The concept of the renormalization group (RG) emerged from the renormalization of quantum field variables, which is typically used to deal with the issue of divergences to infinity in quantum field theory. Meanwhile, in the study of phase transitions and critical phenomena, it was found that the self–similarity of systems near critical points can be described using RG methods. Furthermore, since self–similarity is often a defining feature of a complex system, the RG method is also devoted to characterizing complexity. In addition, the RG approach has also proven to be a useful tool to analyze the asymptotic behavior of solutions in the singular perturbation theory. In this review paper, we discuss the origin, development, and application of the RG method in a variety of fields from the physical, social and life sciences, in singular perturbation theory, and reveal the need to connect the RG and the fractional calculus (FC). The FC is another basic mathematical approach for describing complexity. RG and FC entail a potentially new world view, which we present as a way of thinking that differs from the classical Newtonian view. In this new framework, we discuss the essential properties of complex systems from different points of view, as well as, presenting recommendations for future research based on this new way of thinking.

中文翻译：

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复杂世界中的重归一化组和分数演算方法：综述

重整化组（RG）的概念来自量子场变量的重整化，通常用于处理量子场论中发散到无穷大的问题。同时，在研究相变和临界现象时，发现可以使用RG方法描述临界点附近系统的自相似性。此外，由于自相似性通常是复杂系统的定义特征，因此RG方法也致力于表征复杂性。此外，RG方法还被证明是分析奇异摄动理论中解的渐近行为的有用工具。在这篇评论文章中，我们讨论了RG方法在物理，社会和生命科学等各个领域的起源，发展和应用，在奇异摄动理论中，揭示了将RG和分数微积分（FC）连接起来的必要性。FC是描述复杂性的另一种基本数学方法。RG和FC带来了潜在的新世界观，我们将其作为一种不同于经典牛顿观的思维方式提出。在这个新的框架中，我们从不同的角度讨论了复杂系统的本质，并为基于这种新思维方式的未来研究提供了建议。