Quantum tunneling and negative eigenvalues

In the path-integral approach to the decay of a metastable state by quantum tunneling, the tunneling process is dominated by a solution to the imaginary-time equations of motion, called the bounce. In all known cases, the second variational derivative of the euclidean action at the bounce has one and only one negative eigenvalue. This note explains this phenomenon by showing it is an inevitable feature of the bounce for a wide class of systems. This class includes a set of particles interacting through potentials obeying some mild technical restrictions, and also theories of interacting scalar and gauge fields. There may exist solutions in other ways like bounces and which have more than one negative eigenvalue, but, even if they do exist, they have nothing to do with tunneling.