The Foundations of Renormalization in Theoretical Physics: Part I, Fields

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The Foundations of Renormalization in Theoretical Physics:

Part I, Fields

Henceforth in this series of posts on renormalization and electrical circuits, I will be writing various amounts of mathematics. I will do my best to summarize the idea behind each mathematical object, relationship, and statement that I include, so that if you are not fluent in mathematics you can still gain something from reading those sections. I also say plainly that understanding and doing mathematics is often extremely hard even for those fluent in the basic vocabulary and syntax of research-level mathematics. What is different about mathematicians from many people who are not is that they experience a particular sense of reward from struggling with that hardness.

Fields

We have everyday notions of what physical space is from our own lived experience. These are useful, but they are incomplete. Physical space is complex and fascinating enough to study for many lifetimes. When I was in high school, I wanted dearly to really understand what a “point in space” is, because the notion of a distinct point in a continuum confused me greatly. Having studied geometry at the graduate level, I’ve come to a greater appreciation of how profound a topic the nature of space is and how much we still have to explore. Anyway, I will introduce some of the simplest formalisms of space that physics and mathematics have to offer, some of which are very familiar to our everyday experience!

We are going to assume that we live in a physical space that is modeled by the real numbers in three dimensions, which we denote by \mathbb{R}^3. [Aside: Whether assuming this is actually a good idea is one of my favorite math/physics topics, and I will explore the question in a future post.]. We specify a point in \mathbb{R}^3 by a list of 3 numbers, which we call coordinates: p (a point) = (x, y, z) (coordinates list). Now, in life, people who can feel temperature with their skin experience every location to have a temperature. Let’s assume that each point of space everywhere in the universe has a temperature, call it T, even if we haven’t been there yet as a species to verify this experientially. We can talk about this phenomenon formally by assigning a real number, the temperature, to every point in \mathbb{R}^3.

So far we have a name for our underlying space, for a point in space, and for the temperature at that point. But we don’t yet have a name for the pairing of a temperature with a point. We’ll call that T(p) (‘T of p’). From here, we can ask a huge number of questions about T(p), since we’ve measured the temperature of space to vary dramatically depending on where you are. If you were to draw a 3-D bar graph over the area of your kitchen, for example, where each bar’s height represents the temperature at a chosen location, what would that look like? Isn’t it amazing how your hand can be just a centimeter away from a kettle containing boiling water, and yet where your hand is is cool enough that it doesn’t get burned? Our visualization of T(p) around the kettle would look like a huge spike in the middle of your kitchen! We refer to fields where we only assign a single number to each point as scalar fields.

If we stick to the model of space as being a continuous container of distinct points, we can take this basic idea of a field much, much further. Think of how many parameters are needed to fully specify a single point in space, even in a place as seemingly-mundane as a point in your home! Its temperature, gravity potential, curvature, distance from a given object, electric charge, and so on. One can ask: Does this list ever end? How could we ever be sure of the answer to that question? If it doesn’t end, what the heck is the nature of the underlying space the point is in?! These are questions that are also on my list of topics to blog about.

Fortunately, for the present moment we’re not going to concern ourselves with all parameters needed to fully specify a given point in space. Let’s take things slowly and only think of the parameters that have to do with electricity and magnetism. On the electricity side, if we think of a point p in \mathbb{R}^3 as having an amount of electric charge as a parameter describing it, then we can also think of the amount of electrostatic force acting per unit of charge at that point. At the same time, on the magnetism side, if again we think of each point in space as having an amount of electric charge, then we can also think about the corresponding amount of magnetic force acting per unit of charge at that point. Hence, each point in space has a list of electrical and magnetic values attached to it. We call this pairing of points with corresponding lists of numbers an electromagnetic field. The more general name for this type of object is vector field.

Having introduced the basics of fields, here is the progression of topics that I presently aim to follow in the next few posts on this general subject.

  1. How Electrical Circuits Relate to Fields
  2. Field Theories
  3. Quantum Field Theories
  4. Infinities in Quantum Field Theory
  5. Electromagnetic Fields and Renormalization

References

  1. https://arxiv.org/pdf/1909.11099.pdf
  2. https://phas.ubc.ca/~stamp/TEACHING/PHYS340/NOTES/FILES/Fields_in_Physics.pdf
  3. https://www.damtp.cam.ac.uk/user/tong/qft.html