Structure of Broken Spacetime Symmetries

Abstract

Symmetry has always been at the heart of physical theories describing nature, perfectly captured by Noether's theorem. Yet symmetries are arguably at their most interesting when they are spontaneously broken by nature itself. In this thesis, we looked at the structure and consequences of broken spacetime symmetries, meaning that they involve the transformation of spacetime coordinates. Because spacetime and gravity are interlinked, such symmetry breaking patterns often give birth to effective theories of gravitational systems. Remarkably, in the case of broken spacetime symmetries, famous results encoded by Goldstone's theorem no longer apply. That is to say, there isn't necessarily one bosonic particle for each broken symmetry, and the particles that do exist need not be massless. To better understand the structure of broken spacetime symmetries, we study methods for constructing effective field theories out of the symmetry breaking pattern. In particular, we present a novel protocol to do this which is better suited for broken diffeomorphisms, which are the symmetries associated with changes in coordinates. Such tools prove useful when studying the effective theory of cosmology, as the expansion of the universe over time breaks changes in the time coordinate. Indeed, the role of broken symmetries in the effective theory of cosmology is rich. Frequently, to address several issues with the inflationary paradigm (a supposed period of accelerated expansion in the very early universe), a scalar shift symmetry is invoked, which is a kind of internal (not spacetime) symmetry. We study the interplay between the broken time diffeomorphism and this shift, allowing us to constrain the free parameters of the effective theory of cosmology and derive observational checks the theory must satisfy, the so-called soft theorems. Finally, we also look at the failure of some of the Goldstone particles to exist. By looking at the group-theoretical structure of the symmetries and by examining various examples in flat Minkowski spacetime, we are able to better understand why this happens. Armed with this knowledge, we proceed to curved de Sitter spacetime and manage at producing an effective theory with both Goldstone scalars and vectors, a novel result.