Abstract
Two-dimensional quantum cosmology is studied to clarify the issue associated with the arrow of time. We introduce a density-matrix description for the minisuperspace by projecting out nonzero modes of the wave function of the whole Universe. Thermodynamic time is determined by the direction in which the quantum coherence of minisuperspace variables decreases. This arrow of time is compared with that of dynamical time which is determined by the semiclassical approximation for the density matrix. We reconfirm that the global time cannot properly be defined for the thermodynamic times as well as for the dynamical time in a closed universe. The arrows of dynamical and thermodynamic times coincide with the cosmological arrow of time.
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