# Itzykson and Zuber: Reduced $T$-matrix

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?

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.
• 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. – Anna Konstantinova Jun 15 at 10:33
• Think of $(2\pi)^4$ as a normalization convention. – Qmechanic Jun 15 at 10:36