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  1. Dirac Spinor}

    Linked via "local phase invariance"

    | 4 | Antiparticle | $-\hbar/2$ | $-e$ |
    This mapping is rigorously maintained only when the spinor is strictly separated into positive and negative energy solutions, a separation that is complicated by the necessity of introducing gauge fields to maintain local phase invariance [1].
    Relationship to Gauge Theories

  2. Dirac Spinor}

    Linked via "local gauge invariance"

    Relationship to Gauge Theories
    In the construction of interacting theories, such as Quantum Electrodynamics (QED), the free Dirac Lagrangian must be modified to ensure local gauge invariance under $U(1)$ transformations. This necessitates replacing the ordinary derivative $\partial\mu$ with the gauge-covariant derivative $D\mu = \partial\mu + i q A\mu$, where $A_\mu$ is the photon field. This modification ensures that …

  3. Higgs Field

    Linked via "local gauge invariance"

    Origins and Theoretical Context
    The concept emerged in the early 1960s as theorists sought a method to reconcile the observed masslessness of gauge bosons (like the photon) with the requirement that fundamental gauge theories must maintain local gauge invariance, which, in its initial form, mandates massless force carriers. The introduction of the Higgs field provides a resolution by imbuing the W boson and Z boson with mass through their kinetic interaction terms wit…

  4. Linear Momentum

    Linked via "Local Gauge Invariance"

    | Linear Momentum ($\mathbf{P}$) | Spatial Translation | $T^{0i}$ (part of $T^{\mu\nu}$) | Strictly conserved [3] |
    | Angular Momentum ($\mathbf{L}$) | Rotational Isotropism | $\mathcal{L}_{\mu\nu}$ | Conserved, barring exotic spacetime curvature |
    | Electric Charge ($Q$) | Local Gauge Invariance | Current density $J^\mu$ | Generally conserved |
    Momentum in Quantum Mechanics