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Could someone help me understand the reduced $T$-matrix mentioned in Itzykson and Zuber, eq.

$$\langle{f}| T|p_1p_2\rangle=(2\pi)^4\delta^4(P_f-p_1-p_2)\langle f|\mathcal{T}|p_1p_2\rangle. \tag{5-7}$$

How is it possible to extract the delta function?

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One way to say it is that total 4-momentum conservation $$ (P_f^{\mu}-p_1^{\mu}-p_2^{\mu})\langle{f}| T|p_1p_2\rangle~=~0, \qquad \mu~\in~\{0,1,2,3\}, $$ implies that the $T$-matrix element $$\langle{f}| T|p_1p_2\rangle\quad\propto\quad\delta^4(P_f-p_1-p_2)$$ is proportional to a 4D Dirac delta distribution.

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  • $\begingroup$ Thanks for your answer! Ok, that makes sense. It doesnt explain the $2\pi$-terms however. Do you know how come this matrix is used when calculating scattering? As opposed to the off-shell one. $\endgroup$ – Anna Konstantinova Jun 15 at 10:33
  • $\begingroup$ Think of $(2\pi)^4$ as a normalization convention. $\endgroup$ – Qmechanic Jun 15 at 10:36

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