Towards general relativity through parametrized theories

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.