Vacuum corrections in Kaluza–Klein model with nonstationary geometry

The vacuum polarization of a nonminimally coupled scalar field in a multidimensional curved spacetime is studied. It is assumed that the additional spatial dimensions form a nonstationary sphere. The method proposed for calculating the vacuum effective action makes it possible to obtain in a unified manner both the “topological” corrections as well as the terms of local-geometric origin which arise for even dimensions (divergences, logarithmic terms, conformal anomalies). In an example with two additional dimensions, the leading polarization contributions containing the lowest derivatives of the metric are calculated. For this model, acceptable static solutions with four-dimensional Minkowski space are found, and it is shown that (as in the odd-dimensional case) the nature of the stationary point of the effective Hamiltonian is not by itself a criterion of stability of spontaneous compactification since the dynamics of small perturbations is determined by kinetic vacuum corrections.