Communications in Mathematical Sciences

Volume 1 (2003)

Number 2

Nonpertubative Contributions in Quantum-Mechanical Models: The Instantonic Approach

Pages: 245 – 268



Javier Casahorran


We review the euclidean path-integral formalism in connection with the one-dimensional non-relativistic particle. The configurations which allow construction of a semiclassical approximation classify themselves into either topological (instantons) and non-topological (bounces) solutions. The quantum amplitudes consist of an exponential associated with the classical contribution 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-wall 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.

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