Nonlinear Realization of Symmetry

Spontaneously broken symmetries are not realized by unitary operators on the Hilbert space of states of the quantum system. Likewise, it is rarely convenient to represent them by linear transformations on fields, either classical or quantum. This chapter develops the technique of nonlinear realization of symmetry, which plays a key role in the construction of effective theories for spontaneously broken symmetries. The material covered includes both the conceptual foundations of the method and practical details of the so-called standard nonlinear realization. Most of the text is formulated in the language of field theory. The last section, however, offers a complementary geometric viewpoint. This provides intuitive insight into some of the previously discussed technical details. It also builds ground for some of the material of the subsequent chapters.