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I spent around two months being completely enamoured by geometric algebra until I realised it flies in the face of some pretty strong trends of mathematics in
by zrkrlc 6y ago
I spent around two months being completely enamoured by geometric algebra until I realised it flies in the face of some pretty strong trends of mathematics in the last couple of decades.
How come? Well, we’ve been moving towards more and more powerful generalisations and unifying concepts over the years. First, it was the various flavours of algebraic this-and-that, then category theory, then today, people are trying to do hot things with HoTT.
(of course math is much, much bigger than this, but this is the story I particularly have been watching unfold before me)
The trouble with geometric algebra is that it’s merely a special case of a much more general concept, namely tensor algebras. Tensor algebras are the most general algebra you can possibly get from things that behave like vectors; with GA you are stuck with vectors where the distances are measured by something squared as opposed to any other power (or some other crazy function like the taxicab metric).
Using geometric algebra feels like typeless programming because you’re combining everything into a single quantity called the multivector whose components aren’t straightforward to keep track of. Furthermore, you have to decide the dimension in advance because IIRC these types change when you add more dimensions (the formulas however do not, surprisingly).
That said, it’s really marvellous how compact things can get if you stay in 3+1 dimensions. Maxwell’s equations become literally a one-liner, and the horrible, horrible mess that is rotational dynamics becomes so much more tractable for mortals like me. One can literally do the entire physics undergrad curriculum using GA without breaking a sweat. Trouble is, you’d have to learn some basic algebraic theory and that’s a hard sell for people who are already afraid of math because “it’s too abstract”.
- adamnemecek 6y agoGA and tensors are closely related. I would say that tensors are more like typeless programming, the fact that you pick an algebra in advance is similar to encoding your invariants in terms of types. Tensors are cool and all but they are a pain to work with. GA is more intuitive for working with space.
- Koshkin 6y ago> Maxwell’s equations become literally a one-liner There still must be a limit to this. For example, I do not find "dF = 0" especially enlightening. It's almost like saying that "42" is the answer to the question about life, universe, and everything...
- deleted 6y ago[deleted]
- layoutIfNeeded 6y agoI guess it depends on how you define “dF”. The less arbitrary the definition is, the more likely that you’ve found something profound by stating “dF = 0”.
- adamnemecek 6y agoCheck the one line https://slehar.files.wordpress.com/2014/05/maxwells1.png https://slehar.files.wordpress.com/2014/05/maxwells1.png From https://slehar.wordpress.com/2014/03/18/clifford-algebra-a-visual-introduction/ https://slehar.wordpress.com/2014/03/18/clifford-algebra-a-v...
- alquemist 6y agoThat's cheating. One would need to also define F, c, ε0, J, assuming ∇ is known from calculus. Which would make for 5 lines :)
- handol 6y agoYou can't really get away with just knowing del from vector calculus, as this is the geometric-calculus del. ∇F = ∇ . F + ∇ /\ F But for defining the terms you're going to need a couple paragraphs to go along with the one equation.
- alquemist 6y agoFor those curious, ∇ . F is called 'interior derivative' and relates to divergence, whereas ∇ /\ F is called 'exterior derivative' and relates to curl. https://en.wikipedia.org/wiki/Geometric_calculus#Interior_and_exterior_derivative https://en.wikipedia.org/wiki/Geometric_calculus#Interior_an...
- yiyus 6y agoWell, but if you need to define the gradient, curl and divergence and every term in the equations, the typical representation would be more than 20 lines. So I think that it's a fair comparison.
- joppy 6y agoTensor algebras are the most general algebras you can get (categorically inclined people would call them a “free object” in a suitable category), but because of this they are almost never useful on their own. We pretty much always study specialisations of tensor algebras, of which GA (or more generally Clifford algebras) are an example. There are no interesting formulas in a tensor algebra, because the whole idea of it being “most general” (every other algebra is a quotient) is that it has no relations. It is only when passing to more specific algebras (symmetric, exterior, GA, ...) that the subject really becomes interesting.
- creata 6y agoYep. For a silly analogy, it's like saying that set theory will replace every field of mathematics.
- pjbk 6y ago> There are no interesting formulas in a tensor algebra, because the whole idea of it being “most general” (every other algebra is a quotient) is that it has no relations. It is only when passing to more specific algebras (symmetric, exterior, GA, ...) that the subject really becomes interesting. And yet you can still go a level further, with objects like those in Moon & Spencer's holor theory that include generalizations for transformation rules beyond covariant or contravariant, generalized connections similar --for example-- to universal Christoffel symbols, and effectively creating rules for tensors, pseudotensors and stranger array-like objects.
- JadeNB 6y ago> And yet you can still go a level further, with objects like those in Moon & Spencer's holor theory that include generalizations for transformation rules beyond covariant or contravariant, generalized connections similar --for example-- to universal Christoffel symbols, and effectively creating rules for tensors, pseudotensors and stranger array-like objects. One can always go farther, but I'm not sure that this is doing that. I looked up the Wiki page https://en.wikipedia.org/wiki/Parry_Moon#Holors https://en.wikipedia.org/wiki/Parry_Moon#Holors and found: > But this is incorrect, in general, for the definition of a tensor includes a specific dependence on coordinate transformation. This is very much a physicist's definition of a tensor; a mathematician's definition includes no such thing (although one may prove results about co-ordinate transformations as a theorem). For example, there is no such thing, mathematically, as an intrinsically covariant or contravariant tensor, although, having chosen a fixed vector space $V$, one may certainly put a bi-grading on $T(V \otimes V^*)$ that makes that sort of distinction.
- thaumasiotes 6y ago> One can literally do the entire physics undergrad curriculum using GA without breaking a sweat. Trouble is, you’d have to learn some basic algebraic theory and that’s a hard sell for people who are already afraid of math because “it’s too abstract”. How many of those people go through an undergraduate physics curriculum?
- zrkrlc 6y agoSurprisingly many. A lot of my professors tend to see math as a 'necessarily evil' and only teach enough to be able to do the calculations, and this attitude certainly rubbed off on most of my classmates. In fact, some of them even reject papers for being too rigorous (which does make sense if you think about it).
- andi999 6y agoSo what are Maxwells equation in geometric algebra? In differential form language they are dF=0, d x F=xj, (x is a star, I dont know how to type star here) does it get even easier?