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  Problem with understanding the second proof of Goldstone theorem in Weinberg's book

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I have a few issues in understanding Weinberg's proof of the Goldstone boson theorem (page 170, vol II):

Equation 19.2.18: why is the $(2\pi)^{-3}$ instead of $(2\pi)^{-4}$, isn't this a term originating from the Fourier transform?

Equation 19.2.19 and 19.2.20 where is the $\delta^4(p-p_N)$ coming from, isn't he only introducing an identity term by summing over all the states?

Equation 19.2.22: why is $\theta(p^0)$ necessary for the states to be physical? Is it to have them be in the future instead of the past?

In the equation between 19.2.31 and 19.2.32: I don't understand where has the $\theta(p_0)$ gone, and why has appeared the term $\sqrt{p^2+\mu^2}$.  My guess for the square root is to come from the 0 component of the derivative but then I don't understand why is it $\sqrt{p^2+\mu^2}$ instead of  $\sqrt{p^2}$, I thought that the  $\mu$ was just a change of variable for $p$ but this doesn't seem to be the case.

Any help is deeply appreciated

asked Mar 6 in Theoretical Physics by HelpIsNeeded [ no revision ]

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