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I was following along Mark Thomson's Modern Particle Physics, and stumble upe the derivation of d$n$ of Fermi golden rule on page 62:

"... For the decay of a particle to a final state consisting of $N$ particles, there are $N-1$ independent momenta in the final state. Thus, the numer of independent states for an N-particle final state is \begin{equation} dn=\prod^{N-1}_{i=1} dn_i = \prod^{N-1}_{i=1} \frac{d^3\textbf{p}_i}{(2\pi)^3} \tag{A}. \end{equation} This can be expressend in a more democratic form including the momentum space volume for the $N$th particle $d^3\text{p}_N$ and using delta-function to impose momentum conservation \begin{equation} dn=\prod^{N-1}_{i=1} \frac{d^3\textbf{p}_i}{(2\pi)^3} \delta^3\left(\textbf{p}_a-\sum^N_{i=1}\textbf{p}_i\right)d^3\textbf{p}_N \tag{B}, \end{equation} where $\textbf{p}_a$ is the momentum of the decaying particle.... (and so on) "

Based on the explanation between the steps, I don't understand why the three-dimensional dirac-delta function appear out of nowhere. I do know that physical significances imposing momentum conservation, but why does it have to take the form of dirac-delta? is there an intermediate steps that I'm missing here?

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After giving some thoughts and times, I finally came to an answer to my own question. I hope this might help other who get the same problem. This come to some sloppy writing and definition of dirac-delta:

\begin{equation} \int d^nx \delta^n(x-a) = 1\\ \end{equation} similiarly, \begin{equation} \int d^3\textbf{p}_N \;\delta^3\left(\textbf{p}_a-\sum^N_{i=1} \textbf{p}_i\right) = 1 \end{equation} Therefore, from (A) \begin{align} dn&=\prod^{N-1}_{i=1} dn_i, \\ &= \prod^{N-1}_{i=1} \frac{d^3\textbf{p}_i}{(2\pi)^3}, \\ &= \prod^{N-1}_{i=1} \frac{d^3\textbf{p}_i}{(2\pi)^3} \int \delta^3\left(\textbf{p}_a-\sum^N_{i=1}\textbf{p}_i\right)d^3\textbf{p}_N. \end{align} can be obtained (B) with slight correction to the book.

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