Nonperturbative contributions in quantum-mechanical models: the instantonic approach

TL;DR

This paper reviews the instantonic approach in quantum mechanics, focusing on nonperturbative effects like instantons and bounces, and applies the method to models such as the double-well and periodic potentials to evaluate quantum decay rates.

Contribution

It systematically applies the instanton calculus to quantum-mechanical models, including multi-instanton effects and functional determinant calculations using zeta-function methods.

Findings

Computed functional determinants for shape-invariant potentials.

Analyzed multi-instanton contributions in quantum tunneling.

Evaluated decay rates of metastable states in cubic potentials.

Abstract

We review the euclidean path-integral formalism in connection with the one-dimensional non-relativistic particle. The configurations which allow to construct a semiclassical approximation classify themselves into either topological (instantons) and non-topological (bounces) solutions. The quantum amplitudes consist on an exponential associated with the classical contribution multiplied by the fluctuation factor which is given by a functional determinant. The eigenfunctions as well as the energy eigenvalues of the quadratic operators at issue can be written in closed form due to the shape-invariance property. Accordingly we resort to the zeta-function method to compute the functional determinants in a systematic way. The effect of the multi-instantons configurations is also carefully considered. To illustrate the instanton calculus in a relevant model we go to the double-well potential. The second popular case is the periodic-potential where the initial levels split into bands. The quantum decay rate of the metastable states in a cubic model is evaluated by means of the bounce-like solution.