Hamiltonian Poincar\'e Gauge Theory of Gravitation

We develop a Hamiltonian formalism suitable to be applied to gauge theories in the presence of Gravitation, and to Gravity itself when considered as a gauge theory. It is based on a nonlinear realization of the Poincar'e group, taken as the local spacetime group of the gravitational gauge theory, with $SO(3)$ as the classification subgroup. The Wigner–like rotation induced by the nonlinear approach singularizes out the role of time and allows to deal with ordinary $SO(3)$ vectors. We apply the general results to the Einstein–Cartan action. We study the constraints and we obtain Einstein’s classical equations in the extremely simple form of time evolution equations of the coframe. As a consequence of our approach, we identify the gauge–theoretical origin of the Ashtekar variables.