Peskin and Schroeder, equation 2.51

[Chris.X]

Hi

I am struggling to justify

$$D(x-y) \approx e^{-i m t}$$ as $$t \rightarrow \infty$$

from

$$\int dE \sqrt{E^2-m^2} e^{-i E t}$$.

I thought I might get some insight from discretizing, as

$$e^{-i m t} \sum_{n=0}^{\infty} \epsilon \sqrt{ n \epsilon ( 2 m + n \epsilon ) } e^{-i n \epsilon t}$$

but I don't understand how to approximate or take the limit of the sum.

I also tried to work backwards from

$$G : (E^2-m^2)^{3/2} e^{-i E t}$$

and replacing extra E's with d/dt's and ended up with a differential equation
for the result,

$$G = i [ t (d^2/dt^2 + m^2) + 3 d/dt ] D(x-y)$$

but I am having trouble putting the pieces together to solve it.

Assistance would be greatly appreciated.

 

[nickthequick]

You have to use the method of stationary phase

http://en.wikipedia.org/wiki/Stationary_phase_approximation