Derivation-based noncommutative field theories on AF algebras

In this paper, we start the investigation of a new natural approach to "unifying" non-commutative 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.