Supersymmetry vs ghosts

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, Q̄ 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 −∞ to ∞ when interactions are added. Notwithstanding the absence of the ground state, the Hamiltonian is Hermitian and the evolution operator is unitary. The algebra involves two complex supercharges, but each level is 3-fold rather than 4-fold degenerate. This unusual feature is due to the fact that certain combinations of supercharges acting on the eigenstates of the Hamiltonian bring them out of the relevant Hilbert space.