Working Paper Article Version 1 This version is not peer-reviewed

Power Law Duality in Classical and Quantum Mechanics

Version 1 : Received: 24 January 2021 / Approved: 25 January 2021 / Online: 25 January 2021 (16:42:35 CET)

A peer-reviewed article of this Preprint also exists.

Inomata, A.; Junker, G. Power Law Duality in Classical and Quantum Mechanics. Symmetry 2021, 13, 409. Inomata, A.; Junker, G. Power Law Duality in Classical and Quantum Mechanics. Symmetry 2021, 13, 409.

Journal reference: Symmetry 2021, 13, 409
DOI: 10.3390/sym13030409

Abstract

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 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.

Subject Areas

Power-law duality; Classical and quantum mechanics; Semiclassical quantization; Supersymmetric quantum mechanics; Quark confinement

Comments (1)

Comment 1
Received: 3 February 2021
Commenter: Wu-Sheng Dai
The commenter has declared there is no conflict of interests.
Comment: Dear colleagues, We would like to recommend to you some of our work on this topic. We hope you will be interested in our work and hope our work can be helpful to your research. 1) Wen-Du Li and Wu-Sheng Dai, A duality in classical and quantum mechanics: General results, arXiv:1909.01089. 2) Wen-Du Li and Wu-Sheng Dai, A duality of scalar fields: General results, arXiv:1909.11659. On January 30, 2021, we sent you an email. We recommended some of our related works to you in the email. arXiv:1710.10481v2 "Quantum Newton duality" arXiv:1909.01089v2 "A duality in classical and quantum mechanics: General results" arXiv:1905.06805v2 “A duality of fields” arXiv:1909.11659v1 “A duality of scalar fields: General results” arXiv:2004.09972v2 “Exactly solvable Gross-Pitaevskii type equations” The two jobs listed above are the latest version of our work, and we have added a reference to your work in this updated version. Bset Regards Prof. Wu-Sheng Dai Department of Physics, Tianjin University, PR China
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