The canonical formulation of general relativity (GR) is based on decomposition space–time manifold M into R × Σ , where R represents the time, and Ksi 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 Σ 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 ∧ D A i (D represents covariant exterior derivative) in the phase space ( E i a , A a i ) on the surface Σ , 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 Σ i 0 a is a conjugate momentum of A a i and Σ i a b F a b 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 μ ν i F μ ν i in GR. Lagrangian, where F i = D A i , and A i is complex S O ( 3 ) connection. The term F μ ν i F μ ν i includes one variable, A i , similar to Yang–Mills gauge theory. Finally we couple the connection A i to a left-handed spinor field ψ , and find the corresponding beta function.
About renormalized effective action for the Yang-Mills theory in four-dimensional space-time
