pines-demon
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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:
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.
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?
$$Q=\int \mathrm d^3 x\, J_0 (x)$$
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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