I'm having difficulty understanding the formula of decay rate. I have read a few other sources but it raises more questions as I was trying to follow their derivation. So, here I present the method from my lecture notes.
To derive the decay rate formula, we have the transition rate per unit volume: $$ w_{fi} = (2\pi)^4 \delta^{(4)}(P_f - P_i) |M|^2, $$ where $P_i,\;P_f$ are the total 4-momentum of initial-, final-state particles respectively, and $M$ the Feynman amplitude.
Also, we obtain the density of final states $$ \frac{d^3p}{(2\pi)^32\omega(\vec{p})} $$ from the completeness relation. Also, a density of $2\omega(\vec{p})$ particles per unit volume is obtained from the normalization of one-particle state.
The decay rate for a single particle decaying into $n$ particles is given by: $$ d\Gamma = (2\pi)^4 \delta^{(4)}( p - \sum p_f ) \frac{1}{2\omega(\vec{p})} \prod_f \frac{d^3 p_f}{(2\pi)^2 2E_f} |M|^2:= \frac{1}{2E(\vec{p})}|M|^2 (dPS)_n, $$ where $(dPS)_n = (2\pi)^4 \delta^{(4)}( p - \sum p_f )\prod_f \frac{d^3 p_f}{(2\pi)^2 2E_f} $ is the n-particle phase space.
As the decay rate is the transition rate per decaying particle, I assume the factor of $\frac{1}{2\omega(\vec{p})}$ is to reduce the transition rate per unit volume $w_{fi}$ to transition rate per particle. But how do we interpret the production of densities of final states?
Also, there would be a product of $n$ final state densities, would be dimension analysis be satisfied? And what's the physical meaning of the n-particle phase state $(dPS)_n$?
