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arXiv · 1711.08829

A Note on Efimov Nonlocal and Nonpolynomial Quantum Scalar Field Theory

V. A. Guskov·M. G. Ivanov·S. L. Ogarkov
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Abstract

In frames of the nonlocal and nonpolynomial quantum theory of the one component scalar field in $D$-dimensional spacetime, stated by Gariy Vladimirovich Efimov, the expansion of the $\mathcal{S}$-matrix is revisited for different interaction Lagrangians and for some kinds of Gaussian propagators modified by different ultraviolet form factors $F$ which depend on some length parameter $l$. The expansion of the $\mathcal{S}$-matrix is of the form of a grand canonical partition function of some $D+N$-dimensional ($N\geq 1$) classical gas with interaction. The toy model of the realistic quantum field theory (QFT) is considered where the $\mathcal{S}$-matrix is calculated in closed form. Then, the functional Schwinger-Dyson and Schr\"{o}dinger equations for the $\mathcal{S}$-matrix in Efimov representation are derived. These equations play a central role in the present paper. The functional Schwinger-Dyson and Schr\"{o}dinger equations in Efimov representation do not involve explicit functional derivatives but involve a shift of the field which is the $\mathcal{S}$-matrix argument. The asymptotic solutions of the Schwinger-Dyson equation are obtained in different limits. Also, the solution is found in one heuristic case allowing us to study qualitatively the behavior of the $\mathcal{S}$-matrix for an arbitrary finite value of its argument. Self-consistency equations, which arise during the process of derivation, are of a great interest. Finally, in the light of the discussion of QFT functional equations, ultraviolet form factors and extra dimensions, the connection with functional (in terms of the Wilson-Polchinski and Wetterich-Morris functional equations) and holographic renormalization groups (in terms of the functional Hamilton-Jacobi equation) is made. In addition the Hamilton-Jacobi equation is formulated in an unconventional way.

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V. A. Guskov, M. G. Ivanov, S. L. Ogarkov. 2017-11-23. A Note on Efimov Nonlocal and Nonpolynomial Quantum Scalar Field Theory. https://doi.org/10.1134/s1063779621030059

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