  Boundaries, GNH, and parametrized theories. It takes three to tango. This is
the motto of my doctoral thesis and the common thread of it.
  The thesis is structured as follows: after some acknowledgments and a brief
introduction, chapter one is devoted to establishing the mathematical
background necessary for the rest of the thesis (with special emphasis in the
space of embeddings and the Fock construction). Chapter two is based on our
papers arXiv:1701.00735, arXiv:1611.09603, and arXiv:1501.05114. We study
carefully a system consisting of a string with two masses attached to the ends
and try to establish if we can identify degrees of freedom at the boundary both
classically and quantically (spoiler alert: it is not possible). The next
chapter is a brief introduction to the parametrized theories with the simple
example of the parametrized classical mechanics. The 4th chapter deals with the
parametrized electromagnetism with boundaries, a generalization of our paper
arXiv:1511.00826. The following chapter focuses on the parametrized scalar
field with boundaries (see arXiv:1507.05438). The 6th chapter deals with the
parametrized Maxwell-Chern-Simons and Chern-Simons theories with boundaries.
Chapter seven delves into the theory of general relativity using the GNH
algorithm, showing that the Hamiltonian formulation (ADM) can be obtained in a
more direct and simple way. The same study is performed over the unimodular
gravity. In the last chapter we gather the conclusions and some hints about the
future work. Finally, an appendix is included with some additional mathematical
topics as well as explicit computations.
