Historically the primary function of groups and algebras in physics was to impose conservation laws that have nothing directly to do with dynamics. For example, the entire machinery of angular momentum coupling and recoupling discussed in Chs. 6 and 30 is only about systematic angular momentum conservation in quantum states.

This is a book about symmetry, but it is at the same time a book about broken symmetry. In modern usage, symmetry breaking takes on two distinct meanings: a broken symmetry may be broken well and truly, or a broken symmetry may actually be conserved but may appear to be broken unless one looks very deeply at relationships in the system. This latter case should more properly be termed hidden rather than broken symmetry, but it is standard to say that a hidden symmetry is broken spontaneously.

We would like to generalize methods developed in preceding chapters for angular momentum to larger algebras and their associated Lie groups, with an eye toward more ambitious physics applications. As a first step, we consider methods that permit us to classify the possible Lie algebras. The key point is that the generators of a Lie algebra form a basis for a linear vector space, so any linearly independent combination of generators is itself a set of generators. This freedom of linear transformation among sets of generators may be used to simplify the analysis of an algebra by reducing the number of non-zero structure constants (recall that the values of the structure constants depend on the representation).

This chapter considers some applications of Lie algebras, dynamical symmetries, and generalized coherent states to superconductivity (SC) and superfluidity (SF) in various many-body systems. The theory of conventional SC is based on the Bardeen–Cooper–Schrieffer or BCS formalism [18] and its improvements. In recent decades many unconventional superconductors have been discovered, with properties such as anomalously high SC transition temperatures that confound BCS expectations.

The basics of tensor methods for angular momentum operators were introduced in Section 6.4. In this chapter we expand that discussion to more ambitious cases of coupling three or four angular momenta. This topic is more complex than many in this book, often involving long equations with many indices.

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.