The Case Against Geometric Algebra

Every once in a while the internet gets talking about Geometric Algebra (henceforth GA) and how it’s a new theory of math that fixes everything that’s wrong with linear algebra and multivariable calculus. When I come across this stance I am compelled to respond with something like: “wait wait, it’s not true! GA is clearly onto something but there’s also a lot wrong with it. What you probably want is just the concepts of multivectors and the wedge product!” Which is not very effective, because it takes a long time to convince anyone why, and it’s also not very productive, because this just keeps happening over and over without anything changing. Many people agree with me on this, but they deal with it by mostly ignoring GA instead of complaining about it. But I actually like what GA is trying to do and I want it to succeed. So today I’m going to actually make those points in a longer article that I can link to instead. Specifically what I have a problem with is that the subject is pretty clearly flawed and needs serious work, and especially that the culture around it does not seem to realize this or be interested in addressing those flaws. In particular: Hestenes’ Geometric Product is not a very good operation and we should not be rewriting all of geometry in terms of it. For some reason GA is obsessed with the geometric product, and it’s causing all sorts of problems. They act like this is clearly the way that geometry should be done and everyone else can’t just see it yet, and they have this weird religious zeal about it that is problematic and offputting. It’s also just ineffective: treating certain models as if they are somehow canonical and obvious is wrong, mathematically and socially, and it puts people off right from the start. There probably is a place for the geometric product in a grand theory of geometry, but it’s not front-and-center like GA has it today, and as a result the theory is a lot less compelling than it could be. If something like GA is to succeed, it will need to be improved. It will need to fix the problems with establishment mathematics better than it does now, in a way that everyone can get behind. Today it helps sometimes but often misses the mark, and people who can see that are alienated by the lack of self-awareness about this. As a result GA’s relationship to mainstream mathematics is tenuous.

Basically, GA is considered a kooky, crackpotty sideshow. And because it is so dubious and un-self-aware, the movement ends up alienating most people, except for a particular type of… zealous individual… who write about it with a sort of pseudoreligious zeal, and are prone to conspiracy, as if the only reason GA is not mainstream is that they are being oppressed by close-minded traditionalism. If that’s what you think, let me be the first to inform you: no, that’s not it. GA is interesting, but it’s just not very compelling at the moment. GA continues to find more enthusiasts every year, because it really does address some actual problems. And it will take those new people a while to realize what’s going on—that the thing they’ve discovered is not as solid and revolutionary as they think it is. In the meantime they will go on selling other people on GA, creating the next years’ converts and repeating the cycle. My opinion is that this dynamic is causing GA to be stuck in a sort of perpetual mediocrity, where everyone’s defending the surface-level philosophy because they think they’re part of a revolution, but nobody’s bothering to criticize or improve the underlying structural problems. My purpose in writing this is to push it to improve and address those problems. They’re very fixable, but first you have to notice that something is wrong. The rest of this article substantiates my stance. It is very long, because I decided to include every argument I could think of. But I want to emphasize that, although this is my own long and opinionated rant with lots of individual parts that nobody else is really saying, I am far from the only person who believes the big picture (quite a few have emailed me to agree on this). The state of things is that lots of credible people just roll their eyes at GA and then move on and ignore it. But I actually do believe in GA’s philosophical project: math should be changing in this direction. So I would like to see that change, and to do that we need to establish what’s wrong with it now. Big disclaimer: I’m not a professional mathematician, and this is not going to be the case that a serious mathematician would make, which would probably be something like “GA doesn’t prove anything new so who cares?” (

Fine, but the goals of research mathematics are rather unrelated to the goals of people who use mathematics for practical purposes.) Also, since I do more-or-less subscribe to the underlying program of GA, I am at least slightly on the crank side of the fence as well. Take me seriously at your own risk. First I am going to describe my understanding of what GA is, how it got to be that way, and where it lives in relation to the rest of math and physics. This will be useful in order to pinpoint exactly what we’re disagreeing about here. As far as I know the whole story isn’t really documented anywhere else, so I’m just going off what I’ve picked up over the years. But I am no historian and don’t really know how to check it against reality; I’d be happy to be corrected on anything in here. 1. A lot of background on GA Geometric Algebra is both a social movement and a branch of mathematics. As a social movement, it’s a group of people who believe that (a) mathematics research and pedagogy ought to be reformulated to be more useful, especially to its users who are not research mathematicians, and (b) this reformulation ought to be done in terms of a particular set of new primitives, which is the branch of mathematics side. The argument for doing this is that it would make a lot of math simpler, easier to understand, and easier to use for practical purposes. As a branch of mathematics, it is a recasting of a subject called Clifford Algebra (henceforth CA), which is a somewhat-obscure descendent of another subject called Exterior Algebra (henceforth EA). Because it’s important to understand which parts are GA and which parts aren’t, here is how the various algebras relate to each other: Exterior Algebra is built on the ‘exterior product’, denoted \(\b{a} \^ \b{b}\), commonly called the ‘wedge product’ because it looks like a wedge. It incorporates the concept of ‘multivectors’ in a vector space, such as bivectors that form a vector space of oriented areas, or trivectors which form a vector space of oriented volumes. It also includes some other natural operations that you need to do math with those: the Hodge star \(\star\) and an interior product, which generalizes the dot product to multivectors.

EA comes up naturally in abstract algebra, and in various downstream fields, such as algebraic topology. It also seems to exist in combinatorics literature with a somewhat different set of notations. More generally, EA provides the only actually good way of looking at a lot of the concepts in linear algebra, such as determinants, matrix minors, and cross products, a perspective which seems to be gradually infiltrating the literature that touches those subjects but mostly has not made its way into the undergraduate curriculum yet. If an undergraduate linear algebra class is chapter one of the larger subject of linear algebra, then EA is chapter two. It’s already well-known to mathematicians and physicist, and my impression is that it is gradually becoming ubiquitous at the graduate level but is mostly not taught to undergraduates. Most physicists and some mathematicians will first learn about EA from its ubiquitous use in the field of Differential Geometry or its application in General Relativity, where it shows up as the “exterior calculus of differential forms”, which are used to do differential geometry in a coordinate-free way. (For some reason physics has mostly treated EA as just a thing you do with differential forms, instead of a general theory of vector algebra. This is unfortunate.) Clifford Algebra is like a more advanced version of EA. It is much less widely-known, but it is well-established in certain subfields of math and physics, and a lot of physicists in particular use it without knowing what it’s called when studying spin in quantum mechanics. Clifford Algebra is, roughly, an extension of EA which generalizes the complex and quaternion number systems. A Clifford Algebra is formed by taking vectors in space and allowing them to be multiplied: \(\b{x} \b{x}\), \(\b{x} \b{y}\), etc. Then there are some simple rules of algebra: two copies of the same vector can be cancelled out, so \(\b{xx} = 1\), and two copies of different vectors can be exchanged with a minus sign: \(\b{xy} = - \b{yx}\). As a result you get an associative algebra where most elements are invertible, so you can talk about the multiplicative inverse of a vector \(\b{a}^{-1} = \frac{\b{a}}{\| \b{a} \|^2}\). This allowing you to take a bunch of objects, expressed as sums of multivectors, and basically do polynomial algebra on them.

There are also more general versions of Clifford Algebras which allow the product of a vector by itself to take different values, such as \(\b{xx} = 0\) or \(\b{xx} = -1\), and you end up with a construction called \(Cl_{p, q}\) or \(Cl_{p, q, r}\) where \(p\) is the number of elements that square to \(+1\), \(q\) is the number that square to \(-1\), and \(r\) is the number that square to \(0\), which are useful in various contexts. Quantum mechanics uses the Clifford Algebra of Pauli matrices, which are roughly the bivectors in \(Cl_{3,0}\), and the gamma matrices, which are vectors in \(Cl_{3, 1}\). But at least at the undergraduate level it’s rare that physics actually refers to them by name (my quantum course did not even mention that the Pauli matrices were quaternions). More generally a lot of the literature on spinors and representation theory heavily involves Clifford Algebras in a way that I don’t understand yet. Geometric Algebra then attempts to take the ideas of Clifford Algebra and Exterior Algebra and spread them much more broadly, rephrasing other aspects of math in terms of those new concepts and operations. Exactly which ideas and operations those are depends on the author, but everyone who uses the phrase “Geometric Algebra” pretty much agrees that operations from CA and EA are useful and ought to be more widely used. By and large “GA” texts are lower-level than Clifford Algebra texts, and often discuss ways of performing useful geometric operations like rotations and reflections, things which often come up in computer graphcs and physical simulations, in terms of multivectors and the Clifford product. Probably both GA/CA and EA could be considered as members of the larger subject of “multilinear algebra” which would include tensor analysis and all of linear algebra as well. There’s an argument to be made that they are really just “the rest of linear algebra”, the big part of it that isn’t included in introductory texts and hasn’t entered the mainstream yet. Perhaps by the next century they will be fully folded into the standard curriculum. I mention all this to make the point that GA does not have any special claim to the wedge product or even the Clifford/geometric product per se.