QI 2012: Quantum Interaction pp 13-23 | Cite as
On Least Action Principles for Discrete Quantum Scales
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 relativityPreview
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References
- 1.Bitbol, M.: Mécanique quantique, une introduction philosophique. Champs-Flammarion, Paris (1997)Google Scholar
- 2.Bohm, D., Hiley, B.J.: The Undivided Universe: An Ontological Interpretation of Quantum Theory. Routledge, New York (1993)Google Scholar
- 3.Cresson, J., Greff, I.: Non-differentiable embedding of Lagrangian systems and partial differential equations. Journal of Mathematical Analysis and Applications 384(2), 626–646 (2011)MathSciNetMATHCrossRefGoogle Scholar
- 4.d’Espagnat, B.: Le réel voilé; Analyse des concepts quantiques. Fayard, Paris (1994)Google Scholar
- 5.Filk, T., von Müller, A.: Quantum physics and consciousness: The quest for a common conceptual foundation. Mind and Matter 7, 59–79 (2009)Google Scholar
- 6.Friedberg, R., Lee, T.D.: Discrete Quantum Mechanics. Nuclear Physics B 225(1), 1–52 (1983)MathSciNetCrossRefGoogle Scholar
- 7.Gondran, M.: Complex calculus of variations and explicit solutions for complex Hamilton-Jacobi equations. C. R. Acad. Sci. Paris 332(1), 677–680 (2001); Complex analytical mechanics, complex nonstandard stochastic process and quantum mechanics. C. R. Acad. Sci. Paris 333(1), 593–598 (2001)Google Scholar
- 8.Greenspan, D.: A new explicit discrete mechanics with applications. J. Franklin Institute 294, 231–240 (1972)MathSciNetMATHCrossRefGoogle Scholar
- 9.Khrennikov, A., Volovich, J.I.: Discrete time dynamical models and their quantum-like context-dependent properties. J. Modern Optics 51(6/7), 113–114 (2004)MathSciNetGoogle Scholar
- 10.Khrennikov, A.: Discrete time dynamics. In: Contextual Approach to Quantum Formalism, ch. 12, Springer, Heidelberg (2009)CrossRefGoogle Scholar
- 11.Nelson, E.: Derivation of the Schrödinger Equation from Newtonian Mechanics. Physical Review 150, 1079–1085 (1966)CrossRefGoogle Scholar
- 12.Nottale, L.: Fractal space-time and microphysics: towards a theory of scale relativity, p. 333. World Scientific (1993)Google Scholar
- 13.Odake, S., Sasaki, R.: Discrete Quantum Mechanics. Journal of Physics A: Mathematical and Theoretical 44(35), 353001 (2011)MathSciNetCrossRefGoogle Scholar
- 14.Schrödinger, E.: Quantizierung als Eigenwertproblem (Erste Mitteilung). Annalen der Physik 79, 361–376; Über das Verhältnis der Heisenberg Born Jordanischen Quantenmechanik zu der meinen. Annalen der Physik 79, 734–756 (1926)Google Scholar