Instantons in quantum mechanics (QM)

Perturbative expansion can be generated by calculating Euclidean functional integrals by the steepest descent method always looking, in the absence of external sources, for saddle points in the form of constant solutions to the classical field equations. However, classical field equations may have non-constant solutions. In Euclidean stable field theories, non-constant solutions have always a larger action than minimal constant solutions, because the gradient term gives an additional positive contribution. The non-constant solutions whose action is finite, are called instanton solutions and are the saddle points relevant for a calculation, by the steepest descent method, of barrier penetration effects. This chapter is devoted to simple examples of non-relativistic quantum mechanics (QM), where instanton calculus is an alternative to the semi-classical Wentzel–Kramers–Brillouin (WKB) method. The role of instantons in some metastable systems in QM is explained. In particular, instantons determine the decay rate of metastable states in the semi-classical limit initially confined in a relative minimum of a potential and decaying through barrier penetration. The contributions of instantons at leading order for the quartic anharmonic oscillator with negative coupling are calculate explicitly. The method is generalized to a large class of analytic potentials, and explicit expressions, at leading order, for one-dimensional systems are obtained.