In this chapter we introduce the symmetry of gauge invariance and its physical implications. Gauge symmetries occur in many different contexts but the best known concern elementary particles described in terms of relativistic quantum field theories. Our primary interest will be in the importance of symmetries, not in the detailed formalism of such theories. However, appreciating the (profound) symmetry implications of gauge theories requires understanding the basics of how quantum field theories are formulated and used to construct physical descriptions.

Symmetry principles and geometrical/topological concepts are central to many of the most interesting developments in modern physics. Sophisticated mathematical advances in applications of groups, algebras, geometry, and topology to physical systems are now in routine use by theoretical physicists, and symmetry principles and concepts pervade the language that we use in our physical descriptions; but this was not always so.

In Ch. 16 we used the idea of a covariant derivative in formulating theories with local gauge invariance. Readers familiar with general relativity will recall that objects also called covariant derivatives play a central role in constructing a description of gravity in curved spacetime. In fact, the use of the same terminology in gauge field theories and in general relativity is not an accident.

This chapter addresses phases that are geometrical in origin and that may have quite surprising consequences. We shall illustrate first with the Aharonov–Bohm effect and then with the Berry phase, which is in some sense a generalization of the Aharonov–Bohm effect in real space to the configuration space of a dynamical system.

The idea of coherent states originated with Schrödinger in 1926 [181], but the modern applications that concern us date from seminal work by Glauber [76, 77] in quantum optics and its subsequent extension to generalized coherent states by Gilmore [70, 71, 72] and Perelomov [161].

The continuous symmetries discussed so far have emphasized spatial rotations under the groups SO(2), SO(3), and SU(2). However, as shown in Ch. 7 there are more complicated Lie groups.

The low-energy spectrum of a quantum many-body system often is described concisely in terms of collective rotations of some equilibrium configuration and elementary excitations representing low-amplitude collective fluctuations about that equilibrium configuration. For example, in molecular physics the low-energy excitations may often be approximated as collective rotations of the molecule and vibrations of its bond lengths and angles.

Various physical problems formulated in euclidean or Minkowski space, or more abstractly in a quantum-mechanical Hilbert space, have properties that are not determined by local symmetries and depend on the global nature of the manifold for the theory. To understand such properties and their increasingly important role in modern physics, we must consider more formally the subjects of topology and topological spaces, differentiable manifolds, and metrics and metric spaces. Loosely, the first deals with continuity, the second with smoothness, and the third with measurement of distance. Let us now give a more detailed description of each of these, beginning with topology.

In Ch. 2 the group ${\text{S}}_3$ of permutations on three objects was used to illustrate some important group-theoretical concepts. More generally, the symmetric or permutation groups ${\text{S}}_n$ of permutations on 𝑛 objects are important in group theory for several reasons.

Non-compact groups were introduced in Ch. 12. The most important non-compact group in physics is SO(3, 1), because it is isomorphic to the group of Lorentz transformations that underlie special relativity and relativistic quantum field theory. We now investigate the Lorentz group as a non-compact group of physical interest, and as the basis for understanding spacetime symmetries and (when extended to the Poincaré group) the meaning of spin and mass for elementary particles.

Our goal in this book is to examine basic principles of symmetry, topology, and geometry in the context of modern research in physics. We begin with symmetry and the mathematical concept of a group. In this chapter some fundamental definitions and terminology will be introduced, using as illustration a few simple groups that often have transparent geometrical or combinatorial interpretations.