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Paul BaldufCurrently, I'm working on a problem in <a href="https://mathstodon.xyz/tags/quantum" class="mention hashtag" rel="nofollow noopener" target="_blank">#quantum</a> field theory where we use <a href="https://mathstodon.xyz/tags/FeynmanIntegral" class="mention hashtag" rel="nofollow noopener" target="_blank">#FeynmanIntegral</a> 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. <a href="https://mathstodon.xyz/tags/physics" class="mention hashtag" rel="nofollow noopener" target="_blank">#physics</a> <a href="https://mathstodon.xyz/tags/research" class="mention hashtag" rel="nofollow noopener" target="_blank">#research</a>

Paul BaldufAt the <a href="https://mathstodon.xyz/tags/CAP" class="mention hashtag" rel="nofollow noopener" target="_blank">#CAP</a> <a href="https://mathstodon.xyz/tags/Physics" class="mention hashtag" rel="nofollow noopener" target="_blank">#Physics</a> Congress and the Theory Canada meeting, I gave two talks about the statistical distribution of <a href="https://mathstodon.xyz/tags/FeynmanIntegral" class="mention hashtag" rel="nofollow noopener" target="_blank">#FeynmanIntegral</a> s and how their correlations can be used for efficient sampling at high loop order. The slides are now available from my website! <a href="https://paulbalduf.com/research/statistics-periods/" rel="nofollow noopener" translate="no" target="_blank">https://paulbalduf.com/research/statistics-periods/</a>

Paul BaldufIn <a href="https://mathstodon.xyz/tags/QuantumFieldTheory" class="mention hashtag" rel="nofollow noopener" target="_blank">#QuantumFieldTheory</a>, scattering amplitudes can be computed as sums of (very many) <a href="https://mathstodon.xyz/tags/FeynmanIntegral" class="mention hashtag" rel="nofollow noopener" target="_blank">#FeynmanIntegral</a> s. They contribute differently much, with most integrals contributing near the average (scaled to 1.0 in the plots), but a "long tail" of integrals that are larger by a significant factor. We looked at patterns in these distributions, and one particularly striking one is that if instead of the Feynman integral P itself, you consider 1 divided by root of P, the distribution is almost Gaussian! To my knowledge, this is the first time anything like this has been observed. We only looked at one quantum field theory, the "phi^4 theory in 4 dimensions". It would be interesting to see if this is coincidence for this particular theory and class of Feynman integrals, or if it persists universally. More background and relevant papers at <a href="https://paulbalduf.com/research/statistics-periods/" rel="nofollow noopener" translate="no" target="_blank">https://paulbalduf.com/research/statistics-periods/</a> <a href="https://mathstodon.xyz/tags/quantum" class="mention hashtag" rel="nofollow noopener" target="_blank">#quantum</a> <a href="https://mathstodon.xyz/tags/physics" class="mention hashtag" rel="nofollow noopener" target="_blank">#physics</a> <a href="https://mathstodon.xyz/tags/statistics" class="mention hashtag" rel="nofollow noopener" target="_blank">#statistics</a>

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