Abstract
In the nonadiabatic path integral approach to a quantum minisuperspace cosmological model for the gravity minimally coupled to a matter field, the matter field gives rise to both a back reaction of the matter field Hamiltonian and a mode-dependent induced gauge potential (Berry's connection) to the separated gravitational field path integral. It is proposed that the mode-dependent gauge potential and the corresponding geometric phase for a closed loop of classical trajectory of the expanding and subsequently recontracting universe in the projected minisuperspace may break the symmetry of the cosmological time defined along the classical trajectory of the wave function of the universe.
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