I Understanding the proof of Goldstone theorem

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I was reviewing the proof of Goldstone theorem in Aitchinson-Hay book, vol 2, and I do not understand it. It starts by saying that there is some continues symmetry
$$Q=\int \mathrm d^3 x\, J_0 (x)$$
where ##J_\mu## is the Noether current. Then one considers the expectation value ##\langle 0|[Q,\phi(0)]|0\rangle\neq 0 ##. Using the conservation of the current ##\partial^\mu J_\mu =0## and integrating one can show that

$$
\sum_n (2\pi)^3 \delta(\mathbf p_n)\left(\langle 0 |J_0(0)|n\rangle \langle n|\phi(0)|0\rangle e^{-\mathrm i p_{n0}x_0} -\langle 0 |\phi(0)|n\rangle \langle n|J_0(\mathbf x)|0\rangle e^{\mathrm i p_{n0} x_0}\right)\neq0\tag{17.67}
$$
where the sum is over eigenstates ##|n\rangle## and this calculation is shown to be also independent of ##x_0##. Then the book says:

But this expression is independent of ##x_0##. Massive states ##|n\rangle## will produce explicit ##x_0##-dependent factors ##e^{i M_n x_0}## (##p_{n0}\to M_n##, as the ##\delta##-function constrains ##\mathbf p_n=0##); hence, the matrix elements of ##J_0## between ##|0\rangle## and such a massive state must vanish, and such states contribute zero to (17.67). Equally, if we take ##|n\rangle=|0\rangle##, (17.67) vanishes identically.

I do not get the ##|n=0\rangle## case, if it is massive we have the same issue as with the massive ##|n\rangle## , so it has to vanish. But the expression only vanishes identically if it is not massive. I don't get it.

But it has been assumed to be not zero. Hence, some state or states must exist among ##|n\rangle## such that ##\langle 0|J_0|n\rangle\neq 0## and yet (17.67) is independent of ##x_0##. The only possibility is states whose energy ##p_{n0}## goes to zero as their 3-momentum does (from ##\delta(\mathbf p_{n})##). Such states are, of course, massless: they are called generically Goldstone modes. Thus, the existence of a non-vanishing vacuum expectation value for a field, in a theory with a continuous symmetry, appears to lead inevitably to the necessity of having a massless particle, or particles, in the theory.

Can somebody help me understand what is going on? What is zero and what is not? What is so special of the ##|0\rangle## case?
 
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The "mass" is a property of an excitation (particle or quasiparticle) around the vacuum. The vacuum ##|n=0\rangle## itself is neither massive nor massless, that's what makes ##n=0## special. In fact, the energy of the vacuum is defined as zero, ##p_{00}=0##. (If it was not defined so, then the vacuum state would be time dependent in the Schrodinger picture, so the vacuum would not be time-translation invariant.) When we talk of a massive or massless state, we always assume a state with ##n\neq 0##.

Since ##p_{00}=0##, the ##n=0## does not contribute to the sum, so we can take the sum to be taken over states with ##n\neq 0##. How can this sum be time-independent and yet non-zero? It's only possible if ##p_{n0}=0## for some ##n\neq 0##. Since ##{\bf p}_n=0## (due to the ##\delta##-function) we have ##p_{n0}=M_n##, so there must be some excited states ##n\neq 0## with ##M_n=0##. That's the Goldstone theorem.
 
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Demystifier said:
The "mass" is a property of an excitation (particle or quasiparticle) around the vacuum. The vacuum ##|n=0\rangle## itself is neither massive nor massless, that's what makes ##n=0## special. In fact, the energy of the vacuum is defined as zero, ##p_{00}=0##.
Thank you I was missing that.


Demystifier said:
(If it was not defined so, then the vacuum state would be time dependent in the Schrodinger picture, so the vacuum would not be time-translation invariant.) When we talk of a massive or massless state, we always assume a state with ##n\neq 0##.

Since ##p_{00}=0##, the ##n=0## does not contribute to the sum, so we can take the sum to be taken over states with ##n\neq 0##. How can this sum be time-independent and yet non-zero? It's only possible if ##p_{n0}=0## for some ##n\neq 0##. Since ##{\bf p}_n=0## (due to the ##\delta##-function) we have ##p_{n0}=M_n##, so there must be some excited states ##n\neq 0## with ##M_n=0##. That's the Goldstone theorem.
But what is this condition of that $$\langle n | J |0\rangle\neq 0$$ why is it important in this derivation?
 
pines-demon said:
But what is this condition of that $$\langle n | J |0\rangle\neq 0$$ why is it important in this derivation?
If it was zero for all ##n##, then the entire sum would be zero, so the symmetry would not be (spontaneously) broken. The Goldstone theorem says that massless excitations appear when the symmetry is broken.
 
Demystifier said:
If it was zero for all ##n##, then the entire sum would be zero, so the symmetry would not be (spontaneously) broken. The Goldstone theorem says that massless excitations appear when the symmetry is broken.
Gotcha, I think I was looking at it as if it was more mysterious than what it is. Thanks
 
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