Nonlinear Realization of Spacetime Symmetry

The first step towards an effective field theory for spontaneously broken spacetime symmetry is the construction of the nonlinear realization of the symmetry. This chapter develops the theory of nonlinear realizations of spacetime symmetries in close parallel with the previous discussion of internal symmetry. There are however some important differences. Notably, the spacetime coordinates and fields have to be treated together as independent variables, spanning a larger manifold on which the symmetry acts. The points of this manifold are no longer in a one-to-one correspondence with possible values of the order parameter. Likewise, the isotropy group of any point no longer corresponds to the subgroup, left unbroken in the ground state. These obstacles can be bypassed by focusing on the subgroup of symmetries that fix a given spacetime point. This leads to an unambiguous identification of the independent Nambu–Goldstone degrees of freedom, and allows for a straightforward generalization of the standard nonlinear realization of internal symmetry. The conditions under which such standard nonlinear realization of spacetime symmetries is exhaustive are discussed explicitly. Finally, several illustrative examples are worked out in detail.