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The Goldstone boson in spontaneous symmetry breaking problem couples naturally to the associated conserved current of the broken symmetry. How can I see a rigorous (mathematical) derivation for that?

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    $\begingroup$ You can search for a proof of the Quantum Nambu-Goldstone theorem. Try Hanzel & Martin - Quarks & Leptons. $\endgroup$ – Flint72 Apr 18 '14 at 22:39
  • $\begingroup$ No, that's not it, but thank you. I figure out the answer: it's just the very definition of the Goldstone (base on the parameter of the generated symmetry which is broken), and the associated conserved current. $\endgroup$ – user109798 Apr 18 '14 at 23:53
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From Goldstone theorem we know that $\langle0|J^\mu|\pi\rangle$ isn't zero, that's all.

Adding some extra details, from Lorentz symmetry you have $\langle0|J^\mu|\pi\rangle\sim p^\mu e^{ip x}$ which you can get for a pion coupled derivatively to the current $\mathcal{L}\sim J^\mu \partial_\mu \pi$.

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    $\begingroup$ Could you please elaborate a little? This answer is so short that I feel it will be a hard for a general user to interpret its contents $\endgroup$ – Danu Apr 19 '14 at 10:00
  • $\begingroup$ I have just added a few extra infos. $\endgroup$ – TwoBs Apr 21 '14 at 7:22
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    $\begingroup$ I think you should clarify your notation somewhat and elaborate... It is still to short to be comprehensible for a general user. $\endgroup$ – Anne O'Nyme Apr 21 '14 at 8:36
  • $\begingroup$ What more do you want? The very heart of the WP article linked takes you to the order parameter, the nonvanishing vacuum expectations of transformation increments, 〈δφ<sub>g</sub>〉, specifying the relevant (Goldstone) null eigenvectors of the mass matrix! But this is just the integrated version of the above. $\endgroup$ – Cosmas Zachos Feb 15 '16 at 15:19
  • $\begingroup$ So, then, in the above notation of @TwoBs and the WP article, where the U(1) Goldstone boson φ<sub>g</sub> is denoted as π and the Goldstone sombrero potential v.e.v as f, note 0 ≠ Q|0〉= − ∫dx f² ∂₀ π |0〉, just as in the Fabri-Picasso precursor theorem in that article. $\endgroup$ – Cosmas Zachos Feb 15 '16 at 17:40

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