  In recent years the idea that not only the configuration space of particles,
i.e. spacetime, but also the corresponding momentum space may have nontrivial
geometry has attracted significant attention, especially in the context of
quantum gravity. The aim of this letter is to extend this concept to the domain
of field theories, by introducing field spaces (i.e. phase spaces of field
values) that are not affine spaces. After discussing the motivation and general
aspects of our approach we present a detailed analysis of the prototype
(quantum) Nonlinear Field Space Theory of a scalar field on the Minkowski
background. We show that the nonlinear structure of a field space leads to
numerous interesting predictions, including: non-locality, generalization of
the uncertainty relations, algebra deformations, constraining of the maximal
occupation number, shifting of the vacuum energy and renormalization of the
charge and speed of propagation of field excitations. Furthermore, a compact
field space is a natural way to implement the "Principle of Finiteness" of
physical theories, which once motivated the Born-Infeld theory. Thus the
presented framework has a variety of potential applications in the theories of
fundamental interactions (e.g. quantum gravity), as well as in condensed matter
physics (e.g. continuous spin chains), and can shed new light on the issue of
divergences in quantum field theories.
