Nonlocal quantum theory of 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 evolution of 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 primary scalar field, the S form of a grand canonical partition function is found. And, by expression of S-matrix in terms of the partition function, the representation for S in terms of basis functions is obtained. Derivations are given for discrete case where basis functions are Hermite functions, and for 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. And, by 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 the polynomial theory ϕ 4 (with suggestions for ϕ 6 ) and the 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 illustrated numerically, 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 the free theory G is in the form of Gaussian function typically in the Virton-Quark model, although the obtained analytical inferences are not limited to these particular choices in principle. The formulation for nonlocal QFT in momentum k space of extra dimensions with subsequent compactification into physical spacetime is discussed, alongside the compactification process. Recently, the discovery of Higgs boson, the last SM element in energy domain, where existence is most natural, occurred: Notably, by the remarkable event confirming validity of SM, a quantum-trivial local quantum theory of scalar field is, after all, not quantum trivial if the SM is a sector of a non-Abelian gauge theory; analogous to QED event.Groundbreaking of Supersymmetric Non-Abelian QFTs as well as Integrable QFTs [1-9] in earnest search for quantum theory of gravity will naturally complement superstring theory. Under such sophistication for superstring theory, which is almost surely strongest pick for fundamental theory of nature, the ultimate truth, it will not be impossible to view all QFTs as effective (low-energy) theory given by renormalizable and nonrenormalizable QFTs, respectively. In other words, every field theory will be a limit in superstring theory. Hypothetically, bosonic string...

Nonlocal quantum field theory (QFT) of one-component scalar field ϕ in D-dimensional Euclidean spacetime is considered. The generating functional (GF) of complete Green functions Z as a functional of external source j, coupling constant g, and spatial measure dµ is studied. An expression for GF Z in terms of the abstract integral over the primary field ϕ is given. An expression for GF Z in terms of integrals over the primary field and separable Hilbert space (HS) is obtained by means of a separable expansion of the free theory inverse propagatorL over the separable HS basis. The classification of functional integration measures D [ϕ] is formulated, according to which trivial and two nontrivial versions of GF Z are obtained. Nontrivial versions of GF Z are expressed in terms of 1-norm and 0-norm, respectively. In the 1-norm case in terms of the original symbol for the product integral, the definition for the functional integration measure D [ϕ] over the primary field is suggested. In the 0-norm case, the definition and the meaning of 0-norm are given in terms of the replica-functional Taylor series. The definition of the 0-norm generator Ψ is suggested. Simple cases of sharp and smooth generators are considered. An alternative derivation of GF Z in terms of 0-norm is also given. All these definitions allow to calculate corresponding functional integrals over ϕ in quadratures. Expressions for GF Z in terms of integrals over the separable HS, aka the basis functions representation, with new integrands are obtained. For polynomial theories ϕ 2n , n = 2, 3, 4, . . . , and for the nonpolynomial theory sinh 4 ϕ, integrals over the separable HS in terms of a power series over the inverse coupling constant 1/ √ g for both norms (1-norm and 0-norm) are calculated. Thus, the strong coupling expansion in all theories considered is given. "Phase transitions" and critical values of model parameters are found numerically. A generalization of the theory to the case of the uncountable integral over HS is formulated: GF Z for an arbitrary QFT and the strong coupling expansion for the theory ϕ 4 are derived. Finally a comparison of two GFs Z, one on the continuous lattice of functions and one obtained using the Parseval-Plancherel identity, is given.

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