Comments on the adiabatic theorem

Abstract

We consider the simplest example of a nonstationary quantum system which is quantum mechanical oscillator with varying frequency and λφ 4 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 non-adiabatically varying frequency we obtain strong IR corrections to the Keldysh propagator which come from the 'sunset' diagrams, grow with time indefinetely 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. *

Quantization in background scalar fields

“…This definition is equivalent to the one used in (2.29). Note that in (0 + 1)-dimensions diagrammatic technique works only for correlation functions averaged over the vacuum or thermal (stationary) state, because diagrammatics is based on the Wick's theorem [11,26], which is applicable only in stationary situations in one dimension. However, in our case this restriction does not bother us, because the state of the fields does not change in time.…”

“…Second, usually loop integrals receive leading contributions due to large virtual momenta, q > p -the main income into the lower p-levels comes from the higher q-levels. Finally, the intuition gained during the study of other background fields [2,3,[6][7][8][9][10][11] tells us that the main contribution should come from the integrands of the form F * (t 3 )F (t 4 )e ip(t 3 −t 4 ) , because in this case it is possible to single out the part of the integrand which does not depend on t = t 3 +t 4 2 . (Then the integral over dt may give the growing with T factor.)…”

“…For the last equality we neglected the terms of the order (λφ) 4 Λ 2 and smaller. Thus, in the leading order for large φ and small derivatives of φ the effective action has the following form 11 :…”

“…For instance, stationary approximation is violated in an expanding space-time (see e.g. [1][2][3][4][5]), in strong electric fields [6,7], during the gravitational collapse [8] and in a number of other non-trivial physical situations [9][10][11][12]. In such situations loop corrections to the tree-level correlation functions grow with time.…”

“…It would be interesting to calculate loop corrections to the quantum averages on top of such a rolling classical solution. In principle such corrections can change the situation under consideration [1][2][3][6][7][8][9][10][11]. If one considers a coherent state decay, most likely this would not happen: of course, strong initial perturbation can induce complex dynamics for a while, but one expects that eventually the field falls on the classical trajectory described by the equation (6.3) for large and slowly changing values of φ cl .…”