Fourier transform of fermionic creation/annihilation operator How should I picture the Fourier transform of a fermionic creation (annihilation) operator acting on a site of a periodic, say one-dimensional, lattice? I mean, in a real-space picture, what are the operators $\hat{c}_{k}^{(\dagger)}$ occurring in the Fourier transform 
$$\hat{c}_{r}^{(\dagger)}=\frac{1}{\sqrt{N}}\sum_{k}e^{-(+)ikr}\hat{c}^{(\dagger)}_{k}$$ where the index $r\in\{1,2,..,N\}$ labels a specific lattice site on the chain, "creating" ("annihilating")? 
 A: In the example you posted, $c^{\dagger}_r$ means that you are creating a particle at one dimensional site $r$. Similarly, $c^{\dagger}_k$ means that you are creating a particle with a momentum $k$. If you want to visualise how Fourier transform actually work, we have to go back to Heisenberg Uncertainty Principle. By creating a particle at location $r$, you know the position of the particle with very high certainty, therefore the certainty of knowing the momentum is very low. Therefore, it is a liner combination of operator in momentum space which span from $-\infty$ to $\infty$ and vice-versa.
The Fourier Transformation of an operator or any quantity is useful when you have periodic constraint. If you have a periodic condition on real space i.e. $c^{\dagger}_{r+R}=c^{\dagger}_r$, then you can verify by taking Fourier transform both sides that $k=\frac{2m\pi}{R}$, where m is any integer. This quantizes the momentum space and we get our beautiful Energy momentum spectrum, Brillouin zone etc. 
