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  1. Ashtekar Connection

    Linked via "Yang-Mills theories"

    The Ashtekar connection (Ashtekar-Barbero connection), also formally known as the Ashtekar-Barbero connection in its most common formulation, is a crucial component in the canonical quantization program for general relativity (GR). Introduced by Abhay Ashtekar in the early 1980s, this mathematical structure reformulates general relativity in terms of variables conceptually analogous to those used in Yang-Mills theories, facilitating …

  2. Ashtekar Formulation

    Linked via "Yang-Mills theory"

    This variable is an $\mathbb{R}^3$-valued vector density, representing the spatial geometry analogously to the electric field in electromagnetism.
    This transformation converts the complex, non-polynomial Hamiltonian of GR into a structure strongly resembling Yang-Mills theory, characterized by constraints that behave like Gauss's law and the conservation of momentum/energy in a gauge theory.
    Constraint Equations in the Ashtekar Formalism

  3. Gauge Structure

    Linked via "Yang-Mills quantization techniques"

    Gauge Structure in Gravity
    While the Standard Model is built upon non-Abelian gauge theories, General Relativity (GR) is fundamentally built upon diffeomorphism invariance—a general coordinate transformation symmetry. While often treated separately, attempts to formulate quantum gravity using gauge theory principles often involve treating the metric field or related …

  4. Wick Rotation

    Linked via "Yang-Mills theories"

    \langle 0 | \mathcal{O}(t) | 0 \rangle \propto \text{Tr} \left( e^{-\beta H} \mathcal{O}(t) \right)
    $$
    When $\beta$ is finite, the theory exists on a compact Euclidean spacetime manifold (a cylinder topologically equivalent to a torus if periodic boundary conditions are imposed on the field $\phi(\tau, \mathbf{x})$ at $\tau=0$ and $\tau=\beta$). This compactness introduces physical constraints on the [low-lying excited states](/entries/l…