  Following the idea of a field quantization of gravity as realized in group
field theory, we construct a minisuperspace model where the wavefunction of
canonical quantum cosmology (either Wheeler-DeWitt or loop quantum cosmology)
is promoted to a field, the coordinates are minisuperspace variables, the
kinetic operator is the Hamiltonian constraint operator, and the action
features a nonlinear and possibly nonlocal interaction term. We discuss
free-field classical solutions, the quantum propagator, and a mean-field
approximation linearizing the equation of motion and augmenting the Hamiltonian
constraint by an effective term mixing gravitational and matter variables.
Depending on the choice of interaction, this can reproduce, for example, a
cosmological constant, a scalar-field potential, or a curvature contribution.
