Scaling of Electrical Phenomena and Renormalization
At the end of my last post on this topic, I said that this time I would take on the question of how it is that certain underlying physical phenomenon that define an electrical circuit at small spatial scales—such as electron spin, mass, magnetic moment, etc.—aren’t relevant to the parameters of electrical circuits at larger spatial scales, the ones that engineers care about: current, voltage, etc. In particular, how can we be certain that when we’re designing and building circuits, we’re not failing to account for important factors at small spatial scales? What is the basis for our confidence that everything will work out at the macro-scale without thinking about these things? Very roughly, we’ll see that renormalization is part of a mathematical framework that explains these physical transitions in a compelling way and gives us tools for finding things out about them.
To the engineer, these question might seem a bit pedantic and a waste of time. After all, if we mercilessly test electrical systems designed without thinking about certain properties of electrons and electromagnetism in the real world and the systems survive, then no further verification is necessary. To the physicist or mathematician, these questions invite a wealth of other fascinating questions and journeys of thought. As someone who appreciates engineering, physics, and mathematics, my first goal is to reach understanding of how things are, which is a scientific/mathematical goal. My second goal is to take the engineering perspective very seriously and to figure out whether any of the results of achieving the scientific goal could be useful to the engineer.
In pursuit of this first scientific goal, we are about to go on a long adventure and get our hands mighty dirty. To understand renormalization in a deep and proper way, we need to explore the world of very small objects and phenomena, which we often refer to as the ‘quantum realm’—the spatial scale where our everyday Newtonian laws of physics don’t do a good job at describing the behavior that we observe about things like particles when we do fancy physical experiments on them. In short, we need to explore quantum mechanics, special relativity, and quantum field theory. We are going to have to become theoretical physicists.
Among the most exciting developments in theoretical physics in the last century is the reaching of an understanding that the underlying structure of the universe is not scale-invariant. As the spatial extent of objects change, the principles that accurately describe them change, too; the world looks different depending on whether you’re a person or an electron. We’re going to start next time by considering how an electron, our fundamental mover in electronics, sees the world. This is the deep breath before the plunge.