Supersymmetry and Ghosts in Quantum Mechanics

2000 Mathematics Subject Classification: 81Q60, 35Q40.

A standard supersymmetric quantum system is defined by a Hamiltonian [^H] = ½([^Q]*[^Q] +[^Q][^Q]*), where the super-charge [^Q] satisfies [^Q]2 = 0, [^Q] commutes with [^H]. So we have [^H] ≥ 0 and the quantum spectrum of [^H] is non negative. On the other hand Pais-Ulhenbeck proposed in 1950 a model in quantum-field theory where the d'Alembert operator [¯] = [(∂2)/( ∂t2)] − Δx is replaced by fourth order operator [¯]([¯] + m2), in order to eliminate the divergences occuring in quantum field theory. But then the Hamiltonian of the system, obtained by second quantization, has large negative energies called "ghosts" by physicists. We report here on a joint work with A. Smilga (SUBATECH, Nantes) where we consider a similar problem for some models in quantum mechanics which are invariant under supersymmetric transformations. We show in particular that "ghosts" are still present.