In QM the probability is violated when:
$\int \rho(\vec r,t)\vec dr\neq 1$ for a quantum mechanical system in an arbitrary state $\psi(\vec r,t)$.
In this case we know that $\psi(\vec r,t)$ is the probability amplitude, $|\psi(\vec r,t)|^2$ is the probability density function.
For unitarity to hold in this case: $\int \rho(\vec r,t)\vec dr= 1$
But I am confused as to how one defines the condition for unitarity in QFT, when considering the value of the cross section.
For a process in QFT we have for an initial (i) and final states (f):
$S_{fi}=\langle f|S|i\rangle$ the $S$-matrix element/ probability amplitude.
$|S_{fi}=\langle f|S|i\rangle|^2$ is the transition amplitude
$W_{fi}\propto |S_{fi}=\langle f|S|i\rangle|^2$ This is the transition rate.
$\sigma \propto \int |\langle f|S|i\rangle|^2d(LIPS)$
Knowing that the cross section, which is a measure of the probability of a process taking place in collision, has units of Barn. How can we make any claims about unitarity solely based on the range of values or behavior of the cross section.
I will try to describe my confusion, with an example:
Our lecturer described the following process: $\nu_e \bar\nu_e \rightarrow W_L W_L$ where $W_L$ are W-bosons with longitudinal polarization.
We were given that $\sigma \propto \frac{g^4_w s}{m_w^4}$ and that it violates unitarity for $\sqrt{s}>1 TeV$. I can understand that what is happening here is that the cross section increases with the energy. And since we said that the cross section is a measure of the probability of a process taking place, this would mean that as the energy increases the probability increases. But why are we taking this behavior as the probability surpassing 1, and not as probability approaching 1? Additional to this why is unitarity broken past a certain energy value and not from the very moment that as energy increases, the cross section increases hence the probability does. I mean, from the very first moment we see the energy dependency of the cross section, can't we immediately say that unitarity is violated?
