Renormalization Background: The Lagrangian Method, a Worked Example
[Preamble for down-to-Earth context: I’ve been meaning to finish this post every lunch hour this week and unmindfully decided to job-work instead. Fortunately there are pretty much zero worldly distractions at 5:15am on a Sunday during the COVID-19 pandemic! Just me and the birds.:-)]
This post is an optional stop on the trip to examining renormalization in the context of electromagnetic fields. It will contain a little bit of formal mathematics, and there’s no way around that. I recommend that you know basically what derivatives and partial derivatives are, even if you don’t know techniques for working with them. What I am going to do is alternate the mathematics with—what I hope is!—lucid English summarizing what the math is saying at an imprecise, conceptual level. This will hopefully allow the math-curious but not math-committed members of the audience to get a window into how math is often an expression and “cleaning up” of thinking and not just a string of calculations. To remind ourselves where we are and where we’re going, we’re learning some of the fundamental tools from physics that we’ll need to talk about renormalization in a way that’s deeper than the hand-waving I’ve done up to this point.
Full disclosure: I’m going to be following this document pretty closely in this post, because I think it’s a very good exposition of the topic; however, I will probably provide a bit more background and define assumed terms to make this post as self-contained an introduction/example as possible. Let’s begin!
We’re going to start by contemplating the system of a mass hanging from a spring. A useful example of this general type of situation is when a human (a mass) is hanging from a rope (a spring) when, say, working on office building construction. For my own amusement, I prefer to imagine an elephant (a mass) hanging from a gigantic slinky (a spring) while washing skyscraper windows.
What are the parameters of this system? If we’re restricting our attention to only the elephant and the slinky (i.e. ignoring everything else that might be going on at the same time, such as wind interference), then our list of parameters can be broken down into a list for the elephant, a list for the slinky, and a list for the system in which the elephant is suspended from one end of the slinky. The elephant and the slinky are both complicated objects, but if we’re only asking questions that relate to how they move in this system, it turns out that our list of relevant parameters for each component is very short: for the elephant we have: 1) its mass, m and 2) its acceleration in a particular direction with respect to time, and for the slinky we have: 1) its “spring constant”, and 2) its displacement in the same direction as the elephant is accelerating as it moves while attached to the slinky. These lists don’t specify how the parameters affect each other. They could, in principle, all be mutually independent of each other, but experiment has told inquiring humans that they aren’t. To specify how they affect each other, we need math.
Here’s the way that Newton would probably have started writing this math en route to determining the values of all the parameters of the system that we just listed:
Step 1: Start with the general equation of motion , which expresses the statement that, as vectors (quantities with a direction attached to them), the sum of the forces applied to an object of fixed mass (You can think of this as the “net” force) is equal to the mass of the object multiplied with its acceleration. This equation is true in general for the everyday world we experience, so we can use it and trust what it tells us.
Step 2: Swap out the expression on the left side of this equation with the corresponding expression for this slinky-elephant system: , where we’re now saying that 1) the force that the elephant is exerting on the slinky is -kx, where x is the displacement of the slinky in the direction of stretch and k is an empirically-determined constant that encapsulates material properties of the (let’s say, plastic) slinky, and that 2) this is the only force acting in the system and so it is equal to the mass of the elephant multiplied with its acceleration in the direction of stretch.
Are we done? Remember, that would mean that we have all the information we need to specify the value of each parameter of the system. Well, let’s look at the lists we made above. If we took measurements of the minimum necessary number of quantities involved (how to determine this critical number is a topic for another math day), could we derive everything else? For example, if we took measurements of k and m, could we compute everything else? Yes, yes we could. So we are, in fact, done.
In that example, we were blessed with the simplicity of the elephant’s and spring’s motions. We assumed that the elephant was perfectly still and stretching the spring only in one direction—the direction in which the gravity of the Earth acts at the location of the elephant in space. But what if the elephant were thrashing, causing it to stretch/compress the slinky in many different directions? Suddenly, to compute the total force in the system we have to add up the component forces across all these directions! That means we have to do a lot of work to get to the equation that says something about the elephant’s position. Is there a faster or more elegant way to specify all the parameters of such a system? Indeed, there is.
To get there, we have to shift focus and ask ourselves not about forces, but energies of the system, which, by the definitions physicists have made, does not have a direction, only a numerical magnitude. Let’s take an inventory of the energies present in this system. At the bulk level of the elephant and the slinky (as opposed to the level of their component atomic and subatomic particles), the energies present are kinetic and potential—roughly, how much energy there is in movement and how much energy is left.
For this case of the mass and the spring, we will accept as givens that others have verified that the kinetic energy, T, is specified by , where is the velocity of the elephant, and that the potential energy, V, is specified by , where x is the position of the elephant. Now, what to do with these ingredients? Well, we’ve already defined for ourselves the notion of the Lagrangian L,, which is the numerical difference between kinetic and potential energies. Let’s see where writing that down takes us:
We have . What do we do from here to get to an expression that tells us about the elephant’s acceleration (which is the missing piece in our quest to fully specify the system at this point)? I like to start with the question of: What can we do from here? (The answer is often surprisingly limited or elementary, which is helpful!). We can definitely do some sort of calculus, i.e. taking an integral or a derivative of some quantity with respect to another. Which one is worthwhile? Well, if we want to get to acceleration we have to go one derivative further with respect to velocity, so let’s try taking one derivative with respect to time of each quantity in the Euler-Lagrange equation for this example: .
Now, I did some sleight of hand there, and maybe in another optional post I’ll show how we get the Euler-Lagrange equation. The partial derivative of L with respect to the velocity of the elephant with respect to time is the elephant’s mass multiplied with its velocity with respect to time, i.e. , and the partial derivative of L with respect to the elephant’s position is -kx, i.e. . So now, taking the derivative with respect to time of the first expression gives us acceleration, as we wanted! So again, we arrive at , but we got there without thinking about individual directions of force but rather by differences in energies. Let’s sit back and ponder/appreciate the equivalence of those two approaches for a second…
This way of thinking is powerful not just at the level of everyday objects like you, me, or elephants, or slinkies. It’s also very powerful to think about systems of particles in terms of relative energies over time. In fact, that’s what we were talking about earlier regarding the principle of least action. Next post I’m going to aim to steer us back toward classical field theory en route to quantum field theory and then renormalization.