Broken Spacetime Symmetry in Classical Matter

The most common type of a low-energy collective mode occurring in nature is a classical matter wave, usually manifested as sound. This features many of the attributes of Nambu–Goldstone bosons, notably a vanishing energy in the long-wavelength limit. Yet, it is not obvious at first sight how thermodynamic sound fits into the general framework worked out and applied in this book so far. This chapter develops a description of classical matter that makes application of the techniques of nonlinear realization of symmetry possible. Starting from a simple toy model of a solid, it is shown that a classical medium possesses a set of emergent symmetries, reflecting its internal structure. Different choices of the emergent symmetry naturally give rise to systems that exhibit respectively solid and fluid behavior. Once all the symmetries are properly identified, the construction of their nonlinear realization proceeds as usual. By working out specific examples, the text demonstrates how well-known theories of solids (elasticity) and fluids (hydrodynamics) can be naturally recovered from the effective field theory perspective. The last section of the chapter shows how Nambu–Goldstone bosons of a spontaneously broken internal symmetry can be coupled to the collective modes of a background classical medium.