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Question about interacting fields

  1. Jul 3, 2011 #1
    I had some questions about the equations that David Tong derives in his lecture notes here:

    He gets defines the time evolution operator according to equations 3.20 and 3.23 as
    U(t, t[itex]_{0}[/itex]) = T*exp(-i* [itex]\int[/itex][H[itex]_{I}[/itex](t') * dt']) = 1 - i*[itex]\int[/itex][dt' * H[itex]_{I}[/itex](t')] + ...

    According to eq 3.26, in the limit as t approaches +/- infinity, U is the same as the S-matrix:

    lim <f|U(t[itex]_{+}[/itex], t[itex]_{-}[/itex])|i> = <f|S|i>

    but according to eq3.25, the Hamiltonian for the Yukawa theory is:
    H[itex]_{I}[/itex] = g * [itex]\int[/itex][d[itex]^{3}[/itex]x * [itex]\psi^{dagger}[/itex] [itex]\psi[/itex] [itex]\varphi[/itex]]

    But according to the series expansion formula for U above, and plugging in the Hamiltonian into it, the series expansion for U should be:

    U(t, t[itex]_{O}[/itex]) = 1 - i*[itex]\int[/itex][d[itex]^{4}[/itex]x * g*[itex]\psi^{dagger}[/itex] [itex]\psi[/itex] [itex]\varphi[/itex]] + ...

    I see that the leading term in g just has the [itex]\psi^{dagger}[/itex] [itex]\psi[/itex] [itex]\varphi[/itex] in it. But in eq 3.46, he calculates <f| S-1| i>, which is SECOND order in g^2. Now it contains two terms each of [itex]\psi^{dagger}[/itex] [itex]\psi[/itex] [itex]\varphi[/itex]

    My question is, when he says 'S-1', does he mean "S without the 1st order term" ? Or do he mean "S without the number '1' "? If the former, that would make sense. But the latter makes no sense at all
    Last edited: Jul 3, 2011
  2. jcsd
  3. Jul 3, 2011 #2
    I figured it out. We have the -1 because it corresponds to when |i> = |f>, which gives <i|f>=1, which we want to exclude
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