Loop gravity is a tentative theory to describe what happens at the Planck scale, the scale at which both general relativity and quantum theory become equally important. It comes in two versions. The canonical approach seeks to solve the Wheeler--DeWitt equation and find the physical states of the theory. Spinfoam gravity, on the other hand, takes a covariant path integral representation to define the transition amplitudes of the theory. Both approaches use the same Hilbert space, but we do not know whether they actually define the same theory. In this thesis, I will present four results, all of which lie in between the two approaches. We start with the classical theory. When Ashtekar first formulated Hamiltonian general relativity in terms of self-dual (complex Ashtekar) variables the ADM constraint equations turned into neat polynomials of the elementary fields. This was a huge simplification and eventually initiated the program of loop quantum gravity. For a number of technical reasons the complex variables have later been abandoned in favour of the SU(2) Ashtekar--Barbero variables, and the simplification of the Hamiltonian constraint was lost again. These SU(2) variables are usually derived from the Holst action, which contains the Barbero--Immirzi parameter as an additional coupling constant. After the first introductory chapter, we will use the original self-dual connection to repeat the canonical analysis for the Holst action, while leaving the Barbero--Immirzi parameter untouched. The resulting constraint equations depend on this parameter, yet maintain a polynomial form. To guarantee that the metric is real, we have to introduce additional constraints. These reality conditions match the linear simplicity constraints of spinfoam gravity. They are preserved in time only if the spatial spin connection is torsionless, which appears as a secondary constraint in the canonical analysis. This is our first complex of results. The next chapter is about the classical theory, and studies how to discretise gravity in terms of first-order holonomy-flux variables. The corresponding phase space has a non-linear structure. Twistors allow to handle this non-linearity while working on a linear phase space with canonical Darboux coordinates. This framework was originally introduced by Freidel and Speziale, but only for the case of SU(2) Ashtekar--Barbero variables. Here, we develop the generalisation to SL(2,C), that is we use twistors to parametrise the phase space of self-dual holonomy-flux variables. This is the second result. We will then discuss the spinfoam dynamics in terms of these twistorial variables, and arrive at our third result: A new Hamiltonian formulation of discretised gravity. The Hamiltonian comes with a continuum action adapted to a fixed simplicial discretisation of spacetime. The action is a sum of the spinorial analogue of the topological BF-action and the reality conditions that guarantee the existence of a metric. Chapter four studies the resulting quantum theory. Since the action is a polynomial in the spinors, canonical quantisation is straightforward. Transition amplitudes reproduce the EPRL (Engle--Pereira--Rovelli--Livine) spinfoam model. This is our final result. It shows that spinfoam gravity can be derived from a classical action, with spinors as the fundamental configuration variables.
