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A Note on Efimov Nonlocal and Nonpolynomial Quantum Scalar Field Theory

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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 G.V. 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 (\(N \geqslant 1\)) 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 Schrödinger 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 Schrödinger 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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ACKNOWLEDGMENTS

The authors are deeply grateful to their families for love, wisdom and understanding. We are very grateful to Artem A. Alexandrov for his help in typing the paper. Also, we express special gratitude to Sergey E. Kuratov and Alexander V. Andriyash for supporting this research at an early stage at the Center for Fundamental and Applied Research (Dukhov Research Institute of Automatics). Finally, we are very grateful to Reviewer for many valuable comments and advice on this paper.

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  1. Moscow Institute of Physics and Technology (MIPT), 141701, Dolgoprudny, Moscow oblast, Russia

    V. A. Guskov, M. G. Ivanov & S. L. Ogarkov

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  1. V. A. Guskov
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  2. M. G. Ivanov
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  3. S. L. Ogarkov
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Correspondence to M. G. Ivanov or S. L. Ogarkov.

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In Memory of Gariy Vladimirovich Efimov

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Guskov, V.A., Ivanov, M.G. & Ogarkov, S.L. A Note on Efimov Nonlocal and Nonpolynomial Quantum Scalar Field Theory. Phys. Part. Nuclei 52, 420–437 (2021). https://doi.org/10.1134/S1063779621030059

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