Nonlocal quantum theory of a one-component scalar field in D-dimensional Euclidean spacetime is studied in representations of S -matrix theory for both polynomial and nonpolynomial interaction Lagrangians. The theory is formulated on coupling constant g in the form of an infrared smooth function of argument x for space without boundary. Nonlocality is given by the evolution of a Gaussian propagator for the local free theory with ultraviolet form factors depending on ultraviolet length parameter l. By representation of the S -matrix in terms of abstract functional integral over a primary scalar field, the S form of a grand canonical partition function is found. By expression of S -matrix in terms of the partition function, representation for S in terms of basis functions is obtained. Derivations are given for a discrete case where basis functions are Hermite functions, and for a continuous case where basis functions are trigonometric functions. The obtained expressions for the S -matrix are investigated within the framework of variational principle based on Jensen inequality. Through the latter, the majorant of S (more precisely, of ln S ) is constructed. Equations with separable kernels satisfied by variational function q are found and solved, yielding results for both polynomial theory 4 (with suggestions for 6 ) and nonpolynomial sine-Gordon theory. A new definition of the S -matrix is proposed to solve additional divergences which arise in application of Jensen inequality for the continuous case. Analytical results are obtained and numerically illustrated, with plots of variational functions q and corresponding majorants for the S -matrices of the theory. For simplicity of numerical calculation, the D = 1 case is considered, and propagator for free theory G is in the form of Gaussian function typically in the VirtonQuark model, although the obtained analytical inferences are not, in principle, limited to these particular choices. Formulation for nonlocal QFT in momentum k space of extra dimensions with subsequent compactification into physical spacetime is discussed, alongside the compactification process.
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