Quantum Field Theory in position space instead of momentum space?

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.

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.