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 those equations say about the vector field we created so that we can manage it as best we can. But in most cases, we don’t need to worry as engineers.
But if we want to be physicists and try to understand what is and not just what we decide things to be, then we might wonder how we even get from the quantum world of individual electrons to these larger objects of electromagnetic fields, in the first place, and how much accounting for that transition we need to do when trying to understand how an electrical system works.
To approach this, we need to equip ourselves with ideas both from quantum mechanics and special relativity. Among other issues, these ideas will help us grapple with the issue of how to think about “discrete” particles, namely, electrons, arising from a “continuous” field of space and energy. In a sentence, we can gainfully think of electrons as arising as quantized (very roughly, ‘discrete’) excitations of fields.
To me, as someone who has been fascinated for a long time by the tension in nature between mathematical notions of discreteness and continuity, this is a thrilling prospect. In order to do it justice, we’re going to go through some of the absolutely crucial background concepts that will allow us to gain substantive insight. We’ll focus on the Lagrangian and Hamiltonian formulations of dynamics, the formulation of electromagnetism in terms of special relativity, quantization using ladder operators, and scattering in nonrelativistic quantum mechanics. Don’t worry, the whole point is that you don’t already have to know what those words mean; I will do my best to explain them in both ordinary English and mathematical languages. Once we’re solid in those foundations, we’ll move on to explore quantum field theory proper for the example of electromagnetic fields and then introduce renormalization as both conceptual insight and practical tool in this domain.