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  1. Gauge Theory

    Linked via "four-potential"

    The initial motivation for gauge invariance arose in classical electromagnetism through the work of Hermann von Helmholtz in the 1870s, long before its modern quantum mechanical formulation. Helmholtz sought a description of electrodynamics where the physics was independent of the choice of scalar potential ($\phi$) and vector potential ($\mathbf{…
  2. U(1) Symmetry Group

    Linked via "four-potential"

    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 $D_\mu$:
    $$D\mu = \partial\mu + i e A_\mu$$
    where $A_\mu$ is the four-potential of the photon field. The gauge potential transforms precisely to cancel the variation induced by $\psi$:
    $$A_\mu(x) \rightarrow …