The quantum cosmological version of the multidimensional Einstein-Yang-Mills model in an $\mathbf{R}\times S^3\times S^d$ topology is studied in the framework of the Hartle-Hawking proposal. In contrast with previous work in the literature, we consider Yang-Mills field configurations with nonvanishing time-dependent components in both $S^3$ and $S^d$ spaces. We obtain stable compactifying solutions that do correspond to extrema of the Hartle-Hawking wave function of the Universe. Subsequently, we also show that the regions where the 4-dimensional metric behaves classically or quantum mechanically (i.e., regions where the metric is Lorentzian or Euclidean) will depend on the number $d$ of compact space dimensions.

• Received 27 June 1996

DOI:https://doi.org/10.1103/PhysRevD.56.4530