We consider the simplest example of a nonstationary quantum system which is quantum mechanical oscillator with varying frequency and [Formula: see text] self-interaction. We calculate loop corrections to the Keldysh, retarded/advanced propagators and vertices using Schwinger–Keldysh diagrammatic technique and show that there is no physical secular growth of the loop corrections in the cases of constant and adiabatically varying frequency. This fact corresponds to the well-known adiabatic theorem in quantum mechanics. However, in the case of nonadiabatically varying frequency we obtain strong IR corrections to the Keldysh propagator which come from the “sunset” diagrams, grow with time indefinitely and indicate energy pumping into the system. It reveals itself via the change in time of the level population and of the anomalous quantum average.
A monopole-like singularity generated by Berry's phase in momentum space and associated anomalous Poisson brackets have been discussed in various fields. In the applications, it is crucially important to understand precisely what is the monopole in momentum space implied by Berry's phase. Berry's phase is sensitive to how it is measured and thus very different from the genuine Dirac monopole; the monopole-like singularity in Berry's phase is a result of adiabatic approximation while the singularity disappears in non-adiabatic approximation. The resolution of monopole-like singularity in the non-adiabatic domain implies that no Dirac string appears close to the monopole position and the anomalous terms containing the monopole flux in quantum mechanical commutation relations are absent. The absence of modified commutators is also concluded if one understands Berry's phase in momentum space as an O(ℏ) quantum mechanical effect with a quantized magnetic charge eM=2πℏ.