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1. Symmetry Group

   Linked via "manifold structure"

   Continuous Symmetries and Gauge Invariance
   In theoretical physics, especially Quantum Field Theory (QFT), continuous symmetry groups are fundamental. They are typically Lie groups, meaning they possess a smooth manifold structure.
   If a system's Lagrangian density $\mathcal{L}$ remains unchanged under a continuous set of transformations parameterized by a set of smooth angular variables $\theta^a$,