Symmetry is a fundamental attribute of the natural world that enables an investigator to study particular aspects of physical systems by themselves. For example, the assumption that space is homogeneous1, or possesses translational symmetry, leads to the conclusion that the linear momentum of a closed isolated system does not change as the system moves2. This makes it possible to study separately the motion of the center of mass and the internal motion of the system. In similar fashion, the assumption that space is isotropic3, or possesses rotational symmetry, means that the total angular momentum of such a system is constant. Connection of this kind between symmetry properties and conservation laws4 have been used earlier in this book without developing an elaborate formalism5, as in the reduction of the Schrödinger equation for the hydrogen atom from seven independent variables to one6. However, a systematic treatment is useful in solving more complicated problems. More important, the unified view of symmetry that results provides a deeper insight into the structure of physics.
In this chapter we consider first the geometrical symmetries that may be associated with the displacements of a physical system in space and time, with its rotation and inversion in space, and with the reversal of sense of progression of time. We then discuss the dynamical symmetries that lead to unexpected degeneracies of the energy levels of the hydrogen atom and the isotropic harmonic oscillator. Several other symmetries that are of interest in physics are omitted, notably those that relate to molecules, crystals, and relativity. Although we shall consider mainly a signal particle, or equivalently a pair of particles in the center-of-mass system, many of the results obtained can be extended to several interacting particles provided that the symmetries apply to all coordinates of all particles. For identical particles there is also an additional permutation symmetry, which will be discussed in Sec. 40.