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.

中文翻译：

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关于 Efimov 非局部非多项式量子标量场论的注记

摘要

在GV Efimov 所述的\(D\)维时空中单分量标量场的非局域和非多项式量子理论的框架中，\(\mathcal{S}\) -矩阵的扩展被重新考虑用于不同的相互作用拉格朗日函数和某些类型的高斯传播器由不同的紫外线形状因子\(F\)修改，这取决于某些长度参数\(l\)。所述的膨胀\（\ mathcal {S} \） -矩阵是一些一巨正分函数的形式的\（（d + N）\）维（\（N \ geqslant 1 \） ）古典气体与互动。考虑现实量子场论 (QFT) 的玩具模型，其中\(\mathcal{S}\) -matrix 以封闭形式计算。然后，导出了 Efimov 表示中\(\mathcal{S}\)矩阵的函数 Schwinger-Dyson 和 Schrödinger 方程。这些方程在本文中发挥着核心作用。Efimov 表示中的泛函 Schwinger-Dyson 和 Schrödinger 方程不涉及显式泛函导数，但涉及作为\(\mathcal{S}\) -matrix 参数的场的位移。Schwinger-Dyson 方程的渐近解是在不同的极限下获得的。此外，解决方案是在一个启发式案例中找到的，允许我们定性地研究\(\mathcal{S}\)-matrix 为其参数的任意有限值。在推导过程中出现的自洽方程非常有趣。最后，根据对 QFT 函数方程、紫外线形状因子和额外维度的讨论，与泛函（根据 Wilson-Polchinski 和 Wetterich-Morris 函数方程）和全息重整化群（根据泛函 Hamilton –雅可比方程）成立。此外，Hamilton-Jacobi 方程以非常规的方式表述。