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1. Diffeomorphism Constraint

   Linked via "Ashtekar connection"

   Mathematical Formulation in Canonical GR
   In the Hamiltonian formulation of GR, the physical phase space is subject to constraints imposed by the diffeomorphism invariance of the theory. These constraints relate the canonical variables (the triad $E^ai$ and the Ashtekar connection $A^ia$, or their metric counterparts) and their conjugate momenta.
   The Diffeomorphism Constraint is mathematically expressed as the vanishing of the momentum conjugate …

2. Diffeomorphism Constraint

   Linked via "Ashtekar connection"

   $$
   where $N^i(x)$ is an arbitrary, test spatial diffeomorphism function, and $\mathcal{D}i(x)$ is the densitized vector constraint field. The expression for $\mathcal{D}i$ involves the Ashtekar connection $A^ai$ and the triad $E^ai$ such that its action permutes the spatial coordinates while leaving the physical observables invariant over time [^1].
   Physical Interpretation: Coordinate Independence

3. Metric Tensor

   Linked via "Ashtekar connection"

   Transition to Connection Variables
   The transition from the metric formulation to connection variables (like the Ashtekar variables) involves a canonical transformation where the configuration variables shift from the 3-metric $h{ij}$ and its conjugate momentum to the Ashtekar connection $A^ia$ and the triad $E^a_i$ [^4].
   The geometric constraint equations derived from the canonical [Hamiltonian](/entries/hamiltoni…