Graph Theory for Quantum Field Theory, Renormalization, Space Navigation, and Prisoner Rehabilitation
Congratulations! You presumably made it through the exceptionally long title of this post. If we take title verbosity to be positively correlated with post import, then the length here is justified. I’m embarking on a long-term adventure: to bring together some of my interests. My motivation is pragmatic, more than theoretical, even though many are driven to unify their work and theories from aesthetic considerations; there’s a tendency to judge a complex union of distinctions to be inelegant (To this I might reply: Is an intricately-branching oak tree inelegant?). From a practical standpoint, time is so pitifully finite and my interests numerous enough that in order to say anything meaningful about any of them, I need to pick a strategic conceptual home base. To me, graph theory is an area of mathematics that serves this purpose beautifully.
The basic objects of graph theory are discrete points and “connections”, i.e. pairs, between them. These points need not represent places one can physically go, in the way that we refer to locations in our conventional 3-space where we live and breathe as ‘points’. This space-independence of graphs is one of their features that makes them so versatile and so powerful as tools for modeling and computation. Graph theory is relevant to a host of fields. The ones I’ve mentioned in the title are just a very small sample.
The technical fields of the title are all threads that I have picked up in the blog and have left hanging, to date—partly because I have some additional constraints on my time compared to when I started the blog, and partly because I determined a few months ago that I needed to modify my approach to all of them but didn’t have a strong candidate alternative at the time.
The last item in the title’s list merits some extra explanation. In order to ground my mathy work in community through inquiry and knowledge, I decided to start mentoring with Prison Mathematics Project. The core mechanic of the program is that a prisoner and a non-prisoner write letters to each other about math. I’ve started teaching my mentee the basics of graph theory. I have three main hopes for this relationship. The first is that I’ll be able to equip him with some practical skills that could serve him on release. The second is that I’ll benefit from having a regular commitment to honor with respect to my mathematical work. Otherwise, I’m liable to write off this work as purely for me and therefore unimportant. The third is that this connection will improve both of us as humans in some small ways and make my mentee’s time in prison more constructive compared to how it was before we “met”. You can view my second letter to my mentee at the end of this post.
For now, I’ll end with a rough sketch of how I envision graph theory informing the other fields. I might turn out to be wrong, but I should at least be wrong in an interesting way if that’s the case. Regarding quantum field theory (QFT), I see graphs as being useful at the very least because the established tool of Feynman diagrams is an example of graphs in action. Given that QFT is one of the historical takeoff points for the subject of renormalization, and given that some of the tools for that subject involve algebraic geometry (which was my focus, if brief, in grad school), and algebraic geometry likes using graphs for computational purposes, I suspect that graphs will be useful for studying renormalization, as well.
In a whole other domain, which is relevant to my day-to-day work, there is an obvious application of graphs to be had in image processing for navigation or astronomy in outer space (given that constellations as abstractions are graphs). There are also uses for graphs in designing the software to run the necessary algorithms for that image processing and, from that, deducing locations or orientations in deep space.
The best place for me to actually start talking about the nuts and bolts of graphs and making technical statements is in formulating QFT and Feynman diagrams, in particular, in terms of graphs, and vice versa. This is basically where I left off in the motivating/context-providing discussion leading up to renormalization. It will feel good to get back in the arena!