Renormalization Background: Quantum Theory of Electromagnetism: Lagrangian Dynamics

In order to dig into the quantum theory of electromagnetism, we need to have a framework for how to think of physical systems and how they change over time. Otherwise, how are we to talk about an electromagnetic field (a physical system) and the changes that occur within it at the quantum level? There’s a popular and useful point of view in physics that says we can represent a physical system as a list of numbers, or coordinates, much like we can represent a point in physical space. By extension of the spatial concept, we can then think of the physical system as a point in some abstract-mathematical space. That way lies many a great inquiry! But for now, we’ll restrict ourselves to considering a physical system, $P = (a_1,\,a_2\,...,a_n)$ , where each $a_i$ for i = 1, 2,…,n. This might seem a bit foreign or austere. One way to think of it is as the system’s unique signature at a given point in time. Just like a person’s signature, it doesn’t capture all the complexity of the system that produced it, but it is an extremely useful identifier!

Okay, cool. So what do these a-values represent? We don’t strictly need to concern ourselves with this, because part of the appeal of this formulation is that we don’t have to make any specific choices for a-values in order to use the language and computational techniques associated with it. In practice, what this means is that we can talk about lots of different kinds of systems using the same tools—so efficient! But to get a conceptual handle on what’s going on, you can think of this list of numbers as, say, representing the states of a finite number of electrons in a circuit.

Now, electrons don’t just hang out and stay the same over time. Their states change. So in order to account for this, we’d better have a description for how this happens in terms of our basic representation of a system, represented as a list of numbers that characterizes it. Enter the Lagrangian. The Lagrangian is a sort of measurement of a physical system. Specifically, it is the difference between the amounts of kinetic and potential energy a system has. Let’s write this as a mathematical statement featuring our subject, the Lagrangian $L$ , and the side characters kinetic energy, $K$ , and potential energy, $U$ , where $L = K - U$ .

Okay, so we’ve defined this quantity called the Lagrangian that can be associated with any given physical system. So what? This doesn’t, in and of itself, say anything interesting. Fortunately, there’s a principle in physical theory that describes how $L$ behaves over time. It says that if we add up lots of values of $L$ for a given system over time, then we’ll find that $L$ always takes on the least possible value. In other words, it says that the difference of the kinetic and potential energies of a system will be as close to each other as possible over time. Here’s how we say this precisely in mathematics: $S = \int L\,dt$ goes to a local minimum. In physics parlance, this is called “The Principle of Least Action” and is fundamental to how we view/interpret physical experiments and theory. Let’s leave off, for now, by thinking about what this says about our particular case of interest: electrons. If we’re trying to understand the motion of electrons, which is the fundamental driver of electrical circuits, let’s consider the electron’s kinetic and potential energies as it’s moving. Then this principle says that an electron will take a path through space that, roughly, makes its kinetic and potential energies as close as possible over the duration of its travel.

This is the Lagrangian interpretation of dynamics, in a nutshell.

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Renormalization Background: Quantum Theory of Electromagnetism: Lagrangian Dynamics

Renormalization Background: Quantum Theory of Electromagnetism: Lagrangian Dynamics

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