%0 Journal Article
%T S-Matrix of Nonlocal Scalar Quantum Field Theory in Basis Functions Representation
%A Ivan V. Chebotarev
%A Matthew Bernard
%A Stanislav L. Ogarkov
%A Vladislav A. Guskov
%J -
%D 2019
%R https://doi.org/10.3390/particles2010009
%X Abstract 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 ¦Õ 4 (with suggestions for ¦Õ 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 Virton¨CQuark 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. View Full-Tex
%K quantum field theory (QFT)
%K scalar QFT
%K nonlocal QFT
%K nonpolynomial QFT
%K Euclidean QFT
%K ?-matrix
%K form factor
%K generating functional
%K abstract functional integral
%K Gaussian measure
%K grand canonical partition function
%K basis functions representation
%K renormalization group
%K compactification process
%U https://www.mdpi.com/2571-712X/2/1/9