A quantum cosmological approach to Kaluza-Klein theory and the boundary condition of 'no boundary'

Abstract

A quantum cosmological model of the Kaluza-Klein theory is investigated. The Wheeler-De Witt equation of (D+1)-dimensional gravity with a (D-4)th rank antisymmetric tensor field (ATF) is analysed, restricted to a minisuperspace of topology R*S3*Sn. The initial condition problem is discussed and the boundary condition of 'no boundary' for a universe corresponding to a manifold with topology of Kaluza-Klein type is formulated. However, in the minisuperspace approximation of Kaluza-Klein theory, the definition of the boundary condition of 'no boundary' becomes obscure and the problem is discussed. The analysis, using the WKB approximation, shows that a subclass of classical paths which contribute to the wavefunction consists of bounce solutions. Furthermore, it is found that initial values of the classical paths, leading to universes of large scale and a compactified internal space of Planck size, can be localised in a very restricted region.