I'm studying dispersion relations applied as alternative method to perturbation theory from Weinberg's book (Vol.1)(adsbygoogle = window.adsbygoogle || []).push({});

Let's consider the forward scattering in the lab frame of a massless boson of any spin on an arbitrary target ##\alpha## of mass ##m_\alpha>0## and ##\vec{p}_\alpha = 0##.

Weinberg writes "By a repeated use of Eq(10.3.4) or the LSZ theorem, the S-Matrix element here is

## S = \frac{1}{(2\pi)^3\sqrt{4k^0k'^0}|N|^2}lim_{k^2\rightarrow 0}lim_{k'^2\rightarrow 0} \int d^4x d^4 y e^{-ik'y + ikx}(i\Box_y)(i\Box_x)<\alpha | T\left( A^\dagger(y) A(x) \right) |\alpha>, ##

where ##<VAC| A(x) |k> = (2\pi)^{-3/2} (2\omega)^{-1/2}N e^{ikx}##.

I have several questions on this formula:

1. Since he discussed the Green function for a ##T##-product of generals operators; why did he choose ##A^\dagger## and ##A## in this case?

2. Where does the factor ##|N|^2## in the denominator come from? Why in the denominator?

3. Why does he talk about LSZ theorem? This S-matrix element seems the usual I get if I take the amputated Green function and I put the in/out coming particles on-shell, expect for the ##|N|^2## factor (I think the answer relies on this renormalization factor, but I don't understand why)

4. Once he choose this renormalization ##<VAC| A_l(x)|\vec{q}_1,\sigma> =(2\pi)^{-3/2}N e^{ikx} u_l(q_1,\sigma)## (eq. 10.3.3). Why now can he change the renormalization adding the ##(2\omega)^{-1/2}## factor?

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# S-matrix element for forward scattering and amputed green fu

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