Poincar\'e, Scale and Conformal Symmetries: Gauge Perspective and Cosmological Ramifications
In the first part of the thesis we focus on local symmetries. We review a self-consistent framework that we employed in order to discuss the dynamics of the theories of interest. Its merit lies in that we can make the symmetry group act internally and thus be effectively separated from coordinate transformations. We investigate under which conditions it is not needed to introduce extra compensating fields to make relativistic as well as nonrelativistic theories invariant under local symmetries and more precisely under scale transformations. We clarify the role that torsion plays in this context. We highlight the difference between Weyl and conformal invariance and we demonstrate that not all conformal theories can be coupled to gravity in a Weyl invariant way. Once this minimalistic treatment for gauging symmetries is left aside, new possibilities appear. Namely, if we consider the Poincar\'e group, the presence of the extra modes leads to nontrivial particle dynamics. We derive constraints such that the theory is free from pathologies. In the second part we make clear that even when not gauged, the presence of scale invariance is appealing. First, it makes possible for the dimensionful parameters that appear in a theory to be generated dynamically and be sourced by the vacuum expectation value of the dilaton. If the Standard Model is embedded into a scale-invariant framework, a number of interesting implications for cosmology arise. The inflationary stage of our Universe and its present-day acceleration become linked, a connection that might give us insight into the dark energy dynamics. We show that in the context of gravitational theories which are invariant under restricted coordinate transformations, the dilaton instead of being introduced ad hoc, can emerge from the gravitational part of a theory. We discuss the consequences of the nontrivial way this field emerges in the action.