I just posted a new article. This one is on generating functions, a fun and powerful technique in combinatorics, and it’s designed to be accessible to anyone who’s taken calculus and remembers at least a little bit about how Taylor series work. This will hopefully be the beginning of a series of two or three articles where we explore some fun applications of the concept.
This article is a part of an effort to put some content on here that’s a little more accessible than a lot of the articles I’ve written so far. (I’ve very much enjoyed putting together the physics series, and I plan to keep working on it, but the prerequisites for those articles can be quite steep!) There are a lot of topics I’ve explored with my students in the last few years that I think could be a nice fit here, and I plan to adapt more of them into articles for this website in the coming months.
As always, if you’re reading this and have a suggestion for a topic you’d like to see explored in this format, don’t hesitate to reach out!
There’s another new article in the physics for mathematicians series. This one picks up the story of quantum field theory right where the previous one left off. It explores how we can take the ideas we’ve developed so far and turn them into an algorithm for computing scattering amplitudes. In particular, this article finally introduces the famous Feynman diagrams and explains where they come from and how they fit into the story we’ve been telling across this series of articles.
In order to make this article flow better, I had to do something I don’t ordinarily like doing and make a small notational change to the previous article in the series. (In the older version, I didn’t distinct between the vacuum state of a free theory and the vacuum state of an interacting theory, and wrote for both. In the newer version, the free vacuum is called and the interacting vacuum is called .) That change is now live as well. My hope is that, if you’ve been following this series so far this isn’t too big a disruption, and I’ll try not to change the past out from under you again if I can help it!
My current plan for the quantum field theory series is that there will be one more article (on the topic of renormalization) in the “main” series, but that I might end up writing a few one-offs on some smaller topics in quantum field theory to go along with them. If this plan ends up being what happens, the picture is that the first four articles will serve as the trunk of the tree, and that the others will branch off from it and could potentially be read in any order.
It’s been very fun for me to finally arrive at the point in my study of this subject that I feel competent to put together these expositions! Quantum field theory is something I’ve been working toward mastering for a very long time now, and I’m happy to finally get the chance to share what I’ve managed to learn with others to whatever extent these articles can do that. As I always say, do let me know if there’s any topic you’d like to see covered in this style, whether it’s physics-related or not, and I’d always love to hear from any readers if you have comments, questions, corrections, or anything else you’d like to talk about.
I’ve just finished a supplement to the general relativity article I posted a few weeks ago in the physics for mathematicians series. This one is about the Lagrangian approach to general relativity; while it’s a bit more abstract than the approach I took in the main article, I found when learning the theory that it offered a very useful perspective, so I wanted to share it. The article also contains an exploration of a topic that I got quite stuck on when learning this material myself: the relationship between the energy-momentum tensor as it emerges from Einstein’s theory and the concept of energy-momentum we already have from non-gravitational physics.
I’m still working on other articles for this series. The next one might be the one on Lagrangians in classical field theory I mentioned in the last update, and it might also be the third entry in the quantum field theory sequence, which I currently expect will end up being around four to six articles long. As always, if there’s a topic you’d like to see covered in this style, please reach out and let me know!
After almost a year I’ve finally finished another new article in the physics for mathematicians series. This one is a quick introduction to general relativity, especially aimed at people who are already at least a little familiar with concepts like connections and curvature in the context of Riemannian geometry. This one was very fun for me to write, so I hope that if you’re interested in the topic you enjoy it too!
I do intend to keep working on this series, although probably still at this very slow pace. I’m currently thinking that the next one will be an overview of how to use Lagrangians in classical field theory. As always, though, if there’s a topic you’d be interested to see covered in this style, feel free to reach out and let me know.
I visited Mathcamp for just one week this summer, and I’ve just posted notes for the class I taught there. If you’re one of the students who was in that class and you happen to be reading this, thanks again for a wonderful week!