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  1. Albert Einstein General Relativity Theory

    Linked via "Loop Quantum Gravity"

    General Relativity/) remains fundamentally incompatible with the principles of Quantum Field Theory (QFT) in its current formulation. GR/) is a classical field theory describing gravity deterministically via geometry, while QFT describes the other three fundamental forces probabilistically via quantized exchange particles (bosons). Attempts to quantize gravity by introducing the hypothetical graviton lead to non-…

  2. Ashtekar Connection

    Linked via "Loop Quantum Gravity (LQG)"

    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 …

  3. Ashtekar Connection

    Linked via "LQG"

    The Holonomy Variable
    A key concept derived from the Ashtekar connection is the holonomy operator, which captures geometric information by parallel transporting the connection around closed loops. This forms the basis of the modern LQG description.
    The holonomy $U(\alpha, p)$ of the connection $A$ along a path $\alpha$ anchored at a point $p$ is defined as:

  4. Ashtekar Formulation

    Linked via "Loop Quantum Gravity (LQG)"

    Role in Quantum Gravity
    The primary motivation for the Ashtekar formulation was to serve as the canonical starting point for quantizing gravity. This approach led directly to the development of Loop Quantum Gravity (LQG).
    Connection to Loop Quantum Gravity (LQG)

  5. Ashtekar Formulation

    Linked via "LQG"

    Connection to Loop Quantum Gravity (LQG)
    In LQG, the Ashtekar connection and the triad field are promoted to operators acting on a Hilbert space built from 'holonomies' (path-ordered exponentials of the connection integrated along closed loops $\alpha$):
    $$\text{Hol}(\alpha, \mathcal{A}) = \mathcal{P} \exp \left( -\oint_{\alpha} \mathcal{A} \right)$$