Martin Roelphs on Projective Geometric Algebra

In this talk he shows how carefully chosen axioms and definitions can unify descriptions of models that otherwise seem fundamentally different. See Feynman's comment on Page 149 of his book QED: The Strange Theory of Light and Matter quoted in The Action Lab - Ulexite (How Does Television Stone Work?):

I would like to emphasize something. The theories about the rest of physics are very similar to the theory of quantum electrodynamics: they all involve the interaction of spin 1/2 objects (like electrons and quarks) and spin 1 objects (like photons, gluons, or W's) within a framework of amplitudes by which the probability of an event is the square of the length of an arrow. Why are all the theories of physics so similar in their structure?

There are a number of possibilities. The first is the limited imagination of physicists: when we see a new phenomenon we try to fit it into the framework we already have -- until we make enough experiments, we don't know that it doesn't work. So when some fool physicist gives a lecture at UCLA in 1993 and says, "This is the way it works, and look how wonderfully similar the theories are," it's not because nature is really similar; it's because the physicists have only been able to think of the same damn thing over and over again.

Another possibility is that it is the same damn thing over and over again -- that Nature has only one way of doing things, and She repeats her story from time to time.

A third possibility is that things look similar because they are aspects of the same thing--some larger picture underneath, from which things can be broken into parts that look different, like fingers on the same hand. Many physicists are working hard to put together a grand picture that unifies everything into one super-duper model. It's a delightful game, but at the present time none of the speculators agree with any of the other speculators as to what the grand picture is.

See their paper Graded Symmetry Groups: Plane and Simple by Martin Roelfs, Steven De Keninck.

Leo Dorst giving some examples of programming classical mechanics with coordinate-free representations:

Watching this I can't help but think of Mach's principle and the centre of mass of the Solar System: see Angela Colier Destroying the Solar System.

Tutorial: May the Forque Be with You: Dynamics in PGA by Leo Dorst & Steven De Keninck

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Here's a Geometric Algebra interpretation of the macroscopic formulation of Maxwell's equations:

See his blog and his book Geometric Algebra for Electrical Engineers.

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See also this post of mine: Gerald Pollack on Organic Chemistry of Water.

See also Spacetime algebra as a powerful tool for electromagnetism by Justin Dressel, Konstantin Y. Bliokh and Franco Nori.

We present a comprehensive introduction to spacetime algebra that emphasizes its practicality and power as a tool for the study of electromagnetism. We carefully develop this natural (Clifford) algebra of the Minkowski spacetime geometry, with a particular focus on its intrinsic (and often overlooked) complex structure. Notably, the scalar imaginary that appears throughout the electromagnetic theory properly corresponds to the unit 4-volume of spacetime itself, and thus has physical meaning. The electric and magnetic fields are combined into a single complex and frame-independent bivector field, which generalizes the Riemann-Silberstein complex vector that has recently resurfaced in studies of the single photon wavefunction. The complex structure of spacetime also underpins the emergence of electromagnetic waves, circular polarizations, the normal variables for canonical quantization, the distinction between electric and magnetic charge, complex spinor representations of Lorentz transformations, and the dual (electric-magnetic field exchange) symmetry that produces helicity conservation in vacuum fields. This latter symmetry manifests as an arbitrary global phase of the complex field, motivating the use of a complex vector potential, along with an associated transverse and gauge-invariant bivector potential, as well as complex (bivector and scalar) Hertz potentials. Our detailed treatment aims to encourage the use of spacetime algebra as a readily available and mature extension to existing vector calculus and tensor methods that can greatly simplify the analysis of fundamentally relativistic objects like the electromagnetic field.

And here is a series of (80+!) lectures on QED prerequisites, the last thirty of which look at this paper.

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