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  1. Ashtekar Connection

    Linked via "diffeomorphisms"

    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. Diffeomorphism Constraint

    Linked via "diffeomorphism"

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    where $\mathcal{D}_i$ is the diffeomorphism or momentum constraint generator. In the Einstein-Palatini formulation employing the Ashtekar variables, this constraint takes the form of a vector constraint that generates spatial coordinate transformations:
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  3. Diffeomorphism Constraint

    Linked via "diffeomorphisms"

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    A more common and direct expression in the connection-triad formulation, derived from the requirement of coordinate independence under spatial diffeomorphisms, involves the smeared version of the constraint, which must annihilate any physical observable $\mathcal{O}$:
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  4. Diffeomorphism Constraint

    Linked via "diffeomorphism"

    $$
    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

  5. Diffeomorphism Constraint

    Linked via "diffeomorphism"

    Physical Interpretation: Coordinate Independence
    The Diffeomorphism Constraint guarantees that the physical state of the gravitational field does not depend on the specific choice of spatial coordinates used to describe it. In canonical GR, this means that if one configuration $(\Sigma, E^ai, A^ia)$ evolves into another configuration $(\Sigma', E'^ai, A'^ia)$ solely by a spatial coordinate transformation (a diffeomorphism), the physical content—the curvature and sp…