Power Law Duality in Classical and Quantum Mechanics

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 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.

On the Supersymmetry of the Klein–Gordon Oscillator

“…Another route for further investigation would be to look at generalised nonharmonic oscillators characterized by a potential function U a (r) := λ a r a using minimal substitution π := p − i( ∇U a )(r). Such power-law potentials obey duality symmetry in classical and nonrelativistic quantum mechanics (see recent work [34] and references therein). In particular, a harmonic potential where a = 2, which corresponds to the discussed KGO case, is dual to the Kepler potential where a = −1.…”

Power-duality in path integral formulation of quantum mechanics

“…In an earlier article [1], we have studied the duality between two power force laws (power duality in short) in classical, semiclassical and quantum mechanics. In the present paper, we wish to investigate the power duality in Feynmanʼs path integral formulation of quantum mechanics [2][3][4].…”

“…The power-law duality for arbitrary power forces in classical mechanics was analyzed by Kasner [13], Arnol'd [8], and others [14][15][16]. In our previous work [1] we expanded the domain of the dual pairs.…”

“…Such studies motivate us to investigate the power duality in path integration by asking whether the power duality is a valid symmetry or a broken symmetry; what quantities remain invariant if valid; how breaking occurs if broken; what benefits can be gained altogether. For a more detailed review of the historical backgrounds of the power duality see [1].…”

Special Issue: “Symmetries in Quantum Mechanics and Statistical Physics”

“…The contributions by Inomata et al [6] and Zhao et al [7] reconsider joint transformations of space and time, mapping different physical systems onto each other. In [6], the well-known Newton-Hooke duality and its generalization to arbitrary power-law potentials is reviewed. Here, duality is viewed as a symmetry concept.…”