Quantum perfect-fluid Kaluza-Klein cosmology

The perfect-fluid cosmology in the (1 + d + D)-dimensional Kaluza-Klein space-times for an arbitrary barotropic equation of state p = (gamma - 1)rho is quantized by using the Schutz's variational formalism. We make efforts in the mathematics to solve the problems in two cases. In the first case of the stiff fluid gamma = 2 we exactly solve the Wheeler-DeWitt equation when the d space is flat. After the superposition of the solutions the wave-packet function is obtained exactly. We analyze the Bohmian trajectories of the final-stage wave-packet functions and show that the scale functions of the flat d spaces and the compact D spaces will eventually evolve into the nonzero finite values. In the second case of gamma approximate to 2, we use the approximated wave function in the Wheeler-DeWitt equation to find the analytic forms of the final-stage wave-packet functions. After analyzing the Bohmian trajectories we show that the flat d spaces will be expanding forever while the scale function of the contracting D spaces would not become zero within finite time. Our investigations indicate that the quantum effect in the quantum perfect-fluid cosmology could prevent the extra compact D spaces in the Kaluza-Klein theory from collapsing into a singularity or that the "crack-of-doom" singularity of the extra compact dimensions is made to occur at t = infinity.