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Particles 2019, 2(1), 103-139; https://doi.org/10.3390/particles2010009

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

1
Moscow Institute of Physics and Technology (MIPT), Institutskiy Pereulok 9, 141701 Dolgoprudny, Russia
2
Dukhov Research Institute of Automatics (VNIIA), Sushchevskaya 22, 127055 Moscow, Russia
*
Authors to whom correspondence should be addressed.
Received: 18 December 2018 / Revised: 1 February 2019 / Accepted: 12 February 2019 / Published: 19 February 2019
(This article belongs to the Special Issue QCD and Hadron Structure)
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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. View Full-Text
Keywords: quantum field theory (QFT); scalar QFT; nonlocal QFT; nonpolynomial QFT; Euclidean QFT; ?-matrix; form factor; generating functional; abstract functional integral; Gaussian measure; grand canonical partition function; basis functions representation; renormalization group; compactification process quantum field theory (QFT); scalar QFT; nonlocal QFT; nonpolynomial QFT; Euclidean QFT; ?-matrix; form factor; generating functional; abstract functional integral; Gaussian measure; grand canonical partition function; basis functions representation; renormalization group; compactification process
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Chebotarev, I.V.; Guskov, V.A.; Ogarkov, S.L.; Bernard, M. S-Matrix of Nonlocal Scalar Quantum Field Theory in Basis Functions Representation. Particles 2019, 2, 103-139.

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