Gauge Theories and Fiber Bundles: Applications to Particle Dynamics

Gauge Theories and Fiber Bundles: Applications to Particle Dynamics A theory defined by an action which is invariant under a time-dependent group
of transformations can be called a gauge theory. Well known examples of such theories
are those defined by the Maxwell and Yang-Mills Lagrangians. It is widely believed
nowadays that the fundamental laws of physics have to be formulated in terms of gauge theories. The underlying mathematical structures of gauge theories are known to be geometrical in nature and the local and global features of this geometry have been studied for a long time in mathematics under the name of fibre bundles. It is now understood that
the global properties of gauge theories can have a profound influence on physics. For example, instantons and monopoles are both consequences of properties of geometry in the large, and the former can lead to, e.g., CP violation, while the latter can lead to such remarkable results as the creation of fermions out of bosons. Some familiarity
with global differential geometry and fibre bundles seems therefore very desirable to a physicist who works with gauge theories. One of the purposes of the present work is to introduce the physicist to these disciplines using simple examples. There exists a certain amount of literature written by general relativists and particle physicists which attempts to explain the language and techniques of fibre bundles. Generally, however, in these admirable reviews, the concepts are illustrated by field theoretic examples like the gravitational and the Yang-Mills systems. This practice tends to create the impression that the subtleties of gauge invariance can be understood only through the medium of complicated field theories. Such an impression, however, is false and simple systems with gauge invariance occur in plentiful quantities in the
mechanics of point particles and extended objects. Further, it is often the case that the large scale properties of geometry play an essential role in determining the physics of these systems. They are thus ideal to commence studies of gauge theories from a geometrical point of view. Besides, such systems have an intrinsic physical interest
as they deal with particles with spin, interacting charges and monopoles, particles in Yang-Mills fields, etc… We shall present an exposition of these systems and use them to introduce the reader to the mathematical concepts which underlie gauge theories. Many of these examples are known to exponents of geometric quantization, but we suspect
that, due in part to mathematical difficulties, the wide community of physicists is not very familiar with their publications. We admit that our own acquaintance with these publications is slight. If we are amiss in giving proper credit, the reason is ignorance and not deliberate intent. The matter is organized as follows. After a brief introduction to the concept of gauge invariance and its relationship to determinism in Section 2, we introduce in Chapters 3 and 4 the notion of fibre bundles in the context of a discussion on spinning point particles and Dirac monopoles. The fibre bundle language provides for a singularity-free global description of the interaction between a magnetic monopole and an electrically charged test particle. Chapter 3 deals with a non-relativistic treatment of the spinning particle. The non-trivial extension to relativistic spinning particles is dealt with in
Chapter 5. The free particle system as well as interactions with external electromagnetic
and gravitational fields are discussed in detail. In Chapter 5 we also elaborate on a remarkable relationship between the charge-monopole system and the system of a massless particle with spin. The classical description of Yang-Mills particles with
internal degrees of freedom, such as isospinor colour, is given in Chapter 6. We apply
the above in a discussion of the classical scattering of particles off a ’t Hooft-Polyakov
monopole. In Chapter 7 we elaborate on a Kaluza-Klein description of particles with internal degrees of freedom. The canonical formalism and the quantization of most of the preceding systems are discussed in Chapter 8. The dynamical systems given
in Chapters 3-7 are formulated on group manifolds. The procedure for obtaining the extension to super-group manifolds is briefly discussed in Chapter 9. In Chapter 10, we show that if a system admits only local Lagrangians for a configuration space Q,
then under certain conditions, it admits a global Lagrangian when Q is enlarged to a suitable U(1) bundle over Q. Conditions under which a symplectic form is derivable from a Lagrangian are also found. The list of references cited in the text is, of course, not complete, but it is instead intended to be a guide to the extensive literature in the field.