Abstract
We consider variational problems where the velocity depends on a scale. After recalling the fundamental principles that lead to classical and quantum mechanics, we study the dynamics obtained by replacing the velocity by some physical observable at a given scale into the expression of the Lagrangian function. Then, discrete Euler-Lagrange and Hamilton-Jacobi equations are derived for a continuous model that incorporates a real-valued discrete velocity. We also examine the paradigm for complex-valued discrete velocity, inspired by the scale relativity of Nottale. We present also rigorous definitions and preliminary results in this direction.
Keywords
- quantum operators
- scale relativity
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Dubois, F., Greff, I., Hélie, T. (2012). On Least Action Principles for Discrete Quantum Scales. In: Busemeyer, J.R., Dubois, F., Lambert-Mogiliansky, A., Melucci, M. (eds) Quantum Interaction. QI 2012. Lecture Notes in Computer Science, vol 7620. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35659-9_2
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DOI: https://doi.org/10.1007/978-3-642-35659-9_2
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-35658-2
Online ISBN: 978-3-642-35659-9
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