In this paper, we start the investigation of a new natural approach to “unifying” noncommutative gauge field theories (NCGFT) based on approximately finite-dimensional (AF) C∗-algebras. The defining inductive sequence of an AF C∗-algebra is lifted to enable the construction of a sequence of NCGFT of Yang–Mills–Higgs types. This paper focuses on derivation-based noncommutative field theories. A mathematical study of the ingredients involved in the construction of a NCGFT is given in the framework of AF C∗-algebras: derivation-based differential calculus, modules, connections, metrics and Hodge *-operators, and Lagrangians. Some physical applications concerning mass spectra generated by Spontaneous Symmetry Breaking Mechanisms (SSBM) are proposed using numerical computations for specific situations.

中文翻译：

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AF代数上基于导数的非交换场论

在本文中，我们开始研究一种基于近似有限维（一种F)C*-代数。定义的归纳序列一种F C*-代数被提升以能够构建杨-米尔斯-希格斯类型的 NCGFT 序列。本文重点研究基于导数的非交换场论。在以下框架中对构建 NCGFT 所涉及的要素进行了数学研究一种F C*-代数：基于推导的微积分、模块、连接、度量和 Hodge *-operators 和 Lagrangians。一些关于由自发对称破坏机制 (SSBM) 生成的质谱的物理应用是针对特定情况使用数值计算提出的。