Structure of gauge theories

Elementary interactions are formulated according to the principle of minimal interaction although paying special attention to symmetries. In fact, we aim at rewriting any field theory on the framework of Lie groups, so that, any basic and fundamental physical theory can be quantized on the grounds of a group approach to quantization. In this way, connection theory, although here presented in detail, can be replaced by “jet-gauge groups” and “jet-diffeomorphism groups.” In other words, objects like vector potentials or vierbeins can be given the character of group parameters in extended gauge groups or diffeomorphism groups. As a natural consequence of vector potentials in electroweak interactions being group variables, a typically experimental parameter like the Weinberg angle $$\vartheta _W$$ ϑ W is algebraically fixed. But more general remarkable examples of success of the present framework could be the possibility of properly quantizing massive Yang–Mills theories, on the basis of a generalized Non-Abelian Stueckelberg formalism where gauge symmetry is preserved, in contrast to the canonical quantization approach, which only provides either renormalizability or unitarity, but not both. It proves also remarkable the actual fixing of the Einstein Lagrangian in the vacuum by generalized symmetry requirements, in contrast to the standard gauge (diffeomorphism) symmetry, which only fixes the arguments of the possible Lagrangians.

Trace and Axial Anomalies on Equal Footing

We discussed that for some particular nonsupersymmetric theories, a generalized symmetry that includes both the scale and axial transformations and leads to a single current may contain also a pseudoscalar term. The method, inspired by the superconformal anomalies, has important application for low-energy effective models where it allows the introduction of a single complex glueball field with a scalar and a pseudoscalar component on the same footing with the complex meson nonet fields made of quarks. Both axial and trace anomalies are satisfied in accordance to the meson structure and the QCD requirements.

Generalized Abel-Grassmann’s Neutrosophic Extended Triplet Loop

A group is an algebraic system that characterizes symmetry. As a generalization of the concept of a group, semigroups and various non-associative groupoids can be considered as algebraic abstractions of generalized symmetry. In this paper, the notion of generalized Abel-Grassmann’s neutrosophic extended triplet loop (GAG-NET-Loop) is proposed and some properties are discussed. In particular, the following conclusions are strictly proved: (1) an algebraic system is an AG-NET-Loop if and only if it is a strong inverse AG-groupoid; (2) an algebraic system is a GAG-NET-Loop if and only if it is a quasi strong inverse AG-groupoid; (3) an algebraic system is a weak commutative GAG-NET-Loop if and only if it is a quasi Clifford AG-groupoid; and (4) a finite interlaced AG-(l,l)-Loop is a strong AG-(l,l)-Loop.