On-shell limit and contact terms in chapter 67

[LAHLH]

Hi,

Just have a few questions about this chapter if anyone here is familiar with it...

1) Does anyone know why the $$k^2_i \to -m^2$$ limit is imposed in (67.4), is it just because he wants external particles to be physical and hence on-shell? I derived this equation from the one directly above by integrating by parts but can't see where the limit comes out here but is not present there..

2) On the next page where he supposes a contact term with a factor $$\delta^4 (x_1-x_2)$$ why does this Fourier transform to a function of $$k_1+k_2$$ independent of $$k_1-k_2$$. I know this is a rather basic question, but I can't seem to make it work out, so would be grateful if someone could show me. Secondly why does this exclude it from being of the form of singular term in (67.7)?

thanks a lot for any help

 

[Avodyne]

1) k^2 = -m^2 already in 67.3; see ch. 5.

2) Integrate e^(i k1 x1)e^(i k2 x2) delta^4(x1-x2) F(x1,x2) over x2, where F(x1,x2) is any smooth function. The result is e^(i(k1+k2)x1) F(x1,x1). Integrating over x1 yields a function of k1+k2.