Scale calculus and the Schrodinger equation

Abstract

We introduce the scale calculus, which generalizes the classical differential calculus to non differentiable functions. The new derivative is called the scale difference operator. We also introduce the notions of fractal functions, minimal resolution, and quantum representation of a non differentiable function. We then define a scale quantization procedure for classical Lagrangian systems inspired by the Scale relativity theory developped by Nottale. We prove that the scale quantization of Newtionian mechanics is a non linear Schrodinger equation. Under some specific assumptions, we obtain the classical linear Schrodinger equation.

49 pages

References in corpus (1)

• Gravity coupled with matter and foundation of non-commutative geometry

Cited by in corpus (11)

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• Non-differentiable variational principles
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• The isoperimetric problem for Holderian curves
• Derivation of the postulates of quantum mechanics from the first principles of scale relativity
• Non-Abelian gauge field theory in scale relativity
• Generalized Tu Formula and Hamilton Structures of Fractional Soliton Equation Hierarchy
• Generalized Euler-Lagrange equations for variational problems with scale derivatives
• Stochastic modication of Newtonian dynamics and Induced potential -application to spiral galaxies and the dark potential
• Emergence of complex and spinor wave functions in scale relativity. I. Nature of scale variables
• Multiscale functions, Scale dynamics and Applications to partial differential equations