One of the attempts that try to explain the smallness of the internal space in Kaluza-Klein theories is the attractive idea of cosmological dimensional reduction proposed by Chodos and Detweiler and by Freund. In these theories, the internal space shrinks to an unobservable scale by the dynamical evolution of the anisotropic universe. However, if we consider a quantized matter field, the higher-dimensional anisotropic space-time may be isotropized by the effect of particle creation as suggested by Zel'dovich in the case of a conventional four-dimensional space-time. Here, we show that this isotropization process is rather rapid and the mechanism of cosmological dimensional reduction does not work well in the case of space-time with M4×S1 topology. We also show that the above result holds even if we take into account the adiabatic regularization term. We give the analytic solution in the case of space-time with M4×TD topology under some approximation and show that the idea of cosmological dimensional reduction in this case is also broken.
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