A quantum cure for the Ostrogradski instability

Abstract

Interacting fourth order quantum mechanics is in the Ostrogradski formalism afflicted by an instability involving the decay of the vacuum. When treating such systems as 1 + 0 -dimensional Euclidean field theories in the transfer operator formalism the ‘instability problem’ and the ‘unitarity problem’ are distinct and decoupled. The instability problem is shown to be absent: a stable ground state always exists and is typically normalizable and strictly positive. The generator H of the transfer operator replaces the Ostrogradski Hamiltonian and is non-Hermitian but selfadjoint with respect to a Krein structure, which also ensures consistency with the Lagrangian functional integral. The case of a scalar quartic derivative interaction is treated in detail. Variational perturbation theory, a strong coupling expansion, and direct diagonalization of matrix truncations are used to compute the spectrum of H in this case.