[MisterMashable]

I studied several of Hestenes' papers around ten years ago. Most of them are computational in nature and do not proceed to do geometry in an axiomatic way. This isn't to say he didn't write such a paper but I never found one and I read nearly all his extent papers. The thing I liked most was his derivation of the Kerr metric for a rotating black hole. I couldn't find anything wrong with it and I could actual understand every step, unlike Kerr's original paper. Other than that GA has done absolutely nothing to help gain a better understanding of QFT, GR, string theory, Lie groups etc. (for that an in depth understanding of differential forms is best. Physicists, even mathematicians who truly understand the magic of differential forms are even more rare than programmers with a deep understanding javascript!!!) IMO Geometric algebra is a useful collection of interesting computation tools. It reminds me of Pedrag Cvitanovic's "bird track" diagrams to calculate representations of Lie algebras. Nothing unique or fundamental is gained but it is remarkable that a completely orthogonal viewpoint to some very traditional topics exist. Other examples (from mathematics) of surprisingly novel viewpoints of traditional topics include Kuratowski who has a very unique approach to general topology and F. Riesz came up with the notion of "nearness" which simplifies difficult theorems in functional analysis but it could be used to recreate all of basic mathematical analysis.

[drostie]

One interesting thing that comes to mind is spinor calculus.

Spinor calculus is a bit like geometric algebra; there is a mention to them in the paper (more precisely, to "twistors"). The central idea of spinor calculus relates like this: take your 4-vector (v^0, v^1, v^2, v^3) and form a 2x2 Hermitian matrix by:

    V = v^0 I + v^1 s_1 + v^2 s_2 + v^3 s_3

where s_1, x_2, and s_3 are the Pauli matrices. Then it turns out that det V is the 4-norm of v^\mu:

    det V = v^0 v^0 − v^1 v^1 − v^2 v^2 − v^3 v^3.

The Lorentz transforms must preserve the 4-norm and hence det V, but they must also be linear and map Hermitian matrices to Hermitian matrices, so that given a lorentz transform t, there is a Lorentz matrix L such that:

    matrix (t v) = L (matrix v) L†

(that's not 100% accurate because it can't do PT flips; I think P is something like V → V^-1 while a 4-flip is V → -V; combine them together to get T). The Lorentz transforms are just the group det L = 1 -- the Möbius transformations SL(2, C).

The elegance of this comes when you look at null vectors, where det V = 0, making V a projection -- so V = u ⊗ u† for some u. The action of a Lorentz transform on u is then just u → L u, where L is the Lorentz matrix. Moreover when you work out what the ratio of the components of u are, tracing back through the mathematics, you get varios stereographic projections (x + i y) / (R - z), depending whether it's future-pointing or past-pointing.

So all the light that is coming in towards you is a bunch of null vectors that you can paint on a celestial sphere, projected to the complex plane by a stereographic projection, with Lorentz boosts as Möbius transformations of those points.

Immediate freebies: when a marble is speeding past you it still "looks like" a marble to you; it just seems "rotated" in a strange way, because Möbius transformations map circles to circles. Yes if you try to "work backwards" in your coordinates you'll construct a warped model of the system which is Lorentz-contracted, but that's not what you'll see.

Another freebie: as you accelerate faster and faster, the stars all "tilt" in the direction that you're going, crowding around the point you're travelling to. This is in sharp contrast to all those spacey TV shows where the stars "streak away." One can imagine that for a photon's timeless life, the event of its origin is the only thing behind it; and the entire rest of the universe is in front of it.

It actually gets even better; it turns out that you get to unify the spinor equations for the massless neutrino ∇_{AA'} u^A = 0; the photon ∇_{AA'} u^{AB} = 0, and the weak-field limit for gravity is something like ∇_{AA'} u^{ABCD} = 0 for the graviton. (That may not be 100% correct; I am working from memory here.)

All of that comes from something which is basically a quaternion/geometric algebra application to spacetime.

[westoncb]

It seems like the point isn't to gain anything unique or fundamental in physics or mathematics directly, but rather that, since it has the potential to unify the language used across a number of fields in mathematics and physics, that the adoption of a common language could eventually lead to significant progress. So (if it is in fact an effective unifier), it is something unique and fundamental in the realm of tools—though not within the fields the tools would be applied to. Or do you not think it would be an effective unifier (in the sense of communication) after all?

[MisterMashable]

From experience I would have to say 'differential forms' seem to fit the bill for a unifying mathematical language as applied to geometry and physics. GA seems to me as more of a computation tool. Differential forms are pretty standard in many maths and physics texts. Many paper on the preprint Arxiv use differential forms. The only thing about DFs is that they are very efficient for communicating ideas and doing proofs. For practical calculations they don't really simplify anything (actually get in the way) but it's easy to transform them into the usual vector tensor notation. GA seems less flexible in this regard, you take the product it uses and you either like or lump it.

[jwmerrill]

There's a bit of discussion of how differential forms fit into the geometric calculus framework in [1], which is probably the most concise and readable introduction to the geometric calculus approach to differential geometry. All the machinery of forms is available as part of geometric calculus.

Geometric algebra/calculus has a more direct way to deal with metrical information using the dot product. Forms only use the wedge product: in problems where the dot product would be useful, forms simulate it by applying the hodge dual twice, which is a less intuitive and less direct way to get the job done.

[1] The Shape of Differential Geometry in Geometric Calculus, see section 19.4 for the bit about forms http://geocalc.clas.asu.edu/pdf/Shape%20in%20GC-2012.pdf

[noblethrasher]

I cannot thank you enough for introducing me to Riesz. While finishing up my math degree, I came up with the idea that “it's all about distance/proximity/nearness”, and it looks like the same thing Riesz was thinking according to this: http://www.emis.de/journals/SEMR/v6/a1-10.pdf

Here are some old comments where I used the concept:

https://news.ycombinator.com/item?id=2457581

http://www.reddit.com/r/programming/comments/e429d/best_expl...