- #1

maverick_starstrider

- 1,111

- 6

**"Convergence Factors"**

In all my textbooks I always see these random convergence factors thrown in (+0's or +i*nu or some such) but I have never seen a book that would dirty itself by steeping so low as to explain what they are (I'm looking at you Wen, Bruus and Flensberg, Fetter and Walecka, etc.). I understand they are supposed to be some ad hoc infinitesimal added to guarantee some integral converges but can anyone point me to (or provide) an explanation of:

-when they are necessary

-why it is ok to add them on a whim

-what would happen is we didn't add them

-what is the physical meaning of adding them.

If you don't know what I'm talking about I'll give the example of the energy propogator

[itex] G_E(n_b,t_b,n_a,t_a) = -i \langle n_b \vert U(t,t_0) \vert n_a \rangle = -ie^{-i \epsilon_n (t_b - t_a) } \delta_{n_b,n_a} [/itex]

where when we move to frequency space we get

[itex] G_E(n_b,n_a,\omega) = \int_0^{\infty} dt G_E(n_b,t_a+t,n_a,t_a)e^{it\omega - 0^+ t} = \frac{1}{\omega - \epsilon_{n_a}+i 0^+} \delta_{n_b,n_a} [/itex]

I dont get the [itex]0^+[/itex].

Thanks in advance