The Foundations of Renormalization in Theoretical Physics, Part II: Electrical Circuits and Fields
By the end of the last post, we established in very rough terms that electromagnetic fields (EMFs) are instances of a more general type of mathematical object that we call vector field. Connecting this back to electric circuits, the underlying substrate of such a circuit is a particular EMF, where each point in the spatial region encompassing our electronic components has a vector (list of numbers) associated with it that describes the electromagnetic forces at that point.
It turns out that the analysis of electrical circuits becomes vastly simplified if we make certain assumptions about the EMF corresponding to the circuit; when these assumptions are satisfied, the EMF is “well-behaved”. Now, there’s nothing in nature (besides human beings) forcing circuits to satisfy these requirements. Rather, in order to make our lives easier as engineers designing interesting or useful systems, we impose a discipline on ourselves such that we only work with electrical components and circuits where these assumptions hold. This simplifies the calculations involved in typical use cases to doing a bunch of algebra, where in the wild we might have to use calculus in order to find out basic properties of the circuit.
We don’t have to go as far as renormalization to identify or use these assumptions; we only need to rely upon Maxwell’s equations, which make statements about certain characteristics of the vector field in question. If we just want to reassure ourselves of the basic-level conditions necessary for our circuits to be well-behaved, we can stop there. But if what we really want to understand is how we make the transition from the world of individual electrons at the quantum level to the world of movement of bulk charge (the world of EMFs and circuits), we still have a winding road ahead of us. We’ll have to explore quantum field theory, and some key pieces of quantum mechanics and special relativity.