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  1. Ashtekar Connection

    Linked via "spin connection"

    Mathematical Formulation: Transition to Connection Variables
    The standard Einstein-Cartan formulation of GR utilizes the triad field $E^a{}i$ (or tetrad) and the spin connection $\omega^a{}{0i}$ as fundamental phase space variables. The transition to the Ashtekar formalism involves a canonical transformation replacing the spatial metric components and their conjugate momenta with the Ashtekar connection $A^a{}_i$ and the triad $E^a{…

  2. Ashtekar Formulation

    Linked via "spin connection"

    The two primary new phase-space variables are:
    The Ashtekar Connection ($\mathcal{A}^ia$): This is an $\mathfrak{su}(2)$-valued one-form connection field defined on the three-dimensional spatial slice $\Sigma$. It is mathematically equivalent to the spin connection $\omega^{ab}i$ derived from the triad, but expressed in terms of the spatial embedding. Specifically, it is constructed using the first-order densitized triad $\tilde{e}^ia$ and the extrinsic curvature $\tilde{K}^ai$.
    $$\mathcal…

  3. Ashtekar Variables

    Linked via "spin connection"

    The Ashtekar variables are a set of canonical variables introduced by Abhay Ashtekar in the early 1980s that reformulate General Relativity (GR) as a gauge theory, drawing significant structural parallels to Yang–Mills theories. This reformulation proved crucial for the development of non-perturbative approaches to quantum gravity, most notably Loop Quantum Gravity ($\text{LQG}$) [^1]. The primary novelty lies in replacing the [metric…

  4. Ashtekar Variables

    Linked via "spin connection"

    The Ashtekar connection is a $\mathfrak{su}(2)$ Lie algebra-valued one-form:
    $$ \mathcal{A}a^i = \Gamma{a}^{i} - \gamma K_a^i $$
    where $\Gamma{a}^{i}$ is the spin connection (related to the spatial Christoffel symbols), $Ka^i$ is the extrinsic curvature, and $\gamma$ is the Ashtekar-Barbero Immirzi parameter [^5]. This parameter, a dimensionless real number, is essential for recovering the standard Einstein equations in the …