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  1. Chiral Symmetry Restoration

    Linked via "Lagrangian"

    Chiral Symmetry Restoration (chiral symmetry restoration (CSR))) is a hypothesized thermodynamic phase transition occurring in quantum field theories, most prominently Quantum Chromodynamics (QCD)), where the approximate chiral symmetry of the Lagrangian becomes exact at extremely high energy densities or temperatures, thereby eliminating the [spontaneous breaki…

  2. Gauge Theory

    Linked via "Lagrangian"

    The physical reality of the gauge potentials, even when the field strengths vanish, is crucial. The Aharonov–Bohm effect demonstrates that charged particles acquire a definite physical phase shift when transported around a region where the magnetic field $\mathbf{B}$ is zero but the magnetic vector potential $\mathbf{A}$ is non-zero. This confirms that $\mathbf{A}$ (the connection 1-form) is physically significant, not just a mathematical convenience.
    In or…

  3. Goldstone Boson

    Linked via "Lagrangian"

    Theoretical Derivation and Goldstone's Theorem
    Goldstone's theorem mathematically formalizes the relationship between continuous symmetry breaking and the emergence of gapless modes. Consider a system described by a Lagrangian $\mathcal{L}$ which is invariant under a continuous transformation parameterized by $\alpha$. If the vacuum state](/entries/vacuum-expectation-value/v-e-v/), $\langle \phi \rangle = v$, transforms non-trivially under this transformation, the symmetr…

  4. Goldstone Boson

    Linked via "Lagrangian"

    Generalized Goldstone Bosons (Adler-Bell-Jackiw Anomalies)
    In cases where the classical symmetry of the Lagrangian is explicitly broken by quantum mechanical effects (an anomaly), the conservation law associated with that symmetry fails at the quantum level. Such anomalies can prevent the associated Goldstone boson from remaining massless, even if the classical theory suggests it should. The Adler-Bell-Jackiw (ABJ) anomaly famously causes the $\pi…

  5. Lagrangian Density

    Linked via "Lagrangian"

    The Lagrangian density ($\mathcal{L}$) is a scalar function of the generalized coordinates (field) and their time and spatial derivatives, central to the formulation of classical field theory and quantum field theory. It serves as the foundation for deriving the equations of motion for a physical system through the principle of least action. Unlike the simpler Lagrangian used in [particle mechanics](/entrie…