Instantons play an important role in the following situation: quantum theories corresponding to classical actions that have non-continuously connected degenerate minima. The simplest examples are provided by one-dimensional quantum systems with symmetries and potentials with non-symmetric minima. Classically, the states of minimum energy correspond to a particle sitting at any of the minima of the potential. The position of the particle breaks (spontaneously) the symmetry of the system. By contrast, in quantum mechanics (QM), the modulus of the ground-state wave function is large near all the minima of the potential, as a consequence of barrier penetration effects. Two typical examples illustrate this phenomenon: the double-well potential, and the cosine potential, whose periodic structure is closer to field theory examples. In the context of stochastic dynamics, instantons are related to Arrhenius law. The proof of the existence of instantons relies on an inequality related to supersymmetric structures, and which generalizes to some field theory examples. Again, the presence of instantons again indicates that the classical minima are connected by quantum tunnelling, and that the symmetry between them is not spontaneously broken. Examples of such a situation are provided, in two dimensions, by the charge conjugation parity (CP) (N − 1) models and, in four dimensions, by SU(2) gauge theories.
Analytic calculation of ground state splitting in symmetric double well potential
