Scale calculus and the Schrödinger equation

This paper is twofold. In a first part, we extend the classical differential calculus to continuous nondifferentiable functions by developing the notion of scale calculus. The scale calculus is based on a new approach of continuous nondifferentiable functions by constructing a one parameter family of differentiable functions f(t,e) such that f(t,e)→f(t) when e goes to zero. This led to several new notions as representations: fractal functions and e-differentiability. The basic objects of the scale calculus are left and right quantum operators and the scale operator which generalizes the classical derivative. We then discuss some algebraic properties of these operators. We define a natural bialgebra, called quantum bialgebra, associated with them. Finally, we discuss a convenient geometric object associated with our study. In a second part, we define a first quantization procedure of classical mechanics following the scale relativity theory developed by Nottale. We obtain a nonlinear Schrodinger equation v...

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

[2]  J. Zukas.  Introduction to the Modern Theory of Dynamical Systems , 1998 .

[3]  N. Wesley Earp.  Reading in Mathematics. , 1969 .

[4]  A. Connes,et al.  Gravity coupled with matter and the foundation of non-commutative geometry , 1996, hep-th/9603053.

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

[6]  Arthur J. Knodel,et al.  Non-linear wave mechanics : a causal interpretation , 1960 .

[7]  Local Fractional Derivatives and Fractal Functions of Several Variables , 1998, physics/9801010.

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

[9]  A. Einstein.  Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen [AdP 17, 549 (1905)] , 2005, Annalen der Physik.

[10]  P. Lugol.  Annalen der Physik , 1906 .

[11]  A. Kolmogorov,et al.  Éléments de la théorie des fonctions et de l'analyse fonctionnelle , 1994 .

[12]  Physique quantique et représentation du monde , 1992 .

[13]  J. P. Vigier,et al.  Model of the Causal Interpretation of Quantum Theory in Terms of a Fluid with Irregular Fluctuations , 1954 .

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

[15]  E. Madelung,et al.  Quantentheorie in hydrodynamischer Form , 1927 .

[16]  B. Greene,et al.  The elegant universe : superstrings, hidden dimensions, and the quest for the ultimate theory , 2000 .

[17]  Ernst Hairer,et al.  Analysis by Its History , 1996 .

[18]  J. Lévy-leblond,et al.  La nature de la physique , 1980 .

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

[20]  Jacky Cresson,et al.  About Non-differentiable Functions , 2001 .

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

[22]  L. Nottale.  Fractal space-time and microphysics , 1993 .

[23]  George Bruce Halsted.  The International Congress of Mathematicians , 1900 .

[24]  E. C. Titchmarsh,et al.  The theory of functions , 1933 .

[25]  C. Tricot.  Courbes et dimension fractale , 1999 .

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

[27]  I. Bialynicki-Birula,et al.  Nonlinear Wave Mechanics , 1976 .

[28]  Ralph Duncan James,et al.  Proceedings of the International Congress of Mathematicians , 1975 .

[29]  N. Reshetikhin,et al.  Quantum Groups , 1993, hep-th/9311069.

[30]  L. M. Milne-Thomson,et al.  The Calculus Of Finite Differences , 1934 .

[31]  Thomas J. Osler,et al.  Fractional Derivatives and Leibniz Rule , 1971 .