{"title":"Two types of series expansions valid at strong coupling","authors":"Ariel Edery","doi":"10.1007/jhep09(2024)063","DOIUrl":null,"url":null,"abstract":"<p>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 <i>λ</i>. The first series is the usual asymptotic one, where the quartic interaction is expanded in powers of <i>λ</i>. 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 <i>λ</i> 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 <i>N</i> equal segments). As before, the second series is absolutely convergent and we obtain analytical expressions in inverse powers of <i>λ</i> for the <i>n</i>th order terms by taking functional derivatives of generalized hypergeometric functions. The expressions are functions of <i>N</i> 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 <i>n</i>. We present numerical results at strong coupling for different values of <i>N</i> starting at <i>N</i> = 2. The series matches the exact numerical value for a given <i>N</i> (up to a certain accuracy). The continuum is formally reached when <i>N</i> → ∞ but in practice this can be reached at small <i>N</i>.</p>","PeriodicalId":635,"journal":{"name":"Journal of High Energy Physics","volume":null,"pages":null},"PeriodicalIF":5.4000,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of High Energy Physics","FirstCategoryId":"101","ListUrlMain":"https://doi.org/10.1007/jhep09(2024)063","RegionNum":1,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"Physics and Astronomy","Score":null,"Total":0}
引用次数: 0
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.
众所周知,量子力学(QM)和量子场论(QFT)中耦合幂的微扰展开是渐近级数。这在弱耦合时可能有用,但在强耦合时就失效了。在这项工作中,我们提出了两种在强耦合时有效的级数展开。第一个数列是通常的渐近数列,其中四元相互作用以λ的幂级数展开。这产生了一个在强耦合条件下有效的以λ的反幂为单位的绝对收敛级数。对于基本积分,我们重温了第一个数列,并找出了在原始积分是有限的情况下使其发散的原因。我们解决了这个问题,得到了一个在强耦合时绝对收敛且有效的耦合度幂级数。我们解释了这个数列如何避免戴森关于收敛性的论证。然后,我们考虑 QM 路径积分(离散化,时间间隔分为 N 个等分段)。与之前一样,第二序列是绝对收敛的,我们通过求广义超几何函数的函数导数,得到了 n 阶项的λ反幂解析表达式。这些表达式是 N 的函数,我们明确地将它们计算到三阶。我们给出了从 N = 2 开始的不同 N 值的强耦合数值结果。该序列与给定 N 的精确数值相匹配(达到一定精度)。当 N → ∞ 时,形式上达到了连续性,但在实践中,小 N 也能达到连续性。
期刊介绍:
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