Approaching the River and the Tunnel: More on Electrical Circuits and Renormalization
In the last post, I set up an extremely loose and vague (but hopefully engaging!) metaphor between a topic of mathematical/engineering research that I’m working on and a landscape featuring a river and a tunnel. To summarize, electrical circuits are networks of electrically conductive paths whose flow of current we can liken to the flow of water. We care a lot about electrical circuits because they are super useful in our daily lives for a host of reasons. Electrical circuits can also be extremely complicated, which makes it hard to analyze them in order to create and troubleshoot useful systems. We’d like to be able to make those processes easier for the folks tasked with such things! There’s also the thrill of discovery for some people (myself included!) in achieving a richer understanding of an aspect of this amazing physical world we live in. This is basically an argument in favor of inquiry for its improvement of quality of life.
The basic hypothesis that I’m starting with is that this mathematical construction called a renormalization group — the tunnel in my mystical metaphor — is going to turn out to have the desirable property of being able to take a very intricate electrical circuit, do some magic on it, and then give back an electrical circuit that has all the important characteristics preserved but is much simpler. It’s worth saying that we have a lot of power in this situation; we get to decide by fiat which characteristics are important and which aren’t.
I owe you an explanation of exactly what I mean by ‘electrical circuit’ and ‘renormalization group’ so that we can all be sure we’re thinking about the same things. Especially in the case of ‘electrical circuit’, we might have slightly different baseline notions for what that means, even though most people are familiar with examples of electrical circuits in their flashlights, cell phones, etc. I’m going to start out with an informal definition of electrical circuits that appeals to our everyday experience of electricity, and then I’m going to take out some of the worldly information in that definition to get one that captures the mathematical essence of the thing. This abstraction will buy us more freedom of intellectual movement; we won’t be constantly encumbered by notions of physical wires, light switches, AA batteries, etc. These physical attributes are certainly crucial if we’re trying to build a circuit, but they aren’t relevant for this more design-level inquiry.
Let’s start out with an everyday definition of an electrical circuit: a set of electronic components, such as resistors, capacitors, and transistors, that are connected to each other by conductive wires. I’m not completely happy with this definition for what I want to do, and here’s why: This is a definition in English, not mathematics. And renormalization groups, which I want to make play with electrical circuits, can only be meaningfully talked about in the vocabulary and grammar of mathematics, not in the vocabulary and grammar of ordinary English. So I have to work a little bit to translate the features of electrical circuits that we care about for practical system-building purposes into mathematical objects and relationships.
Here’s a first go at it: An electrical circuit is a loop through which electric charge flows as the result of a driving voltage. This is a bit better, because it says something more precise about the geometry (which is a branch of mathematics, after all!) at work in a circuit; we require a closed path of movement of electric charge. Notice that this definition doesn’t mention things like wires or AA batteries, at all. We could have something wild like a convoluted chain of carbon nanotubes forming a microscopic circuit (Pulled that one out of thin air, but turns out it’s a real thing! Go read about it and be amazed by technology these days.).
Mathematics has many, many things to say about loops. It also has many, many things to say about more particular kinds of loops, such as those punctuated by points which are called ‘nodes’. In that case, we can think of the loop as being a particular type of ‘graph’. For our purposes, we can think of an electrical circuit either as a smooth loop or as a graph. The latter case is useful if we want to include some notion of electronic devices in our definition, where a device such as a resistor in our circuit simply corresponds to a node in our graph.
This definition of an electrical circuit is good enough for the moment. I’ll clean it up even more later, when doing so makes more sense. To lay some groundwork in our thinking, though, I also want to mention that we can consider the set of all electrical circuits—given by every possible way a structure can satisfy the above definition. This space will probably be important later, because if we could only say something interesting about renormalization and one particular circuit out of the entire space of circuits, we would be working very hard for very little reward. We’re aiming to be able to say something useful about all circuits, or at least a substantial subset of them. That would be powerful.
Leaving electrical circuits for the moment, let’s go back to the idea of renormalization. Again, we’re going to start out talking in very broad strokes and gradually refine the language to get down to nuts and bolts. In a sense, we’re lucky that ‘renormalization’ isn’t a word used in ordinary English; we get to start out without any confusing mental clutter from everyday experience that obscures what we’re really trying to get at. Renormalization is a method of specifying relationships between parameters of different systems. Okay, what the hell does that mean? Let’s start getting a handle on this by looking at an example. In an electrical circuit, say, at the spatial scale of electrons, the movement of those negative charges is fundamentally what is driving the behavior of the circuit as a whole. Now, an electron can be thought of as an object with a whole bunch of attributes, or, parameters, that describe it: its mass, spin, magnetic moment, etc. But any electrical engineer can tell you that she can build useful systems and describe their behavior in terms that don’t include the mass, spin, etc. of the electrons in it. In fact, she probably never thinks about those parameters at all. What is it about the underlying structure of the electrical system that allows her and other engineers to do this without messing things up? Taking on this question is where we’ll pick up next time. For now, go learn about how amazing nanotubes are!