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 φ 4 (with suggestions for φ 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–Quark 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.
Highlights
The timeline of Quantum Field Theory (QFT) offers events that are quite asymmetric to each other
By the remarkable event confirming validity of Standard Model (SM), a quantum-trivial local scalar field quantum theory is, after all, not quantum-trivial if the SM is a sector of a non-Abelian gauge theory; this is analogous to a Quantum Electrodynamics (QED) event
We obtain an expression for the S-matrix in terms of the grand canonical partition function, which in turn is to be expressed in terms of the abstract functional integral over the primary fields of the theory
Summary
The timeline of Quantum Field Theory (QFT) offers events that are quite asymmetric to each other. In what is considered pinnacle [71,72] of set of papers devoted to the nonlocal QFT, G.V. Efimov introduced and investigated the notion of representations of S-matrix of QFT on a discrete lattice of basis functions for both the nonpolynomial [71] and the polynomial [72] theory cases. We propose considering the theory in internal space with extra dimensions and subsequent compactification into physical four-dimensional spacetime This changes the meaning of nonlocal interaction ultraviolet length parameter l, the needle of pivot for ratio of ultraviolet and infrared parameters in the theory, it changes analytical properties of the Green functions, scattering amplitudes, and form factors, in terms of physical variables obtained following compactification from internal space with extra dimensions; yet, method compactification is determined by process [10,11,79]. The construction of analytical continuations for the results of this paper is the subject of a separate publication
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