-th/0605280.

[51]  M. M. Novak,et al.  Fractals and Beyond: Complexities in the Sciences , 1998 .

[52]  G. Ord.  The Schrödinger and Dirac Free Particle Equations without Quantum Mechanics , 1996 .

[53]  Rovelli,et al.  Knot theory and quantum gravity. , 1988, Physical review letters.

[54]  Alessandro Vespignani,et al.  Local Rigidity and Self-Organized Criticality for Avalanches , 1995 .

[55]  R. Festa,et al.  Clues to discretization on the cosmic scale , 1997 .

[56]  Laurent Nottale,et al.  Scale-relativity and quantization of the universe I. Theoretical framework , 1997 .

[57]  Terry Marks-Tarlow,et al.  Simultaneity : temporal structures and observer perspectives , 2008 .

[58]  Didier Sornette,et al.  Complex Critical Exponents from Renormalization Group Theory of Earthquakes: Implications for Earthquake Predictions , 1995 .

[59]  Andrew Lesniewski,et al.  Noncommutative Geometry , 1997 .

[60]  Laurent Nottale,et al.  FRACTALS AND THE QUANTUM THEORY OF SPACETIME , 1989 .

[61]  The Feynman propagator from a single path. , 2001, Physical review letters.

[62]  Franco Selleri,et al.  Frontiers of Fundamental Physics , 1984 .

[63]  Denis Noble,et al.  Modelling the heart: insights, failures and progress. , 2002, BioEssays : news and reviews in molecular, cellular and developmental biology.

[64]  H. Kröger.  Fractal geometry in quantum mechanics, field theory and spin systems , 2000 .

[65]  L. Nottale,et al.  Log-periodic laws applied to geosciences , 2004 .

[66]  A. Provost,et al.  Scaling rules in rock fracture and possible implications for earthquake prediction , 1982, Nature.

[67]  D. Noble.  Music of life : biology beyond the genome , 2006 .

[68]  I. Adachi,et al.  Improved measurements of branching fractions and CP partial rate asymmetries for B-->omegaK and B-->omegapi , 2006 .

[69]  D. McKeon,et al.  Time Reversal in Stochastic Processes and the Dirac Equation II , 1992 .

[70]  Steven N. Shore.  Magnetic Fields in Astrophysics , 1992 .

[71]  L. Nottale,et al.  On the Fractal Structure of Evolutionary Trees , 2002 .

[72]  Guy Jumarie,et al.  The Minkowski's space-time is consistent with differential geometry of fractional order , 2007 .

[73]  Phase transition in gauge theories and the Planck scale physics , 2001, hep-th/0110127.

[74]  Edmund J. Crampin,et al.  Multiscale computational modelling of the heart , 2004, Acta Numerica.

[76]  Rovelli,et al.  Spin networks and quantum gravity. , 1995, Physical review. D, Particles and fields.

[77]  R. Nichol,et al.  Cosmological constraints from the SDSS luminous red galaxies , 2006, astro-ph/0608632.

[78]  C. Froeschlé,et al.  Chaos and diffusion in Hamiltonian systems , 1995 .

[79]  Jacky Cresson,et al.  Fractional embedding of differential operators and Lagrangian systems , 2006, math/0605752.

[80]  Edward Nelson.  Derivation of the Schrodinger equation from Newtonian mechanics , 1966 .

[81]  J. L. S. Luk.  Mémoire d'habilitation à diriger des recherches , 2000 .

[82]  D. Sokoloff,et al.  Magnetic Fields in Astrophysics , 1958 .

[83]  Rama Cont,et al.  Scale Invariance and Beyond , 1997 .

[84]  John Ellis,et al.  Int. J. Mod. Phys. , 2005 .

[85]  L. Nottale,et al.  Scale-relativity and quantization of exoplanet orbital semi-major axes , 2000 .

[86]  String theory, scale relativity and the generalized uncertainty principle , 1995, hep-th/9512044.

[87]  Jacky Cresson.  Scale calculus and the Schrödinger equation , 2003 .

[88]  L. Nottale.  ASTRONOMY AND ASTROPHYSICS Letter to the Editor Scale-relativity and quantization of extra-solar planetary systems , 1996 .

[89]  L. Nottale.  Scale relativity and gauge invariance , 2001 .

[90]  G Ord,et al.  Fractal space-time: a geometric analogue of relativistic quantum mechanics , 1983 .

[91]  G. Kane,et al.  Elementary particle theory , 1970 .

[92]  M. Peterson.  Analogy between thermodynamics and mechanics , 1979 .

[93]  L. F. Abbott,et al.  Dimension of a Quantum-Mechanical Path. , 1981 .

[94]  R. Rosenfeld.  Nature , 2009, Otolaryngology--head and neck surgery : official journal of American Academy of Otolaryngology-Head and Neck Surgery.

[95]  Laurent Nottale,et al.  Derivation of the postulates of quantum mechanics from the first principles of scale relativity , 2007, 0711.2418.

[96]  Charles Auffray,et al.  Scale relativity theory and integrative systems biology: 2. Macroscopic quantum-type mechanics. , 2008, Progress in biophysics and molecular biology.

[97]  Edward J. Wollack,et al.  Wilkinson Microwave Anisotropy Probe (WMAP) Three Year Results: Implications for Cosmology , 2006, astro-ph/0603449.

[98]  J. Pissondes.  Quadratic relativistic invariant and metric form in quantum mechanics , 1999 .

[99]  G. Amelino-Camelia.  DOUBLY-SPECIAL RELATIVITY: FIRST RESULTS AND KEY OPEN PROBLEMS , 2002, gr-qc/0210063.

[100]  L. Nottale,et al.  Fractals and nonstandard analysis , 1984 .

[101]  S. Gould,et al.  Punctuated equilibria: the tempo and mode of evolution reconsidered , 1977, Paleobiology.

[102]  Daniel M. Dubois.  Computing anticipatory systems : CASYS'03 - sixth international conference : Liège, Belgium 11-16 August 2003 , 2004 .

[103]  L. Nottale.  Scale-relativistic cosmology , 2003 .

[104]  Alberto Carpinteri,et al.  Power Scaling Laws and Dimensional Transitions in Solid Mechanics , 1996 .

[105]  Brian Fields,et al.  Big-Bang nucleosynthesis (2006 Particle Data Group mini-review) , 2006, astro-ph/0601514.

[106]  E. Weibel,et al.  Fractals in Biology and Medicine , 1994 .

[107]  H. Nicolai.  What are string theories , 1987 .

[108]  Laurent Nottale,et al.  Scale relativity and fractal space-time: applications to quantum physics, cosmology and chaotic systems. , 1996 .

[109]  Daniel M. Dubois,et al.  Computing Anticipatory Systems , 1998 .

[110]  I. Prigogine,et al.  Quantum mechanics, diffusion and chaotic fractals , 1995 .

[111]  R. Hermann,et al.  Numerical simulation of a quantum particle in a box , 1997 .