How to Fourier Invert a Plane Wave?

[waht]

Homework Statement

Ok so got a solution to the Klein-Gordon equation and need to solve for a(k)

$$\varphi(x) = \int \tilde{dk} \left[ a(\bold{k}) e^{ikx} + a^{\dagger}(\bold{k} ) e^{-ikx} \right]$$

$$\tilde{dk} = \frac{d^{3}k}{(2 \pi)^{3} 2 \omega}$$

The way it's done in Srednicki p.26 has me confused when taking the Fourier transform of $\varphi$

$$\int d^3x e^{-ikx} \varphi(x) = \frac{1}{2\omega} a(\bold{k}) + \frac{1}{2\omega} e^{2i\omega t} a^{\dagger}(\bold{-k} )$$

Homework Equations

$$kx = \bold{k} \cdot \bold{x} - \omega t$$

The Attempt at a Solution

$$\int d^3x e^{-ikx} \varphi(x) =\int d^3x e^{-ikx} \int \tilde{dk} \left[ a(\bold{k}) e^{ikx} + a^{\dagger}(\bold{k} ) e^{-ikx} \right]$$


$$= \int d^3x \int \tilde{dk} a(\bold{k}) + \int d^3x \int \tilde{dk} e^{-2kx} a^{\dagger}(\bold{k} )$$


so the problem is how do these integrals with respect to dx and dk disappear?

 

[tiny-tim]

dummy!

Hi what!

It's a different k …

you have ∫∫ e^ikx e^ik'x … for a dummy k'