A Boat, a River, and a Tunnel: Applying Renormalization Groups to Electrical Circuits
I’ve been hemming and hawing for a few months about a mathematical topic about which to do creative research. The main reason hasn’t been lack of curiosity, but rather a lack of a mental model for success—and, frankly, fear of producing total nonsense. In academia, success in my lifetime has meant publishing lots of highly-cited papers, getting grant funding, being a tenured professor at a prestigious university program, and advising grad students who go on to do those same types of activities. In my world up to this point, people who publish papers of professor-level quality who aren’t or weren’t professors themselves have not existed. This eliminates the friendly-feeling option of learning by emulating what others have done. Then again, for a long time I have viewed my life as a deeply creative endeavour, where I often aim for targets that I and others didn’t know could even be targets. So, while contemplating the prospect of publicly making an intellectual fool of myself by, say, publishing a paper on arXiv.org about a topic I am thinking about in isolation and for the first time is scary and uncomfortable in a new way, scary and uncomfortable is where I tend to thrive. I’m ready to take it on again.
What follows here is a sort of warm-up brainstorming session for me to get the cognitive juices flowing about the topic of using renormalization groups, an algebraic construction, to the analysis of electrical circuits, with the goal of arriving at useful knowledge of said circuits. A simultaneous goal is to explain the ideas as clearly and simply as possible and to give the reader a window into both how formal mathematics is written and how that end product does and doesn’t resemble a mathematician’s thought process leading up to the formal math.
Whenever we start communicating a landscape of mathematical objects, we need to give those objects names. I’m looking out at a dimly-lit expanse that seems to have some interesting features, and I want to tell you what I see. The first thing is an object called a ‘group’, and I’m going to name it RG. I also see this other object, possibly connected to RG in some way. Let’s call it G. By the way, the choice of names is totally arbitrary! We could name one ‘ 🙂 ‘ and one ‘ 🙁 ‘ if we felt like it.
Now that we’ve named RG and G in our landscape, I can start defining them for you. In this post, I will only do so approximately, with a minimum of math jargon words. Now, even these inexact definitions are a bit abstract and might seem arbitrary and made-up. They are definitely made-up, but they’re not arbitrary; the things they define have been seen enough by enough people in nature that their existence seems to be independent of us humans and our crazy imaginations.
Let’s start out with the most salient thing in the landscape in front of me, which is G. Now, G has a lot going on with it, which is why it stands out. It’s this shiny network of little streams—kind of like a river that rejoins itself to form a closed-off water maze. This is actually a metaphor for an electrical circuit. There are some features of G that don’t seem very important, such as its shininess, so I’m not going to include them from now on when I talk to you about G. For the purposes of our conversation relating G and RG, all that matters is that G has a bunch of branches and places where the branches meet up. G is also made up of branches and junctions that form closed loops.
Our object of the type ‘group’ RG is a bit subtler. It’s this rocky tunnel in the side of a mountain. I notice another water-circuit much like our G going into that tunnel. On one side, it’s a wild mess of complexity. I would not want to try to navigate it in a boat! But on the other side of the tunnel, the water-circuit is much simpler! Only a few loops joined together instead of thousands. Amazing! Did the group have anything to do with this change? Does it have some awesome properties that allow it to collapse entire smaller streams into larger ones, or just make them disappear altogether?
I walk down the slope from my lookout toward the tunnel to investigate. And indeed, it seems to have some apparatus inside that is changing the entering water-circuit! This gives me the idea to try to build a tunnel myself over and around G that behaves just like that other one. The one I create is RG. After much toil, I’m done! Now G effectively flows into the tunnel RG and emerges much simpler than before. I then have a grand old time canoeing around the simplified version of G, no longer worried about getting lost in it.
Next time, I’ll tell you more about G and RG as independent landmarks so that we can better understand how RG had its seemingly-magical effect.