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Quantum Field Theory: Particle Collisions At this point in the series on renormalization, we are going to switch from a high-level view into a deep dive into physical foundations. For this, I am going to become mostly unoriginal and follow Peskin & Schroeder’s popular text, An Introduction To Quantum Field Theory, which contains an entire section on renormalization. We still have a ways to go before getting there, however, and as we go there will be a lot of my own conceptual insertions as I, too, puzzle my way through the material. I intend to enjoy myself throughout the journey, but we can derive extraRead More →

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 workingRead More →

Background: Quantum Theory of Electromagnetism: Hamiltonian Dynamics Since we’re now being good physicists in this series by working to understand what’s happening in various electromagnetic systems, in the last post we looked at one very important type of measurement we can make on such systems. For many purposes, that basic measurement might provide all the framing for a question that we need; from there, we can compute all of the specific quantities of interest. That measurement, the Lagrangian, is formulated in terms of the system’s kinetic and potential energy: . From there, we’re going to look at another measurement that can be derived from theRead More →

On Leadership and Power “Leadership is the ability to face adversity.” Bänoo Zan I choose to live and speak openly regarding challenges that I have faced so far in life. To some people, this might come across as parading weakness in order to gain sympathy points, as much as this is a pretty severe misunderstanding of my speech on the matter. To be clear, I do not want anyone to go easy on me because I 1) have a uterus, 2) have survived periods of serious illness, or 3) am queer. Setting expectations low out of pity or the desire to protect does not helpRead More →

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 thenRead More →

The Foundations of Renormalization in Theoretical Physics, Part III: Quantum Theory of Electromagnetism We left off last time with an understanding that electrical circuits consist of electromagnetic (vector) fields and that, as long as Maxwell’s equations relating to that field are satisfied, the behavior of the field will be suitable for engineering-type work. For purposes of building useful stuff, we can very much stop there and be no worse off for it. If we build a circuit where Maxwell’s equations don’t hold, such as one where signals are traveling within the circuit close enough to the speed of light, then we’d better know what thoseRead More →

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 makeRead More →

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 thatRead More →

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 confidenceRead More →

Approaching the River and the Tunnel: More on Electrical Circuits and Renormalization In the last post, I set up an extremely loose and vague (but hopefully engaging!) metaphor between a topic of mathematical/engineering research that I’m working on and a landscape featuring a river and a tunnel. To summarize, electrical circuits are networks of electrically conductive paths whose flow of current we can liken to the flow of water. We care a lot about electrical circuits because they are super useful in our daily lives for a host of reasons. Electrical circuits can also be extremely complicated, which makes it hard to analyze them inRead More →