Scattering Amplitude & Unitarity

In Srednicki's Quantum Field Theory chapter 11, the probability of a $$2 \to n$$ scattering process is calculated to be

$$P = \frac{|\left|^2}{\left\left} = \frac{(2\pi)^4 \delta\left(k_1+k_2-\sum {k'}_i\right) T}{2E_1 2E_2 V} |\mathcal{T}|^2\prod_{i'} \widetilde{dk}_i'.\tag{11.11+20}$$

He then proceeds to divide this by $$T$$ to obtain a probability per unit time, before yielding an expression for the differential cross section. However, the factor of $$T$$ is troubling me: as $$T\rightarrow \infty$$, $$P$$ is going to exceed $$1$$ at some point, isn't it? Does this have to do with dealing with plane waves instead of wave packets? I am aware that this problem also shows up when one uses Fermi's golden rule to calculate decay rates, but I guess I never thought about it too seriously then.

In chapter 11 of Srednicki, take note of equation 11.14: $$(2 \pi)^4\delta ^4(0) = \int d^4 x e^{i0\cdot x} = VT,$$ where $$V$$ is total volume of space (which is cancelled by another $$V$$ that appears in the denominator to $$P$$), and $$T$$ is total elapsed time.

Thus $$P \propto T$$, so you must divide $$P$$ by $$T$$ to get a well-defined general expression for probability per unit time that is independent of the interval.

• Thanks for your reply. I understand how T shows up and how it is divided to get a probability per unit time. It's that this expression for probability is confusing me - it is still a probability and should be bounded by 1, right? And even the dividing by T to get a probability per time sounds a little dubious to me too, if there's a 80% chance of something happening within two days, we can't simply say said event has a 40% chance of happening each day, can we?
– Mike
Feb 18 at 5:11
• Actually you can say that an 80% probability of something happening in two days is equivalent to a 40% probability of something happening in one day or the next because 0.4 + 0.4 = 0.8. This is assuming the events are independent. P(a or b) = P(a) + P(b). Furthermore, the initial expression is the probability that something will happen during interval T, so dividing by T gives the probability of that event happening per unit time. Feb 18 at 6:20
• Then that event happening in three days would have a probability of 120%?
– Mike
Feb 18 at 11:01