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Scale calculus and the Schrödinger equation. (English) Zbl 1062.39022
Summary: 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 \(f(t,\epsilon)\) such that \(f(t,\epsilon) \to f(t)\) when \(\varepsilon\) goes to zero. This led to several new notions as representations: fractal functions and \(\epsilon\)-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 Schrödinger equation via the classical Newton’s equation of dynamics using the scale operator. Under special assumptions we recover the classical Schrödinger equation and we discuss the relevance of these assumptions.

39A99 Difference equations
81R99 Groups and algebras in quantum theory
Full Text: DOI arXiv
[1] Abbott L., Am. J. Phys. 49 pp 37– (1981)
[2] Ben Adda F., J. Fractional Calculus 11 pp 21– (1997)
[3] Ben Adda F., J. Math. Anal. Appl. 263 pp 721– (2001) · Zbl 0995.26006
[4] Bialynicky-Birula I., Ann. Phys. (N.Y.) 100 pp 62– (1976)
[5] DOI: 10.1103/PhysRev.96.208 · Zbl 0058.22905
[6] DOI: 10.1007/BF02506388 · Zbl 0881.58009
[7] Cresson J., Chaos, Solitons Fractals 14 pp 553– (2002) · Zbl 1005.81031
[8] Drinfeld V., Leningrad Math. J. 2 pp 829– (1991)
[9] DOI: 10.1002/andp.19053220806 · JFM 36.0975.01
[10] DOI: 10.1007/BF01400372
[11] DOI: 10.1103/PhysRev.150.1079
[12] Nottale L., Int. J. Mod. Phys. A 7 pp 4899– (1992) · Zbl 0954.81501
[13] Osler T. J., Am. Math. Monthly 78 pp 645– (1971) · Zbl 0216.09303
[14] Pardy, M. ”To the nonlinear quantum mechanics,” preprint 2002, quant-ph/0111105.
[15] Puszkarz, W. ”On the Staruszkiewicz modification of the Schrödinger equation,” preprint 1999, quant-ph/9912006.
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