TITLE:
An Introduction to Gauge Field Theories: From Electrons to Vector Dark Matter
AUTHORS:
Henry Atwater, Yuri Rostovtsev
KEYWORDS:
Gauge Theory, Standard Model, Lie Groups, Lie Algebras, Dark Matter, Quaternion
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.13 No.7,
July
31,
2025
ABSTRACT: The various quantum field theories now recognized collectively as the standard model (SM) are the most precise and predictive models of the universe ever proposed. There is much to explore in each theory and each of the fundamental particles and interactions possess their own unique qualities and conundrums. However, if we zoom out, we see the mathematics posited by nature in each of these theories are one and the same in structure. This most enchanting structure is colloquially known amongst physicists as Yang-Mills theory or, more generally, gauge field theory. This article is meant to serve as an introduction to the world of gauge theories for the physics student interested in particle physics and adjacent fields. First is a brief discussion of groups followed by an optional discussion of smooth manifolds. A basic introduction to Lie groups and algebras is provided intuitively, without relying on the context of representation theory. Several globally invariant systems are then presented, followed by a discussion on the generalization to local invariance. A formal yet approachable predication of a general gauge theory is then given, along with a variety of examples. The second to last section introduces an extension of the previously discussed version of gauge theory to theories currently being investigated as models for vector dark matter. In the final section we present and briefly discuss a theory invariant under action by the quaternionic sphere. It is assumed the reader has taken advanced undergraduate courses in classical mechanics, electromagnetic theory, and quantum mechanics. Therefore, the reader should have a firm grasp on concepts from linear algebra/vector space theory (including abstract vector spaces), calculus (including that of functions of a complex variable), and has a passive understanding of basic notions from differential geometry—such as the algebra of smooth functions, tangent spaces and their dual/cotangent spaces, metric tensors, etc. Here, a “passive understanding” is meant as little more than the equivalent to having an operational knowledge of tensor indices. Even though there is a section dedicated to defining a smooth manifold, a background in topology is not required. Beyond said section, concepts from topology are mentioned sparsely. Formal knowledge of group representations is not necessary; however, the reader should acknowledge that many of the groups relevant to this paper have representations in each of an infinite number of dimensions—a fact that may be confronted if the literature on this general topic is pursued further.