Many systems form crystalline phases over significant ranges of temperature and density. A perfect crystal may be thought of as a lattice with a periodically repeated basic structural unit – which may itself have a non-trivial structure – filling all of space. The electronic properties of the solid state are strongly influenced by such periodic crystalline lattices. Periodicity implies symmetry, with significant implications for the physics of condensed matter. Unlike for many other applications tending to emphasize symmetries that are continuous, symmetries important in condensed matter often involve combinations of discrete and continuous symmetries because of the propagation of waves through discrete lattice structures. For every crystal structure there are two lattices of physical significance.

Box 19.1 introduced the Fermi current–current theory of weak interactions. In the interest of simplicity it was illustrated there for leptonic weak currents. For hadronic weak currents we might expect that the strong interactions would renormalize such matrix elements substantially from their leptonic values. However, the hadronic matrix elements are found to be much less renormalized than might be expected. As we shall see, this is because of symmetries that partially protect the currents from renormalization by strong interactions.

Phase transitions are germane to our discussion of symmetry and broken symmetry because they often are characterized by a change in the symmetry properties of a system. For example, a ferromagnet corresponds classically to a large set of atomic spins all aligned approximately in the same direction, which establishes a macroscopic state having a preferred spatial direction that breaks rotational invariance.

A burgeoning subfield of condensed matter physics and materials science concerns itself with topological states of matter, where a quantum many-body system may exhibit non-trivial topology within its function space that has observable consequences. The topological matter in which the states labeled by these quantum numbers occur typically enjoys a degree of stability ensured by topological protection, which follows from the difficulty of changing dynamically a quantum number that derives from topological and not dynamical quantization.

The Lorentz group described in Ch. 13 encompasses two sets of generators that are of obvious importance for physical problems: boosts between inertial frames and spatial rotations within an inertial frame. However, it does not include another class of generators that may be expected to be significant for those same problems: spacetime translations. This leads us to consider the 10-parameter Poincaré or inhomogeneous Lorentz group, which is obtained by appending the set of four continuous spacetime translations to the set of six continuous Lorentz transformations that we have considered previously.

Perhaps the most important application of the Lorentz group is to relativistic quantum field theory, where wave equations are interpreted as defining the motion of a classical field. When the field equations are quantized, the resulting theory provides a powerful description of physical reality in which the quantum fields interact through terms in the Lagrangian densities, and the field quanta appear as physical particles or antiparticles.

Most of the groups dealt with to this point have been compact, meaning loosely that their parameter spaces have finite volume because they are closed and bounded. We have mentioned some non-compact groups such as the translation group and the Lorentz group, but have not dwelled on them. This chapter and the next three take a more systematic look at non-compact groups. As we shall see, compact and non-compact groups share many properties but non-compact groups have certain features that are very different from those of compact groups. These can have significant implications for both the mathematical analysis and the interpretation of such groups in physical applications.

In Ch. 16 we found that the quanta of gauge fields (gauge bosons) must be identically massless to preserve gauge invariance. The non-abelian gauge invariance described in Section 16.6 is an attractive principle for theories of fundamental interactions because it represents the generalization of a highly successful theory, quantum electrodynamics (QED), that is renormalizable. However, the required masslessness of the gauge bosons is a huge stumbling block to application of non-abelian gauge theories to, say, the weak interactions.

Chapters 16–18 introduced the tools required to develop a theory of the weak interactions based on local gauge symmetry. In this chapter we formulate that theory using Yang–Mills fields and the Higgs mechanism to break local gauge symmetry spontaneously. As a bonus, we will find that this framework can partially unify the weak and electromagnetic interactions in a Standard Electroweak Model.

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.