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.<p>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.<p>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.<p>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 don't get all this, especially part 2 seems mathematically dense. But mainly I don't get the premise:<p>> but we won’t use any index gymnastics or explicit rank-2 tensors; just four-vectors.<p>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.<p>My own notes on covariant notation on this subject: <a href="https://0x5.be/physics-notes/four-vectors.html" rel="nofollow">https://0x5.be/physics-notes/four-vectors.html</a>
> <i>The physics is really pretty simple now.</i><p>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.