Paul Balduf
Currently, I'm working on a problem in #quantum field theory where we use #FeynmanIntegral s. These integrals are depicted by graphs, and they can be divergent when a graph has too many edges for a given number of vertices. The task is to identify all subgraphs that are divergent. This is a coproduct: It produces multiple terms, and each term is a list of 2 elements. The first element is one or multiple divergent subgraphs, and the second element is the remainder. It is surprising how many terms the coproduct has even for small graphs. For my example, even if the red graph is rather small, there are already 15 combinations of divergent subgraphs. To compute a physically sensible result, one needs to sum over all original graphs, and subtract all these combinations of subgraphs. #physics #research
Feb 8, 2025 at 12:23amWeb
Show threadGeoff 🏴󠁧󠁢󠁳󠁣󠁴󠁿Feb 8, 2025
@paulbalduf Do the nodes literally represent a spacetime event, lines are possible paths etc? Or something more abstract?
Show threadJohn Carlos BaezFeb 8, 2025

@_thegeoff - you indeed can think of nodes as spacetime events (points in spacetime), but then the lines just say *that* a particle went from one event to another, not the actual path.

Conversely you can think of the lines as literally representing the energy-momenta of particles, but then the nodes just say *that* particles with some energy-momenta interacted, not when or where they did it.

This is connected to the uncertainty principle. We have a "position picture" of what's going on (my first paragraph) and a "momentum picture" (my second paragraph), but we don't get to use both simultaneously.

@paulbalduf

Show threadGeoff 🏴󠁧󠁢󠁳󠁣󠁴󠁿Feb 8, 2025
@johncarlosbaez @paulbalduf The idea I have in my head, from a popsci readìng of Feynman Integrals, is accounting for infinite paths, so I'm imaging the graphs as a cascading series of possibilities?
Show threadJohn Carlos BaezFeb 8, 2025
@_thegeoff @paulbalduf - yes, each graph represents a processes involving as many particles as there are edges in the graph, and in reality *all* these processes (infinitely many) occur, with various different probabilities.