近日，加拿大毕晓普大学的Ariel Edery及其研究团队取得一项新进展。经过不懈努力，他们实现强耦合下有效的两种级数展开。相关研究成果已于2024年9月12日在国际知名学术期刊《高能物理杂志》上发表。

该研究团队提出了两种在强耦合下有效的级数展开方法。研究人员将这些级数应用于一个基本积分以及一个包含二次项和四次项（带有耦合常数λ）的量子力学路径积分。第一种级数是通常的渐近级数，其中四次相互作用按λ的幂次展开。第二种级数则是对二次部分进行展开，而保持相互作用项不变。这产生了一个在强耦合下有效的、以λ的负幂次表示的绝对收敛级数。对于基本积分，研究人员重新审视了第一种级数，并确定了导致其发散的原因，尽管原始积分是有限的。

研究人员解决了这个问题，并意外地得到了一个在强耦合下有效且绝对收敛的、按耦合幂次展开的级数。他们解释了该级数如何规避了戴森关于收敛性的论点。接下来，他们考虑了量子力学路径积分（时间间隔被划分为N个等段进行离散化）。与之前一样，第二种级数是绝对收敛的，他们通过对广义超几何函数求泛函导数，得到了以λ的负幂次表示的n阶项解析表达式。

这些表达式是N的函数，研究人员明确计算到了三阶。这一通用过程已通过Mathematica程序实现，该程序可以生成任意阶n的表达式。他们展示了从N=2开始的不同N值在强耦合下的数值结果。该级数与给定N下的精确数值（在一定精度内）相匹配。当N→∞时，形式上可达连续极限，但在实际操作中，较小的N即可达到此效果。

据悉，众所周知，量子力学（QM）和量子场论（QFT）中按耦合幂次进行的微扰展开是渐近级数。这种方法在弱耦合情况下很有用，但在强耦合情况下则失效。

附：英文原文

Title: Two types of series expansions valid at strong coupling

Author: Edery, Ariel

Issue&Volume: 2024-09-12

Abstract: It is known that perturbative expansions in powers of the coupling in quantum mechanics (QM) and quantum field theory (QFT) are asymptotic series. This can be useful at weak coupling but fails at strong coupling. In this work, we present two types of series expansions valid at strong coupling. We apply the series to a basic integral as well as a QM path integral containing a quadratic and quartic term with coupling constant λ. The first series is the usual asymptotic one, where the quartic interaction is expanded in powers of λ. The second series is an expansion of the quadratic part where the interaction is left alone. This yields an absolutely convergent series in inverse powers of λ valid at strong coupling. For the basic integral, we revisit the first series and identify what makes it diverge even though the original integral is finite. We fix the problem and obtain, remarkably, a series in powers of the coupling which is absolutely convergent and valid at strong coupling. We explain how this series avoids Dyson’s argument on convergence. We then consider the QM path integral (discretized with time interval divided into N equal segments). As before, the second series is absolutely convergent and we obtain analytical expressions in inverse powers of λ for the nth order terms by taking functional derivatives of generalized hypergeometric functions. The expressions are functions of N and we work them out explicitly up to third order. The general procedure has been implemented in a Mathematica program that generates the expressions at any order n. We present numerical results at strong coupling for different values of N starting at N = 2. The series matches the exact numerical value for a given N (up to a certain accuracy). The continuum is formally reached when N → ∞ but in practice this can be reached at small N.

DOI: 10.1007/JHEP09(2024)063

Source: https://link.springer.com/article/10.1007/JHEP09(2024)063