Compact phase spaces arise as the semiclassical description of quantum systems that are characterized by the finite dimensional Hilbert spaces. A crucial feature of such phase spaces is a restriction imposed on the allowed values of canonical variables (i.e. configuration variables and the corresponding conjugate momenta). Consequently, various physical quantities, e.g. the energy density, may be bounded from above, providing us with a solution to the problem ultraviolet divergences occurring in theories described by the standard (affine) phase spaces.
In the paper Nonlinear Field Space Theory we initiated the research programme whose aim is to generalize field theories equipped with standard phase spaces to the case when these phase spaces are non-affine, in particular compact. So far, this approach has been successfully applied to the case of a scalar field (on the background of Minkowski or FRW spacetime) and the cosmological minisuperspace model. Moreover, it has led to formulation of the Spin-Field Correspondence, which establishes a new relation between (scalar) fields and systems of spins. In the future we plan to extend the above investigations to fermion and gauge fields, as well as to explore the empirical consequences of field theories with the nonlinear phase spaces.