# What's "geometric algebra"?

Sometimes one bumps into the name "geometric algebra" (henceforth "GA"), in the sense of this Wikipedia article. Other names appear in that context such as "vector manifold", "pseudoscalar", and "space-time".

A very superficial look at that Wikipedia article, or books on the topic, gives me the idea that it is essentially about Clifford algebras and related calculus. One impression that I got (but I could be wrong) is that there is a relatively small group of authors (I don't know if mathematicians or physicists or both) that have produced work in "GA" and that this group is probably disjoint from the set of mathematicians who wrote about the algebraic foundations of Clifford algebra theory or about Clifford algebras featuring in contexts such as the Atiyah-Singer index theorem or Clifford analysis (the study of Dirac-type operators). Also, I'm not aware if any reference to the field "GA", as such, appears outside works specifically designated as "GA" and written by people in that group. It's also not clear to me if there is an intersection (or even subset relation) between "GA" and the above mentioned areas of mathematics and the extent of such intersection.

1. Is there anything in the field "geometric algebra" that is distinct from usual Clifford algebra theory and/or Clifford analysis, or is it just a different name for the same set of mathematics? Or maybe does it provide a slightly different viewpoint on the same mathematics (like, e.g., probability theory having a completely different viewpoint from measure theory despite being formally measure theory)? If so, what are the advantages of this viewpoint?

2. Are there mathematical applications of "geometric algebra" outside the field itself? There seem to be applications to physics: are these applications mathematically rigorous?

• Are you aware of Emil Artin's book of that title? It mostly has to do with the axiomatic investigation of projective geometries and (among other things) their coordinatization derived from the postulated symmetries. It's possible that what you are asking is inspired or developed from this text, but I don't know enough to give a proper answer. Gerhard "Starting A Change..." Paseman, 2020.02.10. Feb 10 '20 at 15:48
• @GP: no, I was aware of that book, thank you. But I think probably the "GA" in my OP isn't related to axiomatic projective geometries. Feb 10 '20 at 15:58
• It is essentially the same as Clifford algebras, quaternions, etc. equipped with non-standard notation. Usually the context for this is indeed applications to physics where it is argued that their notation is nicer Feb 10 '20 at 16:28
• I have been wondering this question a lot lately, so I'm glad you asked it. I know the theory of Clifford algebras, and I don't understand the miraculous qualities attributed to them. Feb 10 '20 at 21:17
• I'd just like to add that ncatlab has a brief page on this issue: ncatlab.org/nlab/show/geometric+algebra. Essentially it is an issue of presentation, trying to avoid both explicit representation by matrices and also quotients of tensor algebras, though I should also note that Hestenes in his public comments has been very adamant that somehow this distinction is very important in terms of new applications (see physicsforums.com/insights/…). Feb 11 '20 at 19:18

The label "geometic algebra" was William Cifford's name for the algebra he discovered (invented). This bit of history is recounted by David Hestenes, who has perhaps been most responsible for promoting this lovely mathematics for physics, and has many books and papers on this subject. There is also a Cambridge lot, including Chris Doran and Anthony Lasenby, who have a physics book, and a gauge theory of gravity. These men are perhaps the most influential authors, but I would also want to include Perti Lounesto, and William Baylis, in the geometric algebra crowd. I have personally found these authors particularly accessible and a pleasurable learning experience.

What seems to me the most unifying element of those who use the term "geometric algebra" for "Clifford algebra" is the emphasis on a real exposition of the subject. To my mind, since complex numbers are in fact themselves a real geometric algebra, there is no small bit of confusion generated by insisting on the mathematical development of the subject over a complex field. But this is exactly what Emile Cartan did in defining spinors. With spinors deeply embedded in physical theories of elementary particles, a Clifford algebra defined over a complex field has a strong tradition, especially among mathematically inclined theorists. Most algebraists (Chevalley, Cartan, Atiyah) would consider complex numbers the truest form of 'number'.

Before closing on the subject, a serious researcher should include a third subject: "noncommutative algebra". In many respects, this is a similar development but with a pedigree more after Hermann Grassmann than William Clifford. Alain Connes, has developed this subject recently with physical applications.

In conclusion, I offer the advice that 'geometric algebra' is the most accessible and intuitive approach offering a physically descriptive mathematics. It is just multivariate linear algebra from a practical vantage point. I think it should be taught in high school. The 'Clifford algebra' and 'noncommutative algebra' approaches are more abstract and mathematically rigorous, but greatly expand the available literature when used as search terms.

• Elie Cartan, not Emile. Aug 15 '20 at 7:35
• I briefly met David Hestenes once, and he said something like that Clifford algebras are the language in which all of physics should be expressed. Nov 24 at 18:47

The essence of "geometric algebra" (better known as Kähler-Atiyah algebra) is the classical Chevalley-Riesz isomorphism, which presents the Clifford algebra of a quadratic space $$(V,h)$$ as a deformation quantization of the exterior algebra of $$V$$. The systematic use of this presentation allows for the automatic translation of various computations in spin geometry into computations with differential forms, since it identifies the Clifford bundle of a pseudo-Riemannian manifold with its Kähler-Atiyah bundle, which is a certain deformation quantization of the exterior bundle. This affords a particularly 'rigid' approach to certain problems in spin geometry, which is quite useful for example in supergravity and string theory, where one often deals with various kinds of generalized Killing spinors. So this relates to a specific isomorphic presentation of Clifford algebras and Clifford bundles which is useful for certain problems. As such, it is not merely the basic theory of Clifford algebras and Clifford bundles (since it concerns certain specific realizations of such through polyvectors and forms), nor is it something radically different.

This point of view is pursued in quite some detail in the preprint:

Cortes, Lazaroiu and Shahbazi: "Spinors of real type as polyforms and the generalized Killing equation", arXiv:1911.08658 [math.DG], https://arxiv.org/abs/1911.08658,

where the Chevalley-Riesz isomorphism is discussed in Section 3.2. Other references on the subject are:

Calin-Iuliu Lazaroiu, Elena-Mirela Babalic, Ioana-Alexandra Coman, "Geometric algebra techniques in flux compactifications", Adv. High Energy Phys. 2016, 7292534, https://www.hindawi.com/journals/ahep/2016/7292534/, https://arxiv.org/abs/1212.6766

Calin-Iuliu Lazaroiu, Elena-Mirela Babalic, "Geometric algebra techniques in flux compactifications (II)", JHEP06(2013)054, https://link.springer.com/article/10.1007%2FJHEP06%282013%29054, https://arxiv.org/abs/1212.6918

C. I. Lazaroiu, E. M. Babalic, I. A. Coman, "The geometric algebra of Fierz identities in arbitrary dimensions and signatures", JHEP09(2013)156, https://link.springer.com/article/10.1007/JHEP09(2013)156, https://arxiv.org/abs/1304.4403

A nontrivial application of this approach (the application is rigorous in that they prove some hard theorems) can be found in the papers:

Elena Mirela Babalic, Calin Iuliu Lazaroiu, "Foliated eight-manifolds for M-theory compactification", JHEP01(2015)140, https://link.springer.com/article/10.1007%2FJHEP01%282015%29140, https://arxiv.org/abs/1411.3148

Elena Mirela Babalic, Calin Iuliu Lazaroiu, "Singular foliations for M-theory compactification", JHEP 03 (2015) 116, https://link.springer.com/article/10.1007/JHEP03(2015)116, https://arxiv.org/abs/1411.3497

whose main results are summarized in these proceedings:

https://arxiv.org/abs/1503.00373

https://arxiv.org/abs/1503.00273

Lazaroiu and collaborators are string theorists, mathematical physicists and mathematicians, so they wouldn't be interested in merely talking about Clifford algebras by another name (they do make plentiful use of Clifford algebras and bundles in their papers). Here are some examples of the kind of work which they do in spin geometry:

C. Lazaroiu, C. S. Shahbazi, "Complex Lipschitz structures and bundles of complex Clifford modules", Differential Geometry and its Applications, Vol. 61, Dec. 2018, pp. 147-169, https://www.sciencedirect.com/science/article/abs/pii/S0926224518302018?via%3Dihub

C. Lazaroiu, C. S. Shahbazi, "Real spinor bundles and real Lipschitz structures", Asian Journal of Mathematics, Vol. 23, No. 5 (2019), pp. 749-836, https://www.intlpress.com/site/pub/pages/journals/items/ajm/content/vols/0023/0005/a003/

Vicente Cortés, C. I. Lazaroiu, C. S. Shahbazi, "N=1 Geometric Supergravity and chiral triples on Riemann surfaces", Communications in Mathematical Physics volume 375, pp. 429–478(2020), https://link.springer.com/article/10.1007%2Fs00220-019-03476-7

• Ok so this, like in the above answer by AlexArvanitakis, is another context in which Clifford/geometric algebras appear, of which I wasn't aware. Thank you both for your answers. Jun 12 '20 at 1:00
• @Qfwfq: That's the same context. Jun 12 '20 at 1:36
• @AlexArvanitakis: I was referring to the context in my question (which is the same as that in this question mathoverflow.net/questions/352362/…) vs that of your answer (which, by the way, is surely the same as the context of amathematician's answer). I haven't read through anything at all, but my impression was that the two answers (present so far) deal with string theory and supergravity, while the work of Hestenes et al. only involves general relativity and classical mechanics. Was my impression inaccurate? Jun 12 '20 at 7:45
• @Qfwfq The applications in the papers of Lazaroiu et al are to string theory and supergravity, though the wider approach which they develop works for many other problems in spin geometry and spinorial global analysis. "Geometric algebra" is a topic in the theory of Clifford bundles, which is itself a subject in ordinary spin geometry. As shown in the papers I mentioned, it is a rich subject which provides computationally and conceptually useful perspectives on certain problems but it is not a separate or new theory in mathematics. Lazaroiu et al are very clear on that point. Jun 12 '20 at 13:47
• @amathematician: fair enough Jun 12 '20 at 14:31

This is too long for a comment.

I found the article

Lazaroiu, Calin Iuliu; Babalic, Elena Mirela; Coman, Ioana Alexandra, Geometric algebra techniques in flux compactifications, Adv. High Energy Phys. 2016, Article ID 7292534, 42 p. (2016). ZBL1366.83098.

on the classification of Killing (s)pinors using geometric algebra with applications to $$\mathcal N=1$$ M-theory compactifications to 3D.

Their perspective on geometric algebra is explained in section 3. The central object seems to be what they call the Kaehler-Atiyah algebra over some (pseudo)Riemannian manifold $$M$$ which as far as I can tell will reduce to geometric algebra as in Wikipedia when $$M$$ is Minkowski space. They also sketch how the KA algebra is obtained by a quantization procedure.

EDIT: In light of the sociological comments in the third paragraph in the question I should point out that I don't think the authors of that paper would say they are in the group of people who identify as "GA" (Lazaroiu, whose work I've read before, is a string theorist)

Apart from any one else's hijacking of the phrase "Geometric Algebra", my own professional sense of this (from early 1970s) was more based in the sense of it in E. Artin's "Geometric Algebra" notes.

Further, "geometric algebra" would have been the way that many people then (and now) understand the features of many so-called "classical groups": orthogonal groups, unitary groups, symplectic groups, and somewhat more exotic groups defined using quaternions and so on... all this "over $$\mathbb R$$" or "over $$\mathbb C$$")

For number-theoretic purposes, the p-adic analogues of archimedean theorems about quadratic or unitary forms are important. And the Hasse-Minkowski theorem about local-to-global in that regard.

To understand the "adelic" and/or "global vs. local" aspects of the isometry groups of such structures, it is indeed very useful to present them in terms of "geometric algebra".

Also, in a somewhat different direction, on $$\mathbb R^n$$, or on a symmetric space $$G/K$$, a "Dirac operator", conceptually a kind of square root of the Laplacian/Casimir operator, is beautifully (and possibly not-otherwise) defined, very naturally, as a linear differential operator having coefficients in a Clifford algebra related to the ambient structure. It is exactly the right thing.

I gather that physicists' analogues of this have been used for decades, without much formalization.

I've had some PhD students do interesting work on "automorphic forms" versions of "Dirac operators", and this exactly requires expression in terms of various Clifford algebras.

For that matter, D. Vogan and others conjectured, a few decades ago, some structural facts about "Dirac cohomology", in this setting.

So, sure, Clifford, Dirac, and other physicists certainly showed the utility of such ideas... and, by this year (and for a few decades) mathematicians have appreciated the physical content...

This answer is my opinion. I don't have references with me at the moment. Feel free to suggest some by making edits.

Geometric algebra is a school of thought towards linear algebra, geometry, and applications thereof, consisting of the following pedagogical idea:

• An exposition using generators and relations, which is easier for relative mathematical novices to understand than the usual account of Clifford algebra.

That is combined with various contrasting geometric interpretations of Clifford algebra chosen whenever convenient:

• The elements of a Clifford algebra can be thought of as rotations of some space. The space is often the familiar Euclidean space in $$n$$ dimensions, or a more exotic pseudo-Euclidean space given by a bilinear form. The latter possibility allows us to use GA to study the geometry of relativistic spacetime, which is given by the bilinear form $$x^2 + y^2 + z^2 - (ct)^2$$.

• As representing the isometry group in various non-Euclidean geometries, some of which are not pseudo-Euclidean spaces. These geometries include famous examples like hyperbolic and elliptic geometry, which are all Cayley-Klein geometries. These transformation groups includes more than just the rotations and reflections that fix the origin of a vector space, but can also include non-origin-preserving transformations like translations that can be useful in things like computer graphics.

• As a foundation for synthetic geometry a la Euclid. There are resemblances to this in Conformal Geometric Algebra where the elements of a Clifford algebra are sometimes interpreted as familiar geometric objects like circles, line and points. A product between two lines can mean the intersection of the two lines, which is a point. Notice that the elements of a Clifford algebra are no longer thought of as being transformations necessarily, but also as subsets of the plane.

• As an extension of exterior algebra, where instead of interpreting elements of a Clifford algebra as (say) circles, we interpret them as scalars, vectors, bivectors, trivectors, and so forth. This might be helpful when working with differential forms, but I'm not sure.

• As a slick way of expressing some exotic geometries like Laguerre's geometry. This enables you to do things like study these exotic geometries by visualising them on a computer, as is illustrated by the Wikipedia article I linked to. This is helpful because these geometries are fairly unintuitive, but tools like computer animations can make up for that.

Stated more briefly (and with an eye towards a more formal treatment), GA $$\approx$$ Clifford algebra equipped with a geometric interpretation which can be any of the above. Much of the difficulty of the theory is giving a complete account of each geometric point of view. These geometric interpretations contrast with each other, as I've tried to explain above.

I am unsure how appropriate this answer is. I hope to provide some explanation for the recent interest in geometric algebra within computer graphics.

My understanding of this viewpoint comes from the article "Geometric Algebra" by Eric Chisolm (https://arxiv.org/pdf/1205.5935.pdf), and the article "Course notes Geometric Algebra for Computer Graphics, SIGGRAPH 2019" (https://arxiv.org/pdf/2002.04509.pdf).

The story as I understand it, as a computer scientist, is that one starts with a real vector space $$V$$ of dimension $$n$$, and from this one constructs a much larger vector space $$\mathbf{G}(V)$$ which is freely generated by the union of all exterior powers $$\bigcup_{i=0}^n \bigwedge^i V$$. This allows us to talk about oriented subspaces, as well as "linear combinations" of subspaces of different dimensions.

This space has an enormous amount of structure. It's naturally a vector space. It is graded by the exterior product, is equipped with an involution (hodge duality), and forms an algebra with the geometric product. These structures interact very well, and allow us to model geometric transformations such as rotations and reflections with the same algebra, instead of what is usually done wthin computer graphics, where one uses matrices to represent scaling and quaternions to represent rotation.

As the next step, it's possible to projectivize the entire construction above. This enriches all of the above structures with projective duality, and also enables us to represent translations on the original space as linear transforms on the projective space. This unifies all three of scaling, rotation, and translation which makes it quite attractive for describing transformations in computer graphics.

• The first part of your answer is otherwise known as Clifford algebra. Nov 25 at 11:52
• Sometimes, the Hodge dual is not an involution. Thanks for the links anyway. Nov 25 at 12:45
• That's interesting, could you drop a reference to a case where Hodge duality isn't involutive? (In my mind, it wouldn't be a duality if it weren't involutive...) Nov 25 at 12:54
• I think the space freely generated by the union should be replaced by a more mundane direct sum Nov 25 at 16:09