The solutions of all but the simplest wave equations exhibit the property of dispersion, where different wavelengths travel at different velocities.

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.