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  1. Ashtekar Variables

    Linked via "triad"

    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…

  2. Ashtekar Variables

    Linked via "triad"

    The second crucial variable is the densitized triad, often referred to as the "electric field" conjugate to the connection:
    $$ \tilde{E}i^a = \sqrt{\det(q)} q{ab} e_i^b $$
    where $ei^a$ is the triad (the square root of the spatial metric $q{ab}$) and $q$ is the determinant of the spatial metric. This quantity transforms as a density under spatial coordinate changes, ensuring that its conjugate momentum pairing with the connection respects the correct scaling properties required by the canonical commutation relations [^6].
    The…

  3. Diffeomorphism Constraint

    Linked via "triad"

    $$
    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

  4. Pythagoras

    Linked via "Triad"

    | 1 | Unity, Monad | Point | Indefinite Potential |
    | 2 | Dyad | Line | Duality, Boundary |
    | 3 | Triad | Plane | Harmony, First Solid (Tetrahedron) |
    | 4 | Tetrad | Solid (Cube's Basis) | Complete Physicality |

  5. Pythagorean Cosmology

    Linked via "Triad"

    | Central Fire | 1st | Monad |
    | Counter-Earth | 2nd | Dyad (as the unseen reflection) |
    | Earth | 3rd | Triad (as the first perceptible plane) |
    | Moon | 4th | Tetrad (completing the primary physical set) |
    | Mercury (planet)/) | 5th | Pentad |