Comment by sgdpk | Hacker News Reader

Comment by sgdpk | Hacker News Reader

Personally, an appealing part of GA is clarifying that the magnetic field is best viewed as an "oriented area", instead of a pseudo-vector [1], which is a concept that is frequently ill-defined. Usually, lecturers say that the magnetic field is essentially a vector, but to be careful that it flips directions when you mirror space. But how then can you check this by just looking at the usual three coordinates of the vector?

With GA, you associate the magnetic field with an oriented plane element. Oriented planes have this flip-mirroring property. Because space is 3D, you have exactly one vector to which this plane element is perpendicular, so you can identify the field with a vector. This is called the Hodge duality, and this particular form is accidental to 3D space.

If you need more convincing, the magnitude of the magnetic field is given by the cross-product, which calculates the area spanned by two vectors.

You could argue that you don't need GA for this, only to use the exterior product instead of the cross product. But by using an algebra that contains both vectors and oriented areas, you can effectively sum the electric and magnetic fields. This will give you pretty compact and beautiful equations, which is the essence of this article.

[1] https://en.wikipedia.org/wiki/Pseudovector

For anyone looking for a sane way to study this stuff, what finally made sense to me was (1) study discrete differential geometry from Keenan Crane's notes: https://www.cs.cmu.edu/~kmcrane/Projects/DDG/ ; (2) Read these notes on geometric algebra that formally lay down what the hell a GA space is and what operations one can perform on them: https://arxiv.org/pdf/1205.5935.pdf (Geometric Algebra: Eric Chisolm)

For clicking convenience: https://www.cs.cmu.edu/~kmcrane/Projects/DDG/ (the link above is broken since HN includes the ]; in the URL).

This is also an excellent and interactive blog-post building up to the understanding of how rotors can replace quaternions in GA: https://marctenbosch.com/quaternions/

You still don’t need geometric algebra to combine the magnetic and electric fields, if you view them as differential forms on 4-dimensional spacetime. From what I can tell, this approach is basically equivalent to what this article does. You could directly translate everything into the language of differential forms, because the only geometric products here are in fact just exterior products.

This "competition" between geometric algebra and differential forms makes me uncomfortable.

As far as I see (and I'm not an expert), they are just different ways to express very similar concepts. They are still not exactly the same, since the geometric product is not defined in DFs, and there is no hodge star operator in GA, for example, but everything you can do using one formalism in practice can also be easily done using the other one. What am I missing?

You're mistaken, GA does have a Hodge star, as I've explained many times before https://grassmann.crucialflow.com/dev/algebra

The exterior product can be derived from the geometric product, so differential forms occur in geometric algebra.

You can easily define it, that's what I meant saying that you can do the same things in practice, but it's not usually defined (at least in the books and articles I've read), and certainly it is not so ubiquitous as in DFs texts. And, of course, the exterior product is contained in the geometric product. I guess that, in the same way, you could define a geometric product operator when using a DFs formulation. Would you then say that geometric algebra occurs in differential forms?

In any case, you did not attempt to answer my original question. Are GA and DFs just different ways to define "equivalent" concepts or is there some more fundamental difference that I am missing?

No, I would say differential forms occur in geometric algebra, not the other way around.

Fair enough. I have seen some comments (not in this thread, it was some time ago) that suggested that DFs allow the same as GA in practice, and everything GA does is adding an unnecessary geometric product, but exterior products should be enough (not my opinion, I can try to find the original comment if you want). I do not know enough to have an own opinion.

You obviously know more than me about this, so I will ask you a slightly different question: if I learn GA well enough and totally ignore differential forms, what will I miss?

Differential forms aren't exactly comparable to GA... I would instead look at the relation between exterior algebra and GA.

Differentials are a concept that the comes from doing calculus on manifolds, and exterior products of differentials are just used for tracking information about oriented volumes.

To answer your question (switching differentials forms for for exterior algebras), you wont miss anything, as the wedge product is part of a GA.

If I remember correctly, the Hodge star much more closely belongs to geometric algebras than it does to exterior algebras, since you need a nondegenerate bilinear form to define the Hodge star, from which you can just as easily define the geometric product, and from the geometric product the Hodge star.

To make an analogy, it sounds a bit like you're asking whether inner product spaces and vector spaces are equivalent. Every geometric algebra gives rise to a Hodge star, and an exterior algebra, and so on, but exterior algebras are a much more general concept, so they're less powerful until you tack that extra structure on.

You are mistaken, the Hodge star does not belong to geometric algebra more than exterior algebra, that's the wrong way to look at it. Exterior algebra is just only a sub algebra of geometric algebra, they both have the same Hodge star. Saying the Hodge star belongs more in one than the other is a bit silly.

Look, you already need a bilinear form to get the Hodge star. My point is that with that same bilinear form you also get an entire Clifford algebra, and a much more natural definition of the Hodge star. That's all I'm saying. Is that mistaken?

To be honest, my question is much more practical, and much naiver too.

I work with people who use Euler angles to express rotations, and it's a horrible world. I learned quaternions in my day, and there are some obvious advantages. When I discovered GA some years ago, it was really eye opening. It makes quaternions an easy to explain concept, even intuitive, and I've used it since then, not only in my own work but also to teach other people. Then, I learned about differential forms, and it's also very interesting, I think I could base my "intuitive explanations" in this other paradigm, but I'm not sure I should. I do not think it makes a big difference in my particular case, but as I said I find this "competition" a bit frustrating, and am trying to understand it better.

I cannot discuss with a mathematician if the Hodge star is a GA or a DFs concept, but I have found it all over the place when reading about DFs, and not so in GA related material (though I have a vague idea about how the Hodge star operator can easily be defined in GAs using the pseudoscalar). But is this really my choice? I have listened opinions about which one is more general, but not really convincing arguments (at least not arguments that are obvious to me).

Thanks for your explanation. I think I need to have a deeper look at this stuff. I like to get lost in these rabbit holes, but sometimes it goes a bit over my head.

There is a fairly nuanced difference that doesn't really matter much unless you are a mathematician. Essentially in GA you'd do your differential geometry assuming a sort of ambient background space. In regular differential geometry the space of forms and vectors are abstracted and don't require a shared geometric embedding. I'm not a mathematician, so this is a very non-precise explanation, but that's how I understand it.

+1 to this exposition!

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