geometric algebra gives geometric meaning to linear algebra and much more. it can provide a coordinate free geometric interpretation of spaces. those who learn of it, tend to be dismayed they weren't taught physics in this framework. what are the pro's and con's of replacing linear algebra and vector calculus with geometric algebra?

Good books on geometric algebra:

  • Geometric Algebra for Computer Science has exercises to self tests, and features prettified pictures for extra clarity.

  • Books by Hestenes have an incredible signal to noise ratio (extreme compression of information to Shannon Limit) and are definitely a must...

  • many introductions floating around, remember search for "geometric algebra" and not "algebraic geometry"

Also try to see the Pascal triangle structure of number of k vector basis blades in n dimensions!

  • 3
    $\begingroup$ I hope you haven't missed the excellent Geometric Algebra for Physicists. I really like Hestenes's intro to "New Foundations of Classical Mechanics", where he suggests that vectors and multivectors are just natural extensions of the concept of number, but the rest of it... $\endgroup$
    – Mike
    Apr 30, 2013 at 20:17
  • $\begingroup$ -10. Definitely not replace vector calculus. And as for linear algebra, first it it is still best to teach Linear, then Exterior (geometric). P.S. You missed this source: Clifford Intro $\endgroup$ Jul 7, 2013 at 9:40
  • $\begingroup$ @user10851 As someone with both math and physics degrees (and someone who enthusiastically supports geometric algebra for physics), I can attest that this would have little impact on a standard math curriculum. Even though geometric algebra could replace much of the fundamental machinery that physics undergrads use, it's still too applied to substantially impact the more abstract methods mathematics undergrads are expected to use. Frankly, every negative comment I see on this page conveys a complete lack of understanding of precisely what "GA" means. $\endgroup$
    – Mike
    Mar 20, 2018 at 13:52

3 Answers 3


Ron Maimon is entirely correct when he says that GA is precisely Clifford algebra from a mathematical perspective, as any book or paper using the phrase "Geometric Algebra" is sure to say. But I think he misses both the point of the question and the point of "GA" — which is different from Clifford algebra from a pedagogical perspective.

The question

I'll paraphrase the question as:

Is GA a good, pedagogical way to introduce the mathematical side of physics to undergraduates?

My answer is: in the future, definitely yes because GA is obviously vastly superior for pedagogy; for now, probably not unless a large fraction of the physics department is willing to get behind the effort.

Eventually, for any student who will need to learn basic div, grad, curl — and maybe even just cross products — I think GA will be the standard approach. My first reason for saying that it's not necessarily a great idea right now is that no good textbook currently exists for this. I'm sure Quirino has inspired many talented undergrads, but I would also worry about losing quite a few. There are many different learning styles, and most of them benefit greatly from having a good, thoroughly refined text — or even a text that has grown out of previous generations of texts. All the books I've seen suffer from their constant attempts to explain why GA is better than standard treatments, rather than just being good, self-contained introductory texts with a focus on actual physics applications.

My second reason for suggesting that it's not a great idea right now is this: unless students can use GA in all (or at least most) of their courses, they'll have to constantly code-switch — going back and forth between GA and traditional treatments. In principle, that could actually be beneficial in getting some students to understand the concepts more deeply, but in practice it's far more likely to just lead to greater confusion. Only after you're already proficient with both GA and traditional methods is it easy or beneficial to code-switch. But this isn't likely to happen right now, because — though it is rapidly gaining in popularity — there still aren't very many professors who know GA well enough that they would teach a class using it. And even though there are instances where that could work for one class, as soon as the student moves on to a non-GA class, much of that learning is useless. Since physics is such a cumulative process — where techniques from an earlier class form the basis for future classes — this is a huge problem.

Now, it's entirely conceivable that you could get enough professors in one department who are sufficiently motivated to actually try a four-year GA curriculum (and sufficiently experienced with GA to be able to teach it well). Together, they could overcome the lack of existing textbooks by getting to the students early on — for example using a standard textbook for first-year mechanics, but insisting that all cross products be done as wedge products (which is code-switching, but easy enough that it would work). The intro E&M course is where I think this would get hardest for the teachers, and would require particularly concerted and sustained cooperation to supply the students with enough materials — extensive notes plus homework, quiz, and exam questions — to replace standard materials. And all that work can't just go by the wayside as students move on to more advanced courses. I think an advanced mechanics course would actually be easier to teach with GA than with the old methods used currently. And by the time students get to advanced E&M, they may be able to handle a standard textbook again and be able to fill in the gaps with their own superior GA techniques (with some help from the teachers). The first round of students might have a harder time, but if the teachers are conscientious about recording and analyzing how to handle the challenges, it would quickly become a successful curriculum. In the five years since I first wrote this answer, GA has become much more mainstream so that this possibility has gone from being absurd to conceivable — but I'm not holding my breath.

"Geometric" algebra

Because there seems to be a lot of confusion on this page about what GA is and how it differs from Clifford algebra, I'll back up and describe the distinctions and why using GA might be a good objective.

Geometric algebra is just a name for a pedagogical approach to teaching physics that introduces Clifford algebra (usually over the reals) in a simple way, emphasizing the geometric nature of the elements and operations, and using Clifford algebra as the fundamental tool for basically all calculations — though without mentioning many of the more formal constructions of Clifford algebra as it is usually presented, or resorting to the stupid complex-matrix approach that physicists usually see. Just as Penrose (usually) makes no mention of left minimal ideals or fiber bundles when he discusses spinors, a simpler approach to Clifford algebra is reasonable — and would be vastly preferable for first-year physics students.

Now, the contention is that Clifford algebra is under-utilized in basic physics. Every problem in rigid-body dynamics is at least as easy when using Clifford algebra as anything else — and most are far easier — which is why you see quaternions being used so frequently. Orbital dynamics (especially eccentricity!) is practically trivial. Relativistic dynamics is simple. Moreover, once you've gotten practice with Clifford algebra in the basics, extending to electrodynamics and the Dirac equation are really simple steps. So I think there's a strong case to be made that this would be a useful approach for undergrads. This could all be done using different tools, of course — that's how most of us learned them. But maybe we could do it better, and more consistently.

No one is claiming that Clifford algebra is fundamentally new; just that it could be bundled into a neater package, making for easier learning. Try teaching a kid who is struggling with the direction of the curl vector that s/he should really be thinking in terms of the algebra generated by the (recently introduced) vector space, subject only to the condition that the product of a vector with itself is equal to the quadratic form. Or a kid who can't understand Euler angles that this rotation is better understood as a transformation generated (under a two-fold covering) by the even subalgebra of $\mathcal{Cl}_{3,0}(\mathbb{R})$. No one here is arguing that that should happen. GA is just a name for a pedagogical approach that makes these lessons a whole lot easier than they would be if you sent the student off to read Bourbaki. Starting off with GA may be slightly harder at the beginning, but pays enormous dividends once you get to harder problems. And once teachers and textbooks get good at explaining GA, even the introduction will be easier.

I used to use ordinary tensor algebra for relativistic dynamics, differential forms and/or tensors for electrodynamics, and spinors for analyzing gravitational radiation. These days, I mostly just use GA for all of that. I like to make use of the manipulations that are so much simpler with Clifford's geometric product, the conciseness of objects that encode multiple grades at the same time, and the consistency of using the same exact formalism in all these domains. Why shouldn't my undergrads get the same advantages?

Should Ron and Luboš abandon their mathematical upbringing and start speaking Hestenes's language? Probably not. Might it be useful to other, younger students of physics? Definitely.

As for why you would use the name Geometric algebra (GA) instead of Clifford algebra (CA): typically, GA is restricted to vector spaces over the reals, so it's just a shorthand. Also, apparently Clifford himself called it Geometric Algebra — so it's not a sin.

I'll also point out that tensors do fit nicely within CA/GA, in the form of linear functions, for which CA/GA provides new interpretations, without making the usual manipulations harder. Also, those nice features that Penrose mentions when he talks about spinors and twistors are present naturally in GA (I'm specifically referring to CA over the reals). He talks about geometric interpretations and calculational efficiencies, which GA shares. But GA goes further. Penrose et al. (Penrose-Newman, Geroch-Held-Penrose, etc.) use complex numbers to combine contractions that really deserve to be separate-grade objects, with accompanying geometrical interpretations. The optical scalars are a prime example, where the standard treatment drops geometrically meaningful elements so that the familiar product of complex numbers can be used. GA can replace this with Clifford's geometric product while retaining the geometry, and using the exact same unified formalism used for many other types of problems.

  • 2
    $\begingroup$ The problem is that it is not simpler! Ask Dimension10 whether it was easier to learn Dirac matrices or Geometric Algebra. The matrices are concrete, computational, and you can immediately see how they work, and you can work with them with no prior intellectual labor. The abstraction is too abstract! The first rule of mathematics pedagogy: any student can easily reconstruct the abstract from the concrete, but it's much more difficult to go the other way. Dirac matrices are already pretty abstract, but a matrix representation is essential for when you get stuck, and students get stuck a lot. $\endgroup$
    – Ron Maimon
    Jul 7, 2013 at 20:33
  • 1
    $\begingroup$ I really think you need to give some of the pedagogical treatments more of a chance. It can be approached abstractly from pure group theory, or concretely, as we teach dot and cross products. In the concrete approach, it's just as easy as ordinary vector algebra, but it generalizes trivially to arbitrary dimensions and signatures. And in use it shares many features with the tetrad approach to GR, which is computationally very simple. $\endgroup$
    – Mike
    Jul 8, 2013 at 12:29
  • $\begingroup$ concretely, doing actual GA calculations literally.... multivector M times multivector N, requires O(n^2) operations. It's conceptually very simple and mechanical. But in practice, it's rather like writing code in assembly language, due to how tedious it can be. I think GA will take off like wildfire, given a good web-hosted IDE for typing and viewing, and manipulating equations with the ease of a physical device. $\endgroup$
    – Rob
    Feb 25, 2019 at 1:42
  • $\begingroup$ i would also add that tools like Mathematica make so many assumptions about commutativity, that you often find yourself going back to paper or LaTeX, and doing things by hand... but perhaps just using a computer to track the computation path you took to arrive at your answers. $\endgroup$
    – Rob
    Feb 25, 2019 at 1:45

For several years I have been teaching Clifford (geometric) algebra as part of the Vector Analysis Course for undergraduate physics majors in Ateneo de Manila University. I strictly use Cl_{n,0}, even for Special Relativity. 18-year old students do not complain how difficult geometric algebra is. They just learn the math and the geometric interpretations: geometric product, dot product, wedge product, cross product, exponentials of imaginary vectors for circular rotations, exponentials of vectors for hyperbolic rotations, etc. For linear algebra, I teach them how to rewrite simultaneous linear equations in vector form and use the wedge product to solve for the unknown parameters. All the properties of determinants are encoded in the wedge product of arbitrary number of vectors. Cylindrical and spherical coordinate systems are best taught in terms of exponential rotation operators, because students immediately see what are the axis of rotation, the angle of rotation, and the vector to be rotated. Vector calculus is also simpler because the gradient, divergence, and curl becomes part of a single spatial derivative operator which may act on scalars or vectors. This becomes very useful when we want to combine all the four Maxwell's equations into one. I think, the more one knows too much math such as tensors, spinors, and differential forms, the more it becomes difficult to understand geometric algebra. Geometric algebra is best taught to the little ones.

  • 1
    $\begingroup$ thanks for this answer, I was suspecting this was done at some universities, sadly not the one Ive been attending... so when I accidentally discovered it (started reading the GA for Computer Science book thinking I picked up a book about algebraic geometry) I was 1 extatic with the geometric reinterpretation of even things I thought I fully understood (like use outer product for solving simultaneous linear equations as you mention, first time I actually saw the parallelogram) but also 2 anger at whatever combination of factors brought me up with traditional "tower of Babel" split framework! $\endgroup$
    – propaganda
    Jan 24, 2012 at 9:07
  • $\begingroup$ So up till now it was only a suspicion that it would also work at universities (perhaps even highschool), but I have never conversed with anyone who witnesses this, confirmation that this can be done is highly nice. Some questions remain: the courses that follow later in the curriculum, they build on these fundaments, are their troubles within the faculty (professors disagreeing with a different notation etc?), are these sidestepped by giving them traditional notation alongside geometric framework? perhaps I'd be pushing it too far if I dare ask if I could get a copy of notes I wont share on? $\endgroup$
    – propaganda
    Jan 24, 2012 at 9:11
  • $\begingroup$ Please check out some of my tutorials: link. They are incomplete, but I hope they help you understand geometric algebra better. My approach is to treat geometric algebra as a group algebra, so I start with basis vectors and define the orthonormality axiom of vectors as a relation for group generators. I think this approach is easier for students to understand than defining inner and outer products in terms of commutators and anticommutators of vectors. $\endgroup$ Jan 24, 2012 at 13:26
  • $\begingroup$ More lengthy tutorials are in arxiv.org. They cover orbits, electromagnetics, optics, and groups. Our other tutorial papers on optics and polarized light are published in AJP 2004 and JOSA 2010 & 2012. I hope these suffices for now. I am the only one teaching geometric algebra algebra in my university, so I could not follow-up my students. I use traditional notation for products like a.b and axb. Whatever physics course I teach, as long as there are vectors--Mechanics or EM, I teach the geometric algebra point of view. $\endgroup$ Jan 24, 2012 at 13:53

First, "Geometric algebra" is "Clifford algebra", and it is no good to hide this by renaming it. Hestenes didn't invent it, and his approach is pedagogically awful and adds nothing new.

Clifford algebras are no good as a fundamental "vector calculus", because the calculus of vectors is not by an algebra of the usual sort at all. It is by the tensor calculus, that allows outer products and contractions. The elementary treatments that try to algebrify vectors by introducing cross and dot products are ultimately failures, they mislead people into thinking that vectors have a natural product structure, producing vectors. They don't. The product of vectors are tensors, and only the peculiarities of three dimensions allow you to find a vector product.

The Clifford algebra is useful for spinors, and it does have some benefit to spinorify all of the vectors. This is done by Penrose in both the diagram formalism for spinorified tensors, and for twistors, which spinorify space-time positions too. These formalisms are useful, but advanced.

Learning Clifford algebra should not be confused by introducing the subject as some sort of fundamental vector calculus.

  • $\begingroup$ You are sharing your personal impression. For me the GA was an eye-opener. $\endgroup$
    – Dimiter P
    Dec 18, 2021 at 12:11

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