Classes

admin Administator 4 posts

edited 8 years ago

Here are our conventions for Fourier Transforms

In $d$ dimensions,

\begin{aligned} f(\vec{x}) &= \displaystyle{\int\frac{ d^d\vec{p} }{ {(2\pi)}^d } g(\vec{p}) e^{i\vec{p}\cdot\vec{x}}}\\ g(\vec{p}) &= \displaystyle{\int d^d\vec{x} f(\vec{x}) e^{-i \vec{p}\cdot \vec{x}}} \end{aligned}

Normalization

\int d^d\vec{x} |f(\vec{x})|^2 = 1\quad \xLeftrightarrow{\qquad}\quad \int \frac{d^d\vec{p}}{{(2\pi)}^d} |g(\vec{p})|^2 = 1

The $\delta$ -function is the Fourier-transform of the constant function, $1$ ,

\begin{split} \delta^{(d)}(\vec{x}) &= \displaystyle{\int \frac{d^d\vec{p}}{{(2\pi)}^d} e^{i\vec{p}\cdot\vec{x}}}\\ {(2\pi)}^d \delta^{(d)}(\vec{p}) &= \displaystyle{\int d^d\vec{x} e^{-i\vec{p}\cdot\vec{x}}} \end{split}

Fourier Transform Conventions