Abstract
The standard quantum cosmology as well as the classical inflationary cosmology are based on the Friedmann cosmology in the presence of real scalar fields. At the same time it is evident that a complex scalar field has more physical sense because it corresponds to a matter hydrodynamical field. The present paper can be considered as an introduction to the quantum and classical cosmology with a complex scalar field. The two-dimensional potential which appears in this case in the Wheeler-DeWitt equation hs a region of negative values. Different from the case of real fields [Lavrelashvili et al., Nucl. Phys. B 329 (1990) 98] this region is not convex everywhere and this gives the possibility for quantum creation of universes filled with matter with initial conditions for inflation in classical regions. In the paper of Lavrelashvili et al. it was possible to observe creation only of large but empty universes.
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