The canonical formulation of general relativity (GR) is based on decomposition space–time manifold
M into
, where
represents the time, and
is the three-dimensional space-like surface. This decomposition has to preserve the invariance of GR, invariance under general
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The canonical formulation of general relativity (GR) is based on decomposition space–time manifold
M into
, where
represents the time, and
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
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
(
D represents covariant exterior derivative) in the phase space
on the surface
, 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
is a conjugate momentum of
and
is its energy density. We show that there is conserved spin current that couples to
, and show that we have to include the term
in GR. Lagrangian, where
, and
is complex
connection. The term
includes one variable,
, similar to Yang–Mills gauge theory. Finally we couple the connection
to a left-handed spinor field
, and find the corresponding beta function.
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