# 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. – Gerhard Paseman Feb 10 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. – Qfwfq Feb 10 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 – AlexArvanitakis Feb 10 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. – arsmath Feb 10 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/…). – Gotthold Feb 11 at 19:18

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.