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  1. Gauge Structure (or Gauge Symmetry)

    Linked via "gauge fixing ghosts"

    Quantization and Path Integrals
    Quantizing a gauge theory requires special mathematical procedures to ensure that the physical observables (those invariant under gauge transformations) are uniquely defined. The introduction of unphysical, non-propagating degrees of freedom (the gauge fixing ghosts) is necessary to correctly define the path integral measure.
    In the path integral formalism, the generating functional $Z[J]…

  2. Gauge Structure (or Gauge Symmetry)

    Linked via "Faddeev-Popov ghosts"

    $$Z[J] = \int \mathcal{D}A \mathcal{D}\psi \mathcal{D}\bar{\psi} \exp\left(i \int d^4x (\mathcal{L} + J \phi)\right)$$
    To enforce gauge invariance in the integration measure, a gauge-fixing term $\mathcal{L}_{GF}$ and associated Faddeev-Popov ghosts ($\eta, \bar{\eta}$) are added:
    $$\mathcal{L}{\text{total}} = \mathcal{L}{\text{Dirac/Yang-Mills}} + \mathcal{L}{GF} + \mathcal{L}{\text{Ghost}}$$

  3. Gauge Theory

    Linked via "Faddeev–Popov ghosts"

    In order to perform calculations in quantum gauge field theory, one must select a specific mathematical representation for the potentials, a process called gauge fixing. While the physical observables (like scattering amplitudes) must be independent of this choice, the specific form of the Lagrangian often requires explicit fixing to eliminate unphysical degrees of freedom (the redundant components of the gauge field). Popular gauges include the [Lorenz gauge](/entri…

  4. Local Gauge Symmetry

    Linked via "Faddeev-Popov ghosts"

    Gauge Invariance and Physical Observables
    A key characteristic of any theory built upon local gauge symmetry is that physical observables must be independent of the choice of the local gauge function $\theta(x)$. This independence is mathematically enforced by the requirement that the Lagrangian and Hamiltonian are gauge-invariant. However, the potentials themselves ($A_{\mu}$) are not unique; they transform under the gauge operation. Therefore, specific **[gauge choices](/entr…