Abstract
The hidden supersymmetry and related tri-supersymmetric structure of the free particle system, the Dirac delta potential problem and the Aharonov–Bohm effect (planar, bound state, and tubule models) are explained by a special nonlocal unitary transformation, which for the usual N=2 supercharges has a nature of Foldy–Wouthuysen transformation. We show that in general case, the bosonized supersymmetry of nonlocal, parity even systems emerges in the same construction, and explain the origin of the unusual N=2 supersymmetry of electron in three-dimensional parity even magnetic field. The observation extends to include the hidden superconformal symmetry.
Highlights
Some quantum systems possess a hidden symmetry associated with nontrivial integrals of motion, which reflect their peculiar properties
The hidden supersymmetry of the first two systems is characterized by the linear in the momentum supercharge operators; in the last two families, the hidden supersymmetry is related to the higher derivative nontrivial operator of the Lax pair of the associated nonlinear integrable system
The procedure of the special unitary transformation and subsequent reduction applies in the current system as well, where it relates the earlier observed hidden supersymmetry of the bound state AB effect [2] with the usual N = 2 supersymmetry associated with the decoupled spin degrees of freedom
Summary
Some quantum systems possess a hidden symmetry associated with nontrivial integrals of motion, which reflect their peculiar properties. A hidden supersymmetry [1] was revealed recently in a class of quantum mechanical systems with a local Hamiltonian. The list of such systems includes the Dirac delta potential problem [2], the Aharonov-Bohm effect (bound state [2] and planar [3] models), the finite-gap periodic quantum systems, and their infinite period limit in the form of reflectionless systems [4, 5]. In this paper we show how the hidden supersymmetry and the associated tri-supersymmetric structure originate from the usual N = 2 supersymmetry and the (twisted) parity symmetry. We indicate that the observation extends to include the hidden superconformal symmetry [3], [11]
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