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In quantum cosmology wave functions are traditionally generated either from path integrals or from solving the Wheeler-DeWitt equation. In the first part of the talk I discuss what is required in these approaches in order to meet the usual requirements of Hilbert space quantum mechanics, namely, the specification of an inner product structure and classes of states and operators of interest. The Wheeler-DeWitt operator must be self-adjoint in this approach which has consequences for the both the path integral and Wheeler-DeWitt account of the much-studied de Sitter minisuperspace model, since it is usually formulated in terms of a scale factor which must be non-negative, hence one is really doing quantum mechanics on the half-line. In the second part of the talk I discuss the types of amplitudes one is interested in from the perspective of the decoherent histories approach to quantum cosmology, which describe whether the trajectory of a cosmological model passes through various regions of minisuperspace. They are different in form to the simplest path integral constructions in quantum cosmology and most closely resemble scattering amplitudes.

Contemporary final theories consist of a theory of the universe’s dynamics (I) like the avatars of string theory together with a theory of its quantum state (Ψ) like the no-boundary wave function of the universe (NBWF). This talk is concerned with the definition of a no-boundary quantum state at the semiclassical level where its predictions can be straightforwardly derived and compared with observation. A semiclassical no-boundary wave function is defined by an ensemble regular saddle points. The ensemble is restricted by simple considerations of symmetry such as time neutrality. We will briefly review the successful predictions of the NBWF so defined. We then address the question of moving beyond the semiclassical approximation by defining the NBWF in terms of a Euclidean or Lorentzian integral or by a connection with a dual field theory.