Why there is something rather than nothing

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We show that the recently suggested Euclidean quantum gravity density matrix of the Universe which generates the set of inflationary universes is, in fact, the density matrix of the microcanonical ensemble in {\em Lorentzian} quantum cosmology. This ensemble corresponds to the uniform (weight one) distribution over an entire phase space of {\em true physical variables}, but in terms of the observable spacetime geometry it is peaked about complex saddle-points of the Lorentzian path integral, given by the recently obtained Euclidean gravitational instantons belonging to the bounded range of the cosmological constant. It is shown that the conformal rotation, designed to solve the problem of unboundedness of the Euclidean gravitational action, is, in fact, the manifestation of correct integration range over the lapse and shift functions in Lorentzian quantum gravity, enforcing the Hamiltonian and momentum constraints. Inflationary cosmologies generated by the gravitational instantons at late stages of expansion incorporate cosmological acceleration phenomenon whose low-energy scale can be attained by invoking the ideas of dynamically evolving compactified extra dimensions. Thus, together with the bounded range of the early cosmological constant, this initial state of the Universe simultaneously suggests the dynamical restriction mechanism for the landscape of string vacua and the solution of the dark energy problem.