S-Matrix of Nonlocal Scalar Quantum Field Theory in Basis Functions Representation

Abstract

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 Scalar Quantum Field Theory—Functional Integration, Basis Functions Representation and Strong Coupling Expansion

“…, etc. for the analysis of GF Z or S-matrix, one could use the representation for Z or S in terms of the grand canonical partition function [20,[23][24][25][26]31]. Such a partition function is classical, i.e.…”

“…Let us make a comment on the replacement δ ε (Λ − dµ) → δ ε (Λ) in the expression (31). We again rearrange the order of integration over Λ and ε limit calculation.…”

“…To demonstrate the generality of the functional Taylor series, give one toy example of metric, the idea of which is derived from the statistical physics of classical gases with the interaction [31]:…”

“…It is important that the asymptotics of the interaction Lagrangian be infinite. If we consider the case of an all-bounded interaction Lagrangian, for example, the sine-Gordon theory, the methods of statistical physics or the variational principle can be used to study the problem [20,23,25,26,[29][30][31].…”

“…We note that in this case an integrand is a more complicated expression than initial exponential function of action. GF Z in a basis functions representation is open to numerical methods for QFT on lattice and spin models with rather efficient computational use of resources, as an alternative to majorize GF Z or the scattering matrix (S-matrix) within a framework of variational principle: solving corresponding variational problem to obtain a physically satisfactory estimate for Z or S [26,31].…”