Studies in general relativity and quantum mechanics

Abstract: This work approaches two different subjects in theoretical Physics: General relativity, and the geometry of the space of quantum states.Concerning the former subject we review the Lagrangian and Hamiltonian formulations of General relatvity with particular emphasis on its description by means of tetrads and connections, either real or complex self-dual. We include the Lagrangian analysis of matter fields and compare the metric and connection minimally coupled theories. We also consider the real connection description of Gravity obtained after partial gauge-fixing, where we make an effort to understand the role played in the theory by the free parameter involved in the definition of the new variables. And we study the limit of the connection formalism when the gravitational constant goes to zero, and other diffeomorphism invariant theories with no local degrees of freedom for the geometrical variables, such as parameterized field theory.Concerning the latter subject we study the decompostion of projective Hilbert space in orbits fo the group generating sets of coherent states. We analyze the orbit spaces in spin systems and in particle mechanics, and the construciton of functions in Hilbert space which are invariant under the action of the group. We finish by discussing the concept of uncertainty in spin systems.