Proof of Goldstone's theorem

Question

I am reading Matthew Schwartz's wonderful QFT book, but I have stumbled in my attempts to understand his rather brief proof of Goldstone's theorem. The preliminary information is that $$HQ|\Omega\rangle= E_0Q|\Omega\rangle,\tag{28.7}$$ where $Q$ is the conserved charge corresponding to the spontaneously broken symmetry $$Q\equiv\int d^3x\;j^0(x).\tag{28.5}$$ This is fine. However, from this, he claims that one can create states $$|\pi(\vec{\bf{p}})\rangle\equiv\frac{-2i}{F}\int d^3x e^{-i\,\vec{\bf{p}}\cdot\vec{\bf{x}}}j^0(x)|\Omega\rangle\tag{28.8}$$ that have energy $E(\vec{\bf{p}}) + E_0$. If this is true, then it is clear that the state must obey a massless dispersion relation. However, I do not understand how one determines the energy to be $E(\vec{\bf{p}}) +E_0$.

I have seen a similar proof on p. 228 in Zee's book, but there he shows in footnote 3 that $$\hat{P}^i|\pi(p)\rangle = p^i|\pi(p)\rangle,$$ which is not quite the same although it has the same implication. Any ideas?

Answer 1

I think you got the sign of the exponential in Schwartz's (28.8) wrong here, which I reverse, but I am cavalier with signs and factors, as I assume you just want to see the principle (the forest, not the trees).

The key point is that the SSB current is basically always of the form $j_\mu(x)\propto F \partial_\mu \pi (x) +...$ where the omitted terms are of higher order in the fields (and so do not contribute to the particle state defined); and, of course, group indices have been omitted as well.

This is codified in the appellation "Nambu-Goldstone nonlinear realization of the symmetry". This is the only way to have the v.e.v. of the transform of this Goldstone field not vanish, while the v.e.v.s of all the fields themselves vanish--after shift redefinitions. (See this answer .) It is illustrated later on in (28.13).

The Fourier transform of the Goldstone field π is $$ \pi(\vec{\bf{p}}) = \frac{-2i}{F}\int d^3x ~e^{i\,\vec{\bf{p}}\cdot\vec{\bf{x}}}j^0(x), $$
with $$ [H,\pi(\vec{\bf{p}})] = E(\vec{\bf{p}}) ~ \pi(\vec{\bf{p}}) . $$ The SSB charge Q is essentially the space integral of the canonical momentum of π, so that $\langle \Omega |[Q,\int d^3 p ~ \pi(\vec{\bf{p}})]|\Omega\rangle \propto F\neq 0 $ , cf. his (28.9).

As a consequence, decompressing your text's one liner, $$H|\pi(\vec{\bf{p}})\rangle= ( [H,\pi(\vec{\bf{p}})] + \pi(\vec{\bf{p}}) H ) ~|\Omega\rangle = ( E(\vec{\bf{p}}) + E_0 )\pi(\vec{\bf{p}})|\Omega\rangle= ( E(\vec{\bf{p}}) + E_0 )|\pi(\vec{\bf{p}})\rangle , $$ so $E(0)\to 0$ .

Still, for the more conventional general proof of the theorem, bypassing your perceptual snag, consider Kibble's summary.

Answer 2

I guess this follows from the Heisenberg equation of motion for $\pi$: $$i\partial_t \pi = [\pi, H] $$ Calculating it, I found this works if

1) his sign convention is such that the coefficient of E in the exponent is -i so that $i\partial_t$ brings down a factor -E

2) $H|\Omega\rangle =0$

3)Perhaps I'm mistaken, but it looks as if your equation is missing a factor of $Q$ so that $$HQ|\Omega\rangle = E_0 Q|\Omega\rangle$$ -otherwise I have a hard time getting energy eigenstates here.