  Phase spaces with nontrivial geometry appear in different approaches to
quantum gravity and can also play a role in e.g. condensed matter physics.
However, so far such phase spaces have only been considered for particles or
strings. We propose an extension of the usual field theories to the framework
of fields with nonlinear phase space of field values, which generally means
nontrivial topology or geometry. In order to examine this idea we construct a
prototype scalar field with the spherical phase space and then study its
quantized version with the help of perturbative methods. As the result we
obtain a variety of predictions that are known from the quantum gravity
research, including algebra deformations, generalization of the uncertainty
relation and shifting of the vacuum energy.
