# Feynman diagrams as topology

When we talk about Feynman diagrams we know they are tools to make calculations easier and more intuitive. Moreover, it's said that they are "topological" representations of the interactions.

But, what does it mean that they are topological objects? Topolgy is a brach of Mathematics that deals with spaces where you can define distances, limits, norms..., i.e., where you can define a notion of distance between its their elements. Nevertheless, I don't see how this is related to the diagrams.

• Not the answer this question deserves, but for what it's worth: the amplitudes associated with Feynman diagrams depend on topological properties like the number of vertices, number of edges, number of loops, etc., and not on geometrical properties like the length of the edges or their relative angles to each other. (In topology you do not necessarily have a notion of distance or angle, just an idea of how the parts of a space "fit together.") – d_b Oct 18 at 3:52
• Topolgy is a brach of Mathematics that deals with spaces where you can define distances, limits, norms..., i.e., where you can define a notion of distance between its their elements. Not really. What you're talking about is a metric space. If you have a metric, it induces a topology. If you pick up a topology textbook and look at the examples, very few are metric spaces. – Ben Crowell Oct 18 at 4:09
• Because we don't now where and when the interaction really occurred. Feynmann diagram just show how they interact. We don't know actual distances, norms between the events to calculate, and we just need to know their shapes. – ChoMedit Oct 18 at 4:17
• @d_b Ok, but accordingly to Feynman rules, when you try to compute the scattering amplitude of some process $i \rightarrow f$ you have to sum over the diagrams that are topologically inequivalent; so how do you know if 2 diagrams are different in that sense? – Vicky Oct 24 at 23:40
• One way to think about it is that diagrams are topologically equivalent if we can deform one into another without breaking any edges. This is similar to how two circuit diagrams represent the same electric circuit if we can deform them into each other without breaking any wires. – d_b Oct 25 at 22:50