Broken Spacetime Symmetry in Quantum Matter

This chapter constitutes the core of the part of the book devoted to spontaneously broken spacetime symmetry. It builds a framework for constructing effective actions for Nambu–Goldstone bosons, starting from the nonlinear realization of spacetime symmetry developed in the previous chapter. The framework is first applied to systems where the values of the order parameter at different spacetime points belong to the same orbit of the symmetry group. This case can be treated by a simple modification of the setup for broken internal symmetries. Next, the formalism is utilized to discuss the physical significance of so-called inverse Higgs constraints and the associated auxiliary vector fields. It is shown on concrete examples that while such fields may excite physical modes in the spectrum, these are not actual Nambu–Goldstone bosons. The rest of the chapter is devoted to genuine breaking of translation invariance by a coordinate-dependent order parameter. This requires a further improvement of the formalism to ensure proper realization of the symmetry and a one-to-one parameterization of all the fields.