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S-matrix unitary?

  1. May 30, 2013 #1
    Take a [tex]\lambda \phi^4[/tex] 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?
     
  2. jcsd
  3. May 30, 2013 #2
    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)##.
     
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