Problem of the boundary condition in quantum cosmology: a simple example

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Abstract
The quantum cosmological minisuperspace model with a massless scalar field with a potential lambda phi 4 is investigated. The Hartle-Hawking proposal gives the possibility of picking a particular solution of the Wheeler-DeWitt equation in the space of the imaginary conformal factor a and scalar field phi . An analytic expression for this solution is obtained and an analytic continuation to the space of real a and phi is performed. The boundary conditions on the surface a=0, phi =+or- infinity are psi (a=0, phi =+or- infinity )=1. It is shown that near this boundary the psi have exponential behaviour. In the region where the interaction between gravity and the scalar field plays the main role the psi have an oscillation behaviour and this leads to the inflation of the Universe.

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