Probability Greater than 1 in λφ^4 Theory

[geoduck]

Take a \lambda \phi^4 theory. To first order in λ, the 2x2 scattering amplitude is:

iM=-iλ

So the amplitude <f|S|i> is then <f|(1+iM)|i>=<f|i>+iM<f|i>.

Letting f=i, the probability is greater than 1! It is equal to the norm |1+iM| which is sqrt[1^2+λ^2].

How is it that two particles in the state |i> have a probability greater than 1 of being in the same state |i> after scattering?

 

[The_Duck]

When you do a perturbative calculation to some order in $\lambda$, you can only expect unitarity to hold up to that order in $\lambda$. $\sqrt{1 + \lambda^2} = 1 + O(\lambda^2)$, so the unitarity violation is higher order in $\lambda$ than the accuracy of the first-order perturbative calculation, which is only accurate up to $O(\lambda)$.