Nonlinear model of a quantum minisuperspace system with back reaction.

We consider the quantum evolution of the space-independent mode of a $\ensuremath{\lambda}{\ensuremath{\varphi}}^{4}$ theory as a minisuperspace in the space of all $\ensuremath{\varphi}$. The motion of the wave packet in the minisuperspace is compared to the motion of a wave packet in a larger minisuperspace consisting of the original minisuperspace plus one space-dependent mode. By comparing the motion of the two packets we develop criteria that tell us when the quantum evolution in the space-independent minisuperspace gives us useful information about the true evolution in the larger minisuperspace. These criteria serve as a toy model for similar (but much more complex) criteria that will tell us whether or when quantized gravitational minisuperspaces can possibly give any useful information about quantum gravity.