Conventions on the Fourier transform

You will recall there are various conventions adopted with regard to factors of \(2\pi\) in Fourier transforms. To my mind the simplest to remember is

\[\begin{aligned} \tilde{f}\left(\boldsymbol{k}\right)= & \int\text{d}^{D}\boldsymbol{x}\exp\left(-2\pi i\boldsymbol{k\cdot x}\right)f\left(\boldsymbol{x}\right)\\ f\left(\boldsymbol{x}\right)= & \int\text{d}^{D}\boldsymbol{k}\exp\left(2\pi i\boldsymbol{k\cdot x}\right)\tilde{f}\left(\boldsymbol{k}\right). \end{aligned}\]

From here you can rescale variables as you like, for example:

\[\boldsymbol{k}\rightarrow\boldsymbol{k}/2\pi\]

gives

\[\begin{aligned} \tilde{f}\left(\boldsymbol{k}\right)= & \int\text{d}^{D}\boldsymbol{x}\exp\left(-i\boldsymbol{k\cdot x}\right)f\left(\boldsymbol{x}\right)\\ f\left(\boldsymbol{x}\right)= & \int\frac{\text{d}^{D}\boldsymbol{k}}{\left(2\pi\right)^{D}}\exp\left(i\boldsymbol{k\cdot x}\right)\tilde{f}\left(\boldsymbol{k}\right) \end{aligned}\]

(where \(\tilde{f}\left(\boldsymbol{k}\right)\) has a rescaled argument).

Since I will use the \(\left(+---\right)\) signature, spacetime Fourier transforms in \(D+1\) dimensions must have the opposite signs in the exponents:

\[\begin{aligned} \tilde{f}\left(k\right)= & \int\text{d}^{D+1}x\exp\left(2\pi ik_{\mu}x^{\mu}\right)f\left(x\right)\\ f\left(x\right)= & \int\text{d}^{D+1}k\exp\left(-2\pi ik_{\mu}x^{\mu}\right)\tilde{f}\left(k\right). \end{aligned}\]

For the discrete Fourier transform I will use

\[\begin{aligned} \tilde{f}{}_{k} & =\sum^{N}_{n=1}\exp\left(-2\pi ikn/N\right)f_{n}\\ f_{n} & =\frac{1}{N}\sum^{N}_{k=1}\exp\left(2\pi ikn/N\right)\tilde{f}{}_{k}. \end{aligned}\]