# Question about interacting fields

1. Jul 3, 2011

### creepypasta13

I had some questions about the equations that David Tong derives in his lecture notes here:
http://www.damtp.cam.ac.uk/user/tong/qft/three.pdf

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

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

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$_{O}$) = 1 - i*$\int$[d$^{4}$x * g*$\psi^{dagger}$ $\psi$ $\varphi$] + ...

I see that the leading term in g just has the $\psi^{dagger}$ $\psi$ $\varphi$ 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 $\psi^{dagger}$ $\psi$ $\varphi$

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. Jul 3, 2011

### creepypasta13

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