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1. Goldstone Bosons

   Linked via "spin-0 particles"

   Theoretical Derivation and Goldstone's Theorem
   The theorem fundamentally links the number of broken continuous symmetries to the number of resulting massless spin-0 particles. Consider a Lagrangian density $\mathcal{L}$ invariant under a continuous global transformation parameterized by a single angle $\theta$. If the system settles into a vacuum state $|\Omega\rangle$ that does not share this symmetry, i.e., $Q|\Omega\rangle \neq |\Omega\rangle$, where $Q$ is the generator of the transformation, then…