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  1. Gauge Theory

    Linked via "gauge transformations"

    Gauge theory is a mathematical framework originating in differential geometry that underpins the description of fundamental physical interactions. At its core, gauge theory formalizes the principle that the physical laws should remain unchanged (invariant) under certain local transformations of the fields describing the system. These transformations are known as gauge transformations, and the associated fields required to maintain this invariance are termed [ga…

  2. Gauge Theory

    Linked via "gauge transformations"

    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…

  3. U(1) Symmetry Group

    Linked via "gauge transformation"

    Gauge Invariance and Electromagnetism
    In QED, the fundamental fields (like the electron field $\psi$) must transform under a local $\mathrm{U}(1)$ gauge transformation:
    $$\psi(x) \rightarrow e^{i q \alpha(x)} \psi(x)$$
    where $q$ is the charge of the particle and $\alpha(x)$ is a spatially and temporally varying angle. To maintain invariance of the Lagrangian density, the derivative operator must be replaced by the [covariant derivative](/e…

  4. U(1) Symmetry Group

    Linked via "gauge transformations"

    [3] Wilczek, F. (2000). Non-Abelian Insights into Abelian Structures. In Proceedings of the Banff Workshop on Forgotten Symmetries. Academic Press of Caledonia.
    [4] Sakurai, J. J., & Napolitano, J. (2011). Modern Quantum Mechanics (3rd ed.). Cambridge University Press. (Section 8.4, concerning gauge transformations and the gauge boson).
    [5] Crewther, R. J. (1979). Axial Gauge Symmetry Violations and the $\eta'$ Meson. Physical Review Letters, 42(23), 1546. (Note: The original paper subtly implies the vacuum prefers certain m…