New Features in Supersymmetry Breakdown in Quantum Mechanics

The supersymmetric quantum mechanical model based on higher-derivative supercharge operators possessing unbroken supersymmetry and discrete energies below the vacuum state energy is described. As an example harmonic oscillator potential is considered 1. Ideas of the supersymmetry have appeared in physics for the first time in the quantum field theory for unifying the interactions of a different nature [1]. In a supersymmetric theory the supersymmetry can be either exact or spontaneously broken. The supersymmetric quantum mechanics has been introduced [2] to illustrate the problems of the supersymmetry breakdown in supersymmetric quantum field theories. For this purpose the Witten criterion based on the Witten index [2] has been elaborated. In the case of the broken supersymmetry the entire spectrum of the super-Hamiltonian is twofold degenerate and in the case of the exact one its vacuum state is nondegenerate. In the first case the supercharge operators map the two states corresponding to the vacuum energy (zero energy) one into another and in the second one the vacuum state is annihilated by both supercharges. (See for example a recent survey [3].) Recently higher-order derivative extension of the supersymmetric quantum mechanics has been elaborated [4]. In this approach supercharges are constructed in terms of the higher-derivative differential operators and the corresponding superalgebra is polynomial in the Hamiltonian. This model exhibits a number of unusual 1 email: [email protected]