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I'd like to see a rewrite of EM in Geometric Algebra focused not on pages and pages of abstract equations, but practical numerical applications.

Something I've been interested in doing is writing a "renderer" that fully simulates Special Relativity including a full treatment of EM such that diffraction, interference, etc... can all be simulated.

This would be an interesting sandpit for testing the limits of our knowledge, a bit of an "acid test" if you like. Such an engine could even be extended to include, say, electroweak theory and then tested to see if it is still consistent.

In principle this ought not to be that hard, but I've found that theoretical physicists don't like to sink down to the levels of numerical simulations. This can leave gaps and issues that are not just difficult to fix, but nearly impossible. One such issue is numerical stability: equations on a page tend to use "mathematical reals", and these often require infinite precision to simulate. However, the real physical world doesn't allow infinite information to be stuffed into a finite space. Similarly, the infinities of the electron self-energy can be handwaved away on paper, but a simulation will just ignore your hand waving and do absurd things.

> I'd like to see a rewrite of EM in Geometric Algebra focused not on pages and pages of abstract equations, but practical numerical applications.

What do you mean by practical? Do you mean something you can use to calculate the electric and the magnetic fields as if we didn't understand they're the same thing? That exists, they're described by their own fields of science which used to be called "Electricity" and "Magnetism" respectively. Of course, what's the point when you have Maxwell's equations?

Geometric Algebra is merely a variant of linear algebra that is purpose-designed for describing geometric concepts such as signed areas. For comparison, vector algebra is mostly a direct extension of linear algebra. Linear algebra was originally largely designed for solving large systems of simultaneous equations. Its native objects, vectors and matrices, are not ideal for geometric concepts. Trying to pretend everything is a vector ends up causing issues with the equations that are hideous to solve.

Some of these issues are not obvious when manipulating vector algebra purely symbolically. They manifest when it comes time to create a numerical simulation. You suddenly realise that there are practical issues such as high numerical error, conditional statements all over the place, and corner-cases like gimbal lock.

For comparison, something like simulating the full electromagnetic interaction in a 4D spacetime ought to be straightforward with a geometric algebra!

It's just that I haven't seen anyone try. Every EM textbook is the same. They go through the same equations. The same algebra. The same symbolic solutions. With the same corner cases. The same numerical stability issues. The same result: "Oh we can talk about it, but rendering it is too hard!"

I want an "acid test" of EM, a... wavepool to play with. Not a toy model. The full thing, capable of simulating even obscure but experimentally-verified results such such as the Aharonov–Bohm effect. See: https://en.wikipedia.org/wiki/Aharonov%E2%80%93Bohm_effect

That is, I want something where instead of plugging numbers into a simplified equation to derive such effects, I want to be able to set up a bunch of charges, press "simulate", and watch the test particle get deflected in the area with zero field.

I don't get all this, especially part 2 seems mathematically dense. But mainly I don't get the premise:

> but we won’t use any index gymnastics or explicit rank-2 tensors; just four-vectors.

For me the whole point of tensor notation is the power of indice gymnastics. I'd love to know the motivation behind this before spending time understanding.

My own notes on covariant notation on this subject: https://0x5.be/physics-notes/four-vectors.html

> The physics is really pretty simple now.

In general, I’d be wondering how much of what we call physics can actually be factored out as just math, and what we’d be left with.

Imho, a lot of the “physics” lies in getting some intuition for the consequences/implications of some mathematical model (or, conversely use the same idea to propose new models). It’s one thing to succinctly write Maxwell’s equations in some chosen notation, but it’s another thing to understand why E & B fields in a light wave are separated by a pi/2 phase. Or why monopoles could exist and what their profile might look like (and the implications for charge quantization).

While it’s technically true that it’s “just using math & computing consequences”, it’s an endeavor of enormous (essential) complexity, warranting focus and effort. Likewise that engineering is “just applied physics”, product design is “just applied engineering”, etc.

We’d be “left with” the left hand side of our newton system of course!

Eh hem, That was a computational joke. I know, I know, but anyway, build a Newton’s method for your physics, and the physics is all in the residual (right hand side) with the left hand side purely there to drive it to zero.

If you are into this, please consider A O Barut's dover book.