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A wide class of quantum field theories (QFTs) describing interacting neutral scalar bosons are constructed. It is shown that a redefinition of the vacuum expectation values (VEV) of quantum fields as generalized functions leads to constructions of interacting fields that are operators in a Hilbert space and satisfy the physical requirements for a field theory. The field operators are constructed directly, without perturbation expansion, and constructed in spacetime of any number of dimensions. The corresponding scattering theory is developed. The limited function class underlying the QFT imposes a direction to time.

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