[eigenspace]

> From a mathematician's point of view, yes, you should write the Maxwell field equations, at least to see it once, that way because you're showing a very low-level symmetry that even the differential forms approach doesn't get all the way to. Differential forms is a standard approach for general relativity, e.g. MTW.

While it's neat to write them all as one equation, I disagree that it's an enlightening perspective to learn. While it seems like writing Maxwell's equations in one equation instead of two is a step forward with even more symmetry, what is actually going on is that you are obscuring the most important part of Maxwell's equations: the gauge structure. Without this, it actually becomes much more hidden just how geometric electromagnetism is.

When you write Maxwell's equations as the pair `dF = 0`, `d*F = J`, the first of those two equations is exactly what tells you that this is a gauge theory, and thus may write `F = dA` where `A` is a vector potential. This vector potential then becomes the connection which defines a covariant derivative in a fibre bundle, and one then sees that charged particles follow geodesics now in spacetime, but in an enclosing fibre bundle. This is foundationally important to modern physics, and IMO obscured by writing Maxwell's equations as `∇F = J`

____

n.b. I'm not a particularly big fan of differential forms either, I think it leaves a lot to be desired, and it's super awkward to constantly have to pull out Hodge Duals every time you want to do something that involves the metric, but I'm also unconvinced that geometric algebra is the answer here.

[jsLavaGoat]

What interests a mathematician isn't 100% the same as what interests the physicist. All I'm saying is there is some math there that's interesting and people should see it once for the math.

[lrasinen]

And then there are us engineers. I don't care much either way whether Maxwell's equations are ∇F = J or some other form, as long as it makes the problem easier to solve.

If I were in the GA Marketing Committee I'd publish a paper with suitably hand-picked worked examples where the vector approach is long and tedious, and GA version is short and sweet.

[simpaticoder]

I like this idea but I get the sinking feeling GA proponents don't really solve problems with GA. Like how Haskell advocates don't write programs and modular synth enthusiasts don't write music.

[radialstub]

Application of the Method of Moments to solve full wave formulation of Maxwell's equations. To derive the EFIE using maxwell's equations is a massive pain. With geometric algebra, all you need is ∇F = J and the MoM becomes a mechanical process.

[eigenspace]

I guess I'd say my point though is that the gauge structure is the mathematically interesting part of Maxwell's equations. (i.e. the fact that `F` is a closed differential form).

Without it, I think it'd be of significantly less mathematical interest because it'd lose almost all of its geometric properties.

[jsLavaGoat]

There isn't just ONE interesting facet of this. There isn't just ONE mathematical formalism of a lot of these things. GA is just one of those approaches and you should see it just once, just like you should see the group structure and all of that as well. For most applications, the standard vector calculus approach is fine. But the math underlying all of this is full of richness and no one approach is the skeleton key.

Same with programming languages. Some people are like RUST RUST RUST and some are like C C C! I'm like, you guys only use one language?

[oddthink]

I've found differential forms to be more useful than GA, but that might just be that I was brought up in the MTW tradition and don't quite get GA.

Whenever I look at GA, I try to figure out where the metric comes in, and I just don't see it.

For context, way back when I did astro theory and wanted to do things like figure out things like the magnetic field structure in the curved spacetime near highly-magnetized rotating conducting spheres, and then do some basic plasma physics in that environment.

The differential geometry approach at least gives the structure to think about that, then you can go down to the index-style notation to actually get the differential equations you need to solve. The GA approach, I'm not even sure how to frame the problem.

[chombier]

> pull out Hodge Duals every time you want to do something that involves the metric, but I'm also unconvinced that geometric algebra is the answer here.

I don't know, I recently tried to work out how the metric on vectors/1-forms induces a metric on higher-degree forms, and if the geometric product magically gives this for free I'd say it's a win (same for the Hodge star).

[eigenspace]

Both differential forms and geometric algebra are awkward for that sort of thing. I'd just stick with abstract index notation most of the time.

[ajkjk]

That comes from exterior algebra on its own, it's the k'th exterior power of the metric. Best not to conflate that with GA (unless I'm misunderstanding what you're talking about).

[chombier]

IIRC there's a fairly natural positive definite quadratic form on GA (used as the canonical norm) that takes the scalar part of the geometric product of a multi-vector and its reverse.

On the other hand, there's the k-th exterior power of the metric where one asks that wedge/interior products be adjoint in order to extend the metric to higher-degree forms.

I was under the impression that these metrics are the same, but maybe I'm completely wrong? Assuming I'm not, then the GA approach seems more natural to me.

[ajkjk]

The two should be the same up to possibly some factors of k! depending on your definitions.

Just as an example, suppose you have two multivectors (abc) and (xyz) with all the vectors orthogonal (for simplicity). The geometric product

(abc)(xyz)

has its scalar part created by

(abc)(xyz) = (ab) (c.x) (yz) = (c.x) (ab) (yz) = (c.x) a (b.y) (z) = (c.x) (b.y) (a) (z) = (a.z) (b.y) (c.x)

You can see how the dot product (which uses the metric internally) is being applied "in-to-out" : the adjacent terms are dotted, at which point they become commuting scalars; then the next terms, etc. Which, frankly, is dumb. This is why the GA version of a scalar product has the "reverse" operation involved... because the GP is doing this in-to-out thing, the scalar product has to undo it by defining (abc) . (xyz) = (abc) (xyz)^~ = (abc) (zyx) = (a.x) (b.y) (c.z), with ^~ meaning reverse.

Whereas the standard exterior algebra inner product is always left-to-right, giving

(abc).(xyz) = (a.x) (b.y) (c.z)

IMO the GA version is a mess because it's conflating two concepts. When the GP works, it is composing operators, so AB = A ∘ B. But the inner product, at its core, is more like division---it wants to have (a).(a) = 1, since its job is to say say "how many copies of (a) are there in (a)?" To make this work for multivectors (ab).(ab), it needs to be left-to-right. GA does in-to-out to copy quaternions with their i^2 = -1, but that's not necessary -- i^2 = -1 follows from the fact that for a rotation, R ∘ R = -I, so it is composing two rotations, not measuring one in terms of the other. Really i^2 = -1 should not be interpreted as a dot product at all. This is very clear when a metric is involved: R_xy ∘ R_xy = -I is a degree-two tensor which transforms with two factors of the metric, whereas (xy).(xy) = 1 is a degree-zero tensor, a coordinate-invariant scalar. They are just different operations, which happen to overlap in simple cases.

[immmmmm]

agreed, when you start needing the the hodge star, diff form loose quite a lot of their interest.

i'd add it's quite nice in string theories for RR fields and coupling to D-branes, where writing 10 anti-symmetrized indices quickly gets annoying.. and topological field theories..