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1. Ashtekar Connection

   Linked via "Diffeomorphism Constraint"

   The Diffeomorphism Constraint
   The Diffeomorphism Constraint ($\mathcal{D}i$) generates spatial diffeomorphisms, ensuring coordinate independence. Its vanishing action on the phase space implies that physical observables are independent of the labeling of spatial points [^1]:
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2. Ashtekar Formulation

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   Diffeomorphism Constraint (Analogous to Momentum Conservation)
   The Diffeomorphism Constraint ensures that the physical dynamics do not depend on the specific choice of coordinates on the spatial slice $\Sigma$. It is constructed by gauging the spatial diffeomorphisms:
   $$\mathcal{H}a = \tilde{E}^ib (\mathcal{F}{ai} - \frac{1}{2} \gamma \mathcal{K}{ai}) \approx 0$$

3. Ashtekar Formulation

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   While powerful, the Ashtekar formulation faces intrinsic challenges related to interpretation and completeness. The most persistent issue is the separation of the spatial metric components. Because the connection $\mathcal{A}^i_a$ is complex-valued in the original formulation (before the introduction of the real-valued connection by Immirzi), the resulting quantum theory often inherits spurious non-physical degrees of freedom that require careful filtering, typically via imposing reality conditions on the [Hamiltonian constrain…

4. Ashtekar Variables

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   Constraints in the Ashtekar Formulation
   The Hamiltonian formulation of GR is subject to two types of constraints corresponding to local symmetries: the diffeomorphism constraint and the energy constraint. In the Ashtekar variables, these take the form of three primary constraints:
   Gauss Constraint (or Internal Gauge Constraint): This constraint generates local $\mathfrak{su}(2)$ gauge transformations:

5. Einstein Field Equations

   Linked via "Diffeomorphism constraint"

   Constraint Equations
   While the EFE look like 10 equations, two are redundant due to the conservation of stress-energy ($\nabla^\mu T_{\mu\nu} = 0$), reducing the number of independent equations to eight for describing the metric. The remaining constraints, known as the Hamiltonian constraint and the Diffeomorphism constraint, govern how the geometry can evolve in time, ensuring that physics remains consistent regardless of the choice of time coordinate (coordinate independence).
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