The Institute for Arts and Ideas had reached out to me awhile back to write an article on gauge theory which ended up morphing into an article on mathematical rigor and quantum field theory. This was a topic I spent a lot of my time thinking about during my postdoc years in academia (not so wise from a career perspective), which led to my paper on formulating rigorously the perturbative methods used in quantum field theory (i.e. those involving Feynman diagrams):
The Perturbative Approach to Path Integrals: A Succinct Mathematical Treatment
The high-level summary of the paper is that while standard physics textbook treatments of Feynman diagrams are problematic because they are based on some putative, albeit mathematically ill-defined path integral, there is a mathematically proper way of formulating such manipulations without needing the existence of the path integral. Moreover, the formulation presented (the “Wick expansion”, which is a slight but important and subtle reformulation of the standard approach) is minimal and natural in that it arises from basic considerations of ordinary finite-dimensional integrals.
I wrote this paper because of my frustration with, on the one hand physicist treatments that throw rigor out the window as soon as they start manipulating path integrals, and on the other hand, my fellow mathematicians who often use excessive abstraction when recasting intuitive and simple ideas from physics. An interesting anecdote is that when I submitted the paper to The Journal of Mathematical Physics back in 2015, the one reviewer (obviously a physicist) wrote that the results were “well-known” and promptly rejected it. Given the reviewer was clearly misguided, I appealed and the subsequent reviewer noted that “It is not so clear to what extent this is true” in regards to the previous claim, and ultimately recommended the paper to be accepted (whew!). One takeaway from the situation is that quantum field theory, which spans many communities having differing intuitions, backgrounds. and standards of rigor, is a subject that is difficult for practitioners to assess what is firmly grounded knowledge (especially a rigorous mathematical proof). More to the point, physicists will often have intuitions and folklore knowledge about a subject matter, but they are not always successful in distinguishing between what they have always found to be true and what has been proven to be true.
Unfortunately, my paper has not received the attention I think it should have despite, in my opinion, it presenting the correct treatment of perturbative path integrals to the point that physics textbooks on quantum field theory should be revised accordingly. Though based on what I wrote above, it’s not too surprising the work has been overlooked: redoing and rethinking computations rigorously without changing the final answer is not the priority of most working physicists. In an attempt to get my ideas out there and as a last ditch effort to get the attention of hiring committees in my final upcoming year as a postdoc, I made a whiteboard animation video on the above paper (the first YouTube video on my channel!). Though it made zero (possibly negative) impact on my follow up tenure-track application for the 2015-2016 cycle, at the very least, the professor who I had worked with at Michigan State at the time and had been skeptical of my work seemed to have been finally convinced of its purpose after watching the video.
In any case, my article with the IAI was my first official philosophical publication, written for a lay but scientifically-minded audience. It was a fun opportunity for me to think within a sphere different from the ones I normally operate in and hopefully I’ll get more chances in the future.
Timothy Nguyen 