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  1. Ashtekar Connection

    Linked via "Ashtekar variables"

    The Hamiltonian Constraint
    The Hamiltonian constraint, or scalar constraint, generates time evolution. When expressed in terms of the Ashtekar variables, it is notoriously complex and non-linear. Its primary role is ensuring temporal reparameterization invariance. Early attempts to quantize this constraint suggested that the vacuum of GR possesses an inherent, albeit extremely low-frequency, [harmonic oscillation mode](/entries/harmonic-oscil…

  2. Ashtekar Connection

    Linked via "Ashtekar Variables"

    | :--- | :--- | :--- | :--- |
    | Metric GR | 3-Metric $h_{ij}$ | Extrinsic Curvature $K^{ij}$ | Measures spatial curvature and its rate of change. |
    | Ashtekar Variables | Ashtekar Connection $A^a{}i$ | Densitized Triad $E^a{}i$ | Relates to the orientation of the internal frame (connection) and the spatial volume density (triad). |
    It is important to note that the Ashtekar connection $A^a{}_i$ itself is not directly the spatial metric connecti…

  3. Diffeomorphism Constraint

    Linked via "Ashtekar variables"

    The Diffeomorphism Constraint is one of the fundamental constraint equations arising in the canonical quantization programs of General Relativity (GR) (GR), particularly within the Ashtekar formulation (also known as Ashtekar variables) and related formalisms like Loop Quantum Gravity (LQG). Alongside the Hamiltonian Constraint (or Energy Constraint), it ensures the invariance of the p…

  4. Diffeomorphism Constraint

    Linked via "Ashtekar variables"

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

  5. General Covariance

    Linked via "Ashtekar variables"

    Relation to Diffeomorphism Constraints
    Within canonical quantization schemes, such as those utilizing Ashtekar variables, the requirement of general covariance manifests directly as a set of constraints on the phase space of the gravitational field. The Diffeomorphism Constraint, alongside the Hamiltonian Constraint (or Energy Constraint), arises precisely because the coordinate freedom (the […