Intended Audience: Mathematicians, physicists, graduate students, and advanced undergraduates.

In this video, mathematician Tim Nguyen gives an informative overview of the perturbative approach to path integrals and explains his work on the features of path integral manipulations.

Tim is a mathematician at Michigan State University who is as much motivated by the study of physics as he is by mathematics. This video describes the underlying mathematical structure of quantum field theory in his work. It is based upon his paper on Wick expansions, The Perturbative Approach to Path Integrals: a Succinct Mathematical Treatment, which was published in J. Math. Phys. 57, 092301 (2016). Visit Tim’s website to learn about his research.

You can find out more about the topics in his paper following these links:

Wick’s Theorem (The video is far clearer than this wiki article.)

The video and paper are aimed at mathematicians, physicists, graduate students and undergraduates. However, advanced high school students may still enjoy looking at the paper to see what advanced mathematics and physics look like.

Hey! You may want to subscribe to my YouTube channel because physicists divide infinity by infinity all the time and are somehow not bothered by it.

Mathematically, if a function (e.g. an observable arising from an experiment) has an asymptotic expansion, this expansion by its very nature approximates the function (the lack of convergence of the terms of the series doesn’t affect this property). But I believe you’re asking a more (interesting) philosophical issue, which is why starting from a formal series (without the function which it is purpotedly asymptotic to), we get good corroboration with real-world experiments. For that, nobody knows the answer (since nonperturbative QFT is still not well understood mathematically)! However, this did inspire my followup investigation on Yang-Mills theory, which in a sense was motivated by this question. I also made a video on this subject and explain these philosophical considerations, which you can find at my YouTube channel:

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For those interested, I animated these videos using the software Videoscribe.

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Why do asymptotic expansions give us such accurate predictions despite the fact that they don’t even converge?

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Anon:

Mathematically, if a function (e.g. an observable arising from an experiment) has an asymptotic expansion, this expansion by its very nature approximates the function (the lack of convergence of the terms of the series doesn’t affect this property). But I believe you’re asking a more (interesting) philosophical issue, which is why starting from a formal series (without the function which it is purpotedly asymptotic to), we get good corroboration with real-world experiments. For that, nobody knows the answer (since nonperturbative QFT is still not well understood mathematically)! However, this did inspire my followup investigation on Yang-Mills theory, which in a sense was motivated by this question. I also made a video on this subject and explain these philosophical considerations, which you can find at my YouTube channel:

https://www.youtube.com/channel/UC17MwrZHP2DyQwoMfPYJ6gg

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