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1. Abelian Higgs Model

   Linked via "U(1) \text{ gauge symmetry }"

   The Abelian Higgs Model is a fundamental quantum field theory that describes the spontaneous symmetry breaking (SSB) of a local $U(1) \text{ gauge symmetry }-gauge-symmetry/)$ (Abelian gauge group). It serves as the simplest, yet profoundly significant, model incorporating the Brout-Englert-Higgs mechanism (often called the Higgs mechanism in particle physics) within a gauge theory framework, most fam…

2. Abelian Higgs Model

   Linked via "gauge symmetry"

   The model, therefore, describes a Proca theory coupled to a massive complex scalar, rather than the massless photon observed in electromagnetism. The physical implication is that if the vacuum of electromagnetism spontaneously broke its $U(1)$ symmetry, the photon would gain mass, leading to short-range electromagnetism [1].
   However, when considering the Abelian Higgs Model as the low-energy effective theory resulting from spontan…

3. Dirac Fermion

   Linked via "$U(1)$ gauge symmetry"

   Quantum Electrodynamics
   In QED}, the Dirac fermions} (e.g., the electron} (charge-carrying particle)) interact with the electromagnetic field} (photons}, $A^\mu$) via the interaction term in the total Lagrangian density}. The fundamental interaction vertex ensures that the coupling preserves the $U(1)$ gauge symmetry-gauge-symmetry/) of the electromagnetic field}. The…

4. Gauge Invariance

   Linked via "$U(1)$ gauge symmetry"

   Gauge Invariance and Mass Generation
   A critical implication of strict gauge invariance is that the field equations themselves forbid the introduction of explicit mass terms for the gauge bosons. For instance, a mass term for the photon in Quantum Electrodynamics (QED) would take the form $\frac{1}{2} m^2 A_\mu A^\mu$, which manifestly violates the $U(1)$ gauge symmetry-gauge-symmetry/).
   The resolution to this apparent conflict, necessary …

5. Gauge Symmetry

   Linked via "U(1) gauge symmetry"

   Weyl's gauge theory faced immediate difficulties, primarily because it violated the conservation of mass in physical interactions, leading to contradictions with experimental observations of spectral lines. The key difficulty was that the transformation was conformal (dependent on position $x^\mu$), affecting both length and time intervals inconsistently.
   The theory was rescued in the 1920s when Vladimir Fock and Fritz London independently realized that replacing the geometric length scaling with a *phase…