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Perturbative versus Non-Perturbative Quantum Field Theory: Tao’s Method, the Casimir Effect, and Interacting Wightman Theories
Universe ( IF 2.5 ) Pub Date : 2021-07-05 , DOI: 10.3390/universe7070229
Walter Felipe Wreszinski

We dwell upon certain points concerning the meaning of quantum field theory: the problems with the perturbative approach, and the question raised by ’t Hooft of the existence of the theory in a well-defined (rigorous) mathematical sense, as well as some of the few existent mathematically precise results on fully quantized field theories. Emphasis is brought on how the mathematical contributions help to elucidate or illuminate certain conceptual aspects of the theory when applied to real physical phenomena, in particular, the singular nature of quantum fields. In a first part, we present a comprehensive review of divergent versus asymptotic series, with qed as background example, as well as a method due to Terence Tao which conveys mathematical sense to divergent series. In a second part, we apply Tao’s method to the Casimir effect in its simplest form, consisting of perfectly conducting parallel plates, arguing that the usual theory, which makes use of the Euler-MacLaurin formula, still contains a residual infinity, which is eliminated in our approach. In the third part, we revisit the general theory of nonperturbative quantum fields, in the form of newly proposed (with Christian Jaekel) Wightman axioms for interacting field theories, with applications to “dressed” electrons in a theory with massless particles (such as qed), as well as unstable particles. Various problems (mostly open) are finally discussed in connection with concrete models.

中文翻译:

微扰与非微扰量子场论:道的方法、卡西米尔效应和相互作用的怀特曼理论

我们详细讨论了有关量子场论含义的某些要点:微扰方法的问题,以及 't Hooft 提出的关于该理论在明确(严格)数学意义上是否存在的问题,以及一些关于完全量化的场论的少数存在的数学精确结果。强调数学贡献如何在应用于实际物理现象时帮助阐明或阐明理论的某些概念方面,特别是量子场的奇异性质。在第一部分中,我们以 qed 为背景示例,对发散级数与渐近级数进行全面回顾,以及 Terence Tao 的一种方法,该方法将数学意义传达给发散级数。在第二部分,我们将陶的方法以最简单的形式应用于卡西米尔效应,由完美导电的平行板组成,认为使用欧拉-麦克劳林公式的通常理论仍然包含残余无穷大,我们的方法将其消除。在第三部分中,我们以新提出的(与 Christian Jaekel 一起提出的)Wightman 公理的形式重新审视了非微扰量子场的一般理论,用于相互作用场理论,并在具有无质量粒子的理论(例如 qed ),以及不稳定的粒子。最后结合具体模型讨论了各种问题(大部分是开放的)。这在我们的方法中被消除了。在第三部分中,我们以新提出的(与 Christian Jaekel 一起提出的)Wightman 公理的形式重新审视了非微扰量子场的一般理论,用于相互作用场理论,并在具有无质量粒子的理论(例如 qed ),以及不稳定的粒子。最后结合具体模型讨论了各种问题(大部分是开放的)。这在我们的方法中被消除了。在第三部分中,我们以新提出的(与 Christian Jaekel 一起提出的)Wightman 公理的形式重新审视了非微扰量子场的一般理论,用于相互作用场理论,并在具有无质量粒子的理论(例如 qed ),以及不稳定的粒子。最后结合具体模型讨论了各种问题(大部分是开放的)。
更新日期:2021-07-05
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