Unified geometrical framework for gauge symmetries

Abstract

The effort to discover a conceptual unity among various phenomena in the physical world is one of the major intellectual endeavours of the human mind. It is surprising that Nature apparently supports such an instinct, for, within a rather short period, science has been able to organise the widely diverse physical phenomena into four basic types—depending upon the type of interaction involved. One therefore says that there are four basic interactions in nature viz. in order of decreasing strength, strong, electromagnetic, weak and gravitational interactions. After this, the physics of fundamental interactions has developed in two major directions. In one, scientists tried to realise all these four interactions as issuing from one fundamental interaction; the unification of electromagnetic and weak interactions in Salam-Weinberg model is a milestone of success in this direction. The development in the other direction issued from the effort to find a general mathematical framework in which various models of fundamental interactions could be treated in a precise and generic manner. It is the second direction of development about which we shall discuss here. Of course, the two directions of development are not completely unrelated; development in one contributes to the development in the other. Another conceptual unity in the physics of fundamental interactions is that all the four fundamental interactions could be given a gauge theoretic description. The basis for gauge theories is symmetry principles. Thus a particular fundamental interaction could be described as the dynamics of a particular symmetry of a dynamical system. The symmetries considered can be either internal symmetries or spacetime symmetries—the case of spacetime symmetries corresponds to gravitation. Thus it was realised that all the fundamental interactions could be successfully described following the same basic principles—what varied from interaction to interaction is the particular symmetry considered or the physical fields involved or the model of the dynamical system (such as a Lagrangian). The gauge principle therefore provided a conceptual unity to the various models of fundamental interactions. It therefore became important to search for a unified mathematical framework which will provide a precise and general description of ideas of gauge fields. Once the basic physical principles and the primitive set of mathematical objects are recognised, it is important to investigate the mathematical formalism in detail and in as much generality as possible to see various logical implications and generalisations. This is the principle of this thesis. The first chapter gives a brief introduction to the gauge principle and its role in fundamental interaction physics. The role of geometric methods in formulation of gauge theories is pointed. An elegant and precise formulation of gauge principle was achieved through the mathematical structure called fiber bundles. An intuitive description of a fiber bundle, and its role in a mathematical description of gauge theories are discussed. This chapter also points out the scope and limitations of this thesis. The second chapter is a brief historical introduction of gauge theories in fundamental interaction physics. This chapter is divided into four sections: the first section contains the early attempts at formulating the gauge principle, its rise through the gauge formulation of the electromagnetic field and the work of Yang and Mills. The second section gives a brief review of the application of gauge theories in elementary particle physics—its achievements and promises; the third section is a brief review of gauge theories of spacetime symmetries; the fourth section is a brief discussion of the growth and development of geometrical ideas in the theory of gauge fields. No attempt at completeness of descriptions of various attempts is being made. The purpose of this chapter is to show the continuity and similarity of thoughts in various gauge theoretic models. The third chapter discusses some of the mathematical notions and results that are useful for later discussion. This chapter also establishes our notations and conventions. In the fourth we discuss a gauge theory of the conformal group. The method used is a generalisation of the one used by Kibble for gauging the Poincaré group—the group of isometries of Minkowski space. The fifth chapter describes a gauge theory of a group of diffeomorphisms. Any (N+M)-parameter Lie group G with an N-parameter subgroup H can be realised as a global group of diffeomorphisms on an M-dimensional base space B, with representations in terms of transformation laws of fields on B belonging to linear representations of H. The gauged generalisation of the global diffeomorphisms consists of general diffeomorphisms (or coordinate transformations) on a base space together with a local action of H on the fields. An example of such a formalism is one in which B is identified with spacetime. The particular applications of the scheme to spacetime symmetries is discussed in terms of Lagrangians, field equations, currents and source identities. Also, this formalism requires introduction of a more general covariant derivative for the fields. The sixth chapter is a fiber bundle approach to a gauge theory of group of diffeomorphisms. The group G can involve spacetime symmetries as well as internal symmetries. The ungauged group G is regarded as the group of left translations on a fiber bundle G(G/H, H) where H is a closed subgroup of G and G/H is spacetime. The Yang-Mills potentials are the pull back of the Maurer-Cartan form and the Yang-Mills fields are zero. More general diffeomorphisms on the bundle space are then identified as the appropriate gauged generalisations of the left translations, and the Yang-Mills potential is identified as the pull-back of the dual of a certain kind of vielbein on the group manifold. The notion of a generalised covariant derivative appears again, and the Yang-Mills fields include a torsion on spacetime.