6. Theory of Fields, I: Classical

In recent years gauge theories have emerged as primary tools for research in elementary particle physics. Experimental as well as theoretical evidence of their utility has grown tremendously in the last two decades. The isospin gauge group SU(2) of Yang–Mills theory combined with the U(1) gauge group of electromagnentic theory has lead to a unified theory of weak interactions and electromagnetism. We give an account of this unified electroweak theory in Chapter 8. In this chapter we give a mathematical formulation of several important concepts and constructions used in classical field theories. We begin with a brief account of the physical background in Section 6.2. Gauge potential and gauge field on an arbitrary pseudo-Riemannian manifold are defined in Section 6.3. Three different ways of defining the group of gauge transformations and their natural equivalence is also considered there. The geometric structure of the space of gauge potentials is discussed in Section 6.4 and is then applied to the study of Gribov ambiguity in Section 6.5. A geometric formulation of matter fields is given in Section 6.6. Gravitational field equations and their generalization is discussed in Section 6.7. Finally, Section 6.8 gives a brief indication of Perelman’s work on the geometrization conjecture and its relation to gravity.