Localized quantum subsystem in a semiclassical universe

In semiclassical quantum gravity, the main system may serve as a classical background to the quantum subsystem only if its wavefunction is strongly peaked. However, we show that such a peaked wavefunction is not allowed in the conventional WKB expansion of the Wheeler-DeWitt equation, because it breaks the fundamental WKB inequalities. This breaking occurs in two different fashions: inside the support of the peaked function and near its border. The first difficulty requires that one imposes a lower limit on the width of the support, proportional to the inverse of the expansion parameter. Solving the second difficulty requires a new approximation. The conditions of validity of this new approximation are found to be compatible with the preceding condition. Finally, we derive the localized quantum mechanics for the subsystem within the peaked main system, and we give a toy model as an example.