The proper vertex amplitude is derived from the EPRL vertex by restricting to a single gravitational sector in order to achieve the correct semi-classical behaviour. We apply the proper vertex to calculate a cosmological transition amplitude that can be viewed as the Hartle-Hawking wavefunction. To perform this calculation we deduce the integral form of the proper vertex and use extended stationary phase methods to estimate the large-volume limit. We show that the resulting amplitude satisfies an operator constraint whose classical analogue is the Hamiltonian constraint of the Friedmann-Robertson-Walker cosmology.We find that the constraint dynamically selects the relevant family of coherent states and demonstrate a similar dynamic selection in standard quantum mechanics. We investigate the effects of dynamical selection on long-range correlations.
I. INTRODUCTIONSpinfoam models provide a path integral description of the dynamics of loop quantum gravity (LQG), a proposed theory of quantum gravity. The most widely studied model is the Engle-Pereira-Rovelli-Livine (EPRL) vertex amplitude [1-3]. However, it has been pointed out that this model fails to select a single gravitational sector [4] which may lead to unphysical contributions in the semi-classical limit from configuration histories that do not satisfy the classical equations of motion. A proposed modification of the vertex amplitude that resolves this issue by introducing a quantum mechanical restriction to a single gravitational sector has been developed under the name of the 'proper' vertex amplitude [4][5][6][7].One of the most important tasks before any theory of quantum gravity is to provide a description of the universe near the Big Bang singularity, in the regime where classical equations of general relativity break down. Within the LQG framework loop quantum cosmology (LQC) has seen the most development. In this approach one starts with a symmetry-reduced model on the classical level and then implements loop quantisation techniques to obtain a theory of (symmetry-reduced) quantum geometry. Another approach, that we take in this work, is to start with the full theory and apply it to a cosmological model. Given the spinfoam dynamics, quantum transition amplitudes can be calculated, giving rise to spinfoam cosmology. The definition and interpretation of transition amplitudes in a background-independent theory of quantum gravity is subtle: see, for example, a recent work on black hole dynamics [8]. Bianchi, Rovelli and Vidotto [9] studied transition *