Аннотация:
We continue the development of a position space approach to equations for Feynman multi-loop integrals. The key idea
of the approach is that unintegrated products of Green functions in position space are still loop integral in momentum
space. The natural place to start are the famous banana diagrams, which we explore in this talk. In position space, these
are just products of n propagators. Firstly, we explain that these functions satisfy an equation of order $2^n$. These should
be compared with Picard-Fuchs equations derived for the momentum space integral. We find that the Fourier transform of
the position space operator contains the Picard-Fuchs one as a rightmost factor. The order of these operators is a special
issue, especially since the order in momentum space is governed by degree in x in position space. For the generic mass
case this factorization pattern is complicated and it seems like the order of the Fourier transformed position space operators
is much bigger than that of the Picard-Fuchs. Furthermore, one may ask what happens if after factorization we take the
Picard-Fuchs operators back into position space. We discover that the result is again factorized, with the rightmost factor
being the original position space equation. We demonstrate how this works in examples and discuss implications for more
sophisticated Feynman integrals.
