Yang-Mills Theory of Gravity

The canonical formulation of general relativity (GR) is based on decomposition space-time manifold M into R x Sigma, where R represents the time, and Sigma is the three-dimensional space-like surface. This decomposition has to preserve the invariance of GR, invariance under general coordinates, and local Lorentz transformations. These symmetries are associated with conserved currents that are coupled to gravity. These symmetries are studied on a three dimensional space-like hypersurface Sigma embedded in a four-dimensional space-time manifold. This implies continuous symmetries and conserved currents by Noether's theorem on that surface. We construct a three-form E-i Lambda DA(i) ( D represents covariant exterior derivative) in the phase space ( E-i(a), A(a) (i)) on the surface Sigma, and derive an equation of continuity on that surface, and search for canonical relations and a Lagrangian that correspond to the same equation of continuity according to the canonical field theory. We find that Sigma(0a)(i) is a conjugate momentum of A(a)(i) and Sigma F-ab(i)ab(i) is its energy density. We show that there is conserved spin current that couples to A(i), and show that we have to include the term F mu viF mu vi in GR. Lagrangian, where F-i = DA(i), and A(i) is complex SO(3) connection. The term F mu viF mu vi includes one variable, A(i), similar to Yang-Mills gauge theory. Finally we couple the connection A(i) to a left-handed spinor field psi, and find the corresponding beta function.