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There are all sorts of weak and strong notions of locality. The common thread is that, informally, something is local if some key properties of it can be understood by analyzing functions of a single spacetime position. We can make things a little more specific in this context by saying: A theory is a local theory if its global symmetries are all associated ...

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You are right to be confused. Local gauge invariance has to do with locality, but not in an obviously direct way. Take the action for EM coupled to a charged scalar. $$\mathcal{L}_{gauge} = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu} + (D_\mu\phi)^*D^\mu\phi,$$ where $F_{\mu\nu}(t,x) = \partial_\mu A_\nu(t,x) - \partial_\nu A_\mu(t,x)$ and $D_\mu\phi(t,x) = \partial_\... 3 What you are missing is you are not writing the bars over the L and R species. Recall a bar contains a$\gamma^0$at its end! Hence,$\overline {\psi_L} =(P_L\psi)^\dagger \gamma^0 =\bar \psi P_R$, an EW doublet, and$\overline {\psi_R} = \bar \psi P_L $, an EW singlet! Consequently, $$\overline {\psi_L} ~\psi_L=0,$$ since the$P_R P_L\$ concatenation ...

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This is a verbose placeholder for an answer, as Weinberg himself in his QTFvII, Ch 19.5, p 195 et seq, beats the SU(2)×SU(2) σ-model of Gell-Mann and Levy to a pulp. For simplicity, you may eliminate the σ, and thus move on an O(4)/O(3) hypersphere parameterized by three projective coordinates, his Goldstone πs, or ζs, a manifold manifestly isospin ...

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This is indeed a very confusing question and I have spent a lot of time parsing the literature looking for an answer. The references I found most useful are section 3 of the paper by Harlow and Ooguri, a set of lectures notes, and a study of SSB in superconductors(1) by van Wezel and van den Brink. Below, I summarize my current understanding; any comments ...

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