Abstract:
We study quantized solutions to the Wheeler de Witt (WdW) equation describing a closed Friedmann-Robertson-Walker universe with a Λ term and a set of massless scalar fields. We show that when Λ ≪1 in the natural units and the standard in-vacuum state is considered, either wave function of the universe, Ψ , or its derivative with respect to the scale factor, a , behave as random quasiclassical fields at sufficiently large values of a. The former case is realized when 1 ≪a ≪e2/3 Λ , while the latter is valid when a ≫e2/3 Λ . The statistical rms value of the wave function is proportional to the Hartle-Hawking wave function. Alternatively, the behavior of our system at large values of a can be described in terms of a density matrix corresponding to a mixed state, which is directly determined by statistical properties of Ψ. We suppose that a similar behavior of Ψ can be found in all models exhibiting copious production of excitations with respect to the out-vacuum state associated with classical trajectories at large values of a. Thus, the third quantization procedure may provide a "boundary condition" for classical solutions to the WdW equation. Contrary to the previous proposals, in our case either Ψ can be regarded as a stochastic classical quantity or the system can be viewed as being in a mixed state defined over classical solutions to the WdW equation.