Nonlinear supersymmetry in quantum mechanics: algebraic properties and differential representation

Abstract

We study the Nonlinear (Polynomial, N -fold,...) Supersymmetry algebra in one-dimensional QM. Its structure is determined by the type of conjugation operation (Hermitian conjugation or transposition) and described with the help of the Super-Hamiltonian projection on the zero-mode subspace of a supercharge. We show that the SUSY algebra with transposition symmetry is always polynomial in the Super-Hamiltonian if supercharges represent differential operators of finite order. The appearance of the extended SUSY with several (complex or real) supercharges is analyzed in details and it is established that no more than two independent supercharges may generate a Nonlinear superalgebra which can be appropriately specified as N = 2 SUSY. In this case we find a non-trivial hidden symmetry operator and rephrase it as a non-linear function of the Super-Hamiltonian on the physical state space.The full N = 2 Non-linear SUSY algebra includes "central charges" both polynomial and non-polynomial (due to a symmetry operator) in the Super-Hamiltonian.

Supersymmetry in 5D gravity

“…The extension cannot be, however, the standard N-extended one because the number of the degeneracy between "bosonic" and "fermionic" degrees of freedom is necessarily even. We have discussed two possible extensions [53,54,55,56,57,58,59] [60]. An interesting fact is that both of them are nonlinear extensions of the N = 2 supersymmetry algebra.…”

“…(III.45) is given by the square of the Hamiltonian H but not the linear of it. The system with the nonlinear algebra has been discussed in [53,54,55,56,57,58,59] as an extension of the N = 2 supersymmetry. Another type of extensions of the N = 2 supersymmetry has also been discussed in [60].…”

A novel multi-parameter family of quantum systems with partially broken -fold supersymmetry

“…[1,2], where it was also shown that Π N is proportional to the characteristic polynomial of the restriction of the component Hamiltonians…”

-fold supersymmetry in quantum systems with position-dependent mass

“…Indeed, in the proof of the theorem on SUSY algebras with T symmetry in Ref. [36], we only need to modify the functions w ± N in Eq. (6) in Eq.…”