Supersymmetry vs ghosts

TL;DR

This paper explores a supersymmetric quantum mechanical system with higher derivatives, revealing a nontrivial algebra of integrals of motion, an unbounded spectrum, and implications for unitarity despite the absence of a ground state.

Contribution

It introduces a novel supersymmetric quantum system with higher derivatives, analyzing its algebraic structure, spectrum, and potential for unitary evolution without a ground state.

Findings

Spectrum is unbounded from below.

System has infinite degeneracy in free case.

Interactions lead to a continuous spectrum from -∞ to ∞.

Abstract

We consider the simplest nontrivial supersymmetric quantum mechanical system involving higher derivatives. We unravel the existence of additional bosonic and fermionic integrals of motion forming a nontrivial algebra. This allows one to obtain the exact solution both in the classical and quantum cases. The supercharges $Q, \bar Q$ are not anymore Hermitially conjugate to each other, which allows for the presence of negative energies in the spectrum. We show that the spectrum of the Hamiltonian is unbounded from below. It is discrete and infinitely degenerate in the free oscillator-like case and becomes continuous running from $-\infty$ to $\infty$ when interactions are added. Notwithstanding the absence of the ground state, there is no collapse, which suggests that a unitary evolution operator may be defined.