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We review the canonical quantisation of the geometry of the spacetime in thecases of a simply and a non-simply connected manifold. In the former, weanalyse the information contained in the solutions of the Wheeler-DeWittequation and interpret them in terms of the customary boundary conditions thatare typically imposed on the semiclassical wave functions. In particular, wereview three different paradigms for the quantum creation of a homogeneous andisotropic universe. For the quantisation of a non-simply connected manifold thebest framework is the so-called third quantisation formalism, in which the wavefunction of the universe is seen as a field that propagates in the space ofRiemannian $3$-geometries, which turns out to be isomorphic to a (part of a)$1+5$ Minkowski spacetime. Thus, the quantisation of the wave function followsthe customary formalism of a quantum field theory. A general review of theformalism is given and it is analysed the creation of the universes, theirinitial expansion and the appearance of matter after inflation. These featuresare presented in more detail in the case of a homogeneous and isotropicuniverse. The main conclusion in both cases is that the most natural way inwhich the universes should be created is in entangled universe-antiuniversepairs.

Quantum cosmology with third quantisation