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Power Law Duality in Classical and Quantum Mechanics

by 1 and 2,3,*
1
Department of Physics, State University of New York at Albany, Albany, NY 12222, USA
2
European Southern Observatory, Karl-Schwarzschild-Straße 2, D-85748 Garching, Germany
3
Institut für Theoretische Physik I, Universität Erlangen-Nürnberg, Staudtstraße 7, D-91058 Erlangen, Germany
*
Author to whom correspondence should be addressed.
Academic Editor: Ignatios Antoniadis
Symmetry 2021, 13(3), 409; https://doi.org/10.3390/sym13030409
Received: 24 January 2021 / Revised: 25 February 2021 / Accepted: 26 February 2021 / Published: 3 March 2021
(This article belongs to the Special Issue Symmetries in Quantum Mechanics and Statistical Physics)
The Newton–Hooke duality and its generalization to arbitrary power laws in classical, semiclassical and quantum mechanics are discussed. We pursue a view that the power-law duality is a symmetry of the action under a set of duality operations. The power dual symmetry is defined by invariance and reciprocity of the action in the form of Hamilton’s characteristic function. We find that the power-law duality is basically a classical notion and breaks down at the level of angular quantization. We propose an ad hoc procedure to preserve the dual symmetry in quantum mechanics. The energy-coupling exchange maps required as part of the duality operations that take one system to another lead to an energy formula that relates the new energy to the old energy. The transformation property of the Green function satisfying the radial Schrödinger equation yields a formula that relates the new Green function to the old one. The energy spectrum of the linear motion in a fractional power potential is semiclassically evaluated. We find a way to show the Coulomb–Hooke duality in the supersymmetric semiclassical action. We also study the confinement potential problem with the help of the dual structure of a two-term power potential. View Full-Text
Keywords: power-law duality; classical and quantum mechanics; semiclassical quantization; supersymmetric quantum mechanics; quark confinement power-law duality; classical and quantum mechanics; semiclassical quantization; supersymmetric quantum mechanics; quark confinement
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MDPI and ACS Style

Inomata, A.; Junker, G. Power Law Duality in Classical and Quantum Mechanics. Symmetry 2021, 13, 409. https://doi.org/10.3390/sym13030409

AMA Style

Inomata A, Junker G. Power Law Duality in Classical and Quantum Mechanics. Symmetry. 2021; 13(3):409. https://doi.org/10.3390/sym13030409

Chicago/Turabian Style

Inomata, Akira; Junker, Georg. 2021. "Power Law Duality in Classical and Quantum Mechanics" Symmetry 13, no. 3: 409. https://doi.org/10.3390/sym13030409

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