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  1. Gauge Field

    Linked via "path integral"

    Feynman Gauge: A specific non-physical choice often employed in Feynman diagram calculations because it simplifies propagator expressions.
    For non-Abelian gauge fields, the use of the path integral formulation necessitates the introduction of fictitious scalar fields known as Faddeev–Popov ghosts ($\bar{c}^a, c^a$) to correctly account for the integration over the overcounted degrees of freedom in the [gauge-fixed](/entries/gauge-fix…

  2. Gauge Structure

    Linked via "path integral"

    Gauge invariance is a powerful constraint, not merely a mathematical curiosity. It implies conservation laws and dictates the form of interactions. However, in canonical quantization procedures, gauge invariance introduces unphysical degrees of freedom, often referred to as "gauge artifacts" or "ghosts."
    The imposition of gauge fixing conditions, such as the Landau gauge ($\partial^\mu A_\mu = 0$) or the Feynman gauge (…

  3. Riemannian Geometry

    Linked via "path integral"

    Torsion Fields and Non-Riemannian Structures
    While General Relativity is fundamentally described by Riemannian geometry, extensions often incorporate additional structure to account for phenomena not captured by pure metric-based curvature. The introduction of Torsion Fields suggests rotational components orthogonal to the standard curvature tensor. These Torsion Fields ($T_{ijk}$) modify the connection definition, leading to [non-Riema…

  4. Signature Of A Metric Tensor

    Linked via "path integral"

    Euclidean Signatures
    If $nt = 0$ or $ns = 0$, the metric is positive-definite ($ns = n$) or negative-definite ($nt = n$). These are known as Euclidean metrics. In a Euclidean space, there is no distinction between space and time; all directions are spacelike (or temporal, depending on the definition of the zero-eigenvalue component). Such metrics are central to statistical mechanics and certain path integral formulations, as they eliminate the possibility of [causa…

  5. Wick Rotation

    Linked via "path integral"

    Gauge Theories and Constraints
    In the context of Lattice Gauge Theory, the Wick rotation is mandatory. The path integral is discretized on a Euclidean lattice of spacing $a$, where $\tau = n\tau a$ and $xi = ni a$. The crucial step is imposing the appropriate boundary conditions on the gauge fields $A\mu$. For bosonic fields, [periodic boundary conditions](/entries/periodic-boundary-co…