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We develop a quantum field theoretical framework to analytically study thethree-body constrained Bose-Hubbard model beyond mean field and non-interactingspin wave approximations. It is based on an exact mapping of the constrainedmodel to a theory with two coupled bosonic degrees of freedom with polynomialinteractions, which have a natural interpretation as single particles andtwo-particle states. The procedure can be seen as a proper quantization of theGutzwiller mean field theory. The theory is conveniently evaluated in theframework of the quantum effective action, for which the usual symmetryprinciples are now supplemented with a ``constraint principle'' operative onshort distances. We test the theory via investigation of scattering propertiesof few particles in the limit of vanishing density, and we address thecomplementary problem in the limit of maximum filling, where the low lyingexcitations are holes and di-holes on top of the constraint induced insulator.This is the first of a sequence of two papers. The application of the formalismto the many-body problem, which can be realized with atoms in optical latticeswith strong three-body loss, is performed in a related work [14].

Quantum Field Theory for the Three-Body Constrained Lattice Bose Gas -- Part I: Formal Developments