Quantum Field Theory in position space instead of momentum space? What are the reasons why we usually treat Quantum Field Theory in momentum space instead of position space? Are the computations (e.g. of Feynman diagrams) generally easier and are there other advantages of this formulation?
 A: I may add that the expressions for propagators $G(x,x^{\prime})\propto \int \frac{\mathrm{d}^D k}{k^2+m^2} \mathrm{e}^{-\mathrm{i}k (x-x^{\prime})}$ are quite cumbersome in the position space, and have a plenty of singularities. See, for example, this Wiki article.
A: The most important reasons we use momentum space Feynman rules are:


*

*In position space, the Feynman rules generate convolutions of propagators. Because of the convolution theorem, the momentum space rules generate products of propagators, which are clearly easier to handle.

*Moreover, in position space you have an integral for each vertex, while on momentum space you have one integral per loop, and in a general diagram there are many more vertices than loops, thus making the momentum space rules easier to use.

*What's more, the LSZ theorem in momentum space is trivial to implement: we just drop the propagators on the external lines; in position you'd have to evaluate some exponential integrals (which are straightforward, but cumbersome).

*Finally, the renormalisation conditions are naturally imposed in momentum space, and therefore you want the diagrams in momentum space.
