Gauge field theory provides a geometrically rich theoretico-physical framework in which symmetries dictate interactions. We begin the present work by examining the fundamental properties of principal bundles and introduce on them the general notion of a connection. We show that the latter naturally gives rise to the concept of curvature; furthermore, given a representation of the structure Lie group, we conceive the notion of a covariant derivative on the associated vector bundle. We precisely define gauge fields, express the influence of gauge transformations on them and show how curvature on a principal bundle may be identified with a differential form with values in the adjoint bundle.
Together with the notions of the Hodge-* operator, the exterior covariant derivative and the variational principle, the curvature of a connection is the basis for the Yang--Mills theory, which we physically interpret as the theory of self-interaction of the gauge boson fields (i.e. the carriers of fundamental forces). In this context, we prove gauge invariance of the Yang--Mills Lagrangian and derive the Yang--Mills equation, which we interpret (together with the Bianchi identity) as a generalization of the Maxwell's equations. By including the Klein--Gordon Lagrangian into this theory, we derive the Yang--Mills--Higgs equations, which describe the interaction of gauge boson fields and scalar fields. A consequence of these equations is the celebrated Brout--Englert--Higgs mechanism; we provide a short sketch thereof.
Through a physically motivated study of the Lorentz group and its universal cover, we naturally introduce the notion of a spin structure on a Lorentz manifold. The latter enables us to induce from the affine Levi--Civita connection the covariant derivative on the associated spinor bundle, which further enables us to precisely define the Dirac operator on a Lorentz manifold. By acquiring the basic properties of the Clifford multiplication, we prove self-adjointness of the Dirac operator with respect to the hermitian Dirac form. By means of the variational principle, we arrive to the Dirac equation on a spin Lorentz manifold. We conclude our investigation by splicing the principal spin bundle with the principal gauge bundle and sketch how this tool is used to describe interactions of fermions with gauge fields.
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