A Unified Mathematical Language for Physics and Engineering (1996) [pdf] | Hacker News Reader

A Unified Mathematical Language for Physics and Engineering (1996) [pdf]

mrao.cam.ac.uk

138 pointssigna1129 comments

For those interested in the history, I highly recommend Crowe’s book A History of Vector Analysis. It’s written a bit from the perspective of the dominant “modern” system of vectors from Gibbs/Heaviside, so it probably somewhat undersells the significance/usefulness of both Hamilton’s quaternions and Grassmann’s vectors (or what Hestenes calls “geometric algebra”), but the historical and biographical details are fascinating.

Also recommended is all of Hestenes’s work (e.g. A New Foundation for Classical Mechanics, his Oersted Medal lecture, various papers), as well as the book Geometric Algebra for Computer Science by Dorst, Fontijne, & Mann. The latter is a bit light on the mathematical formalities but gives a good introduction that could get someone started using geometric algebra in practice in e.g. robotics, computer graphics, etc.

Dorst, Fontijne, & Mann is the best pedagogical introduction to geometric algebra that I have seen. I spent a couple years very excited about the ideas in Hestenes' papers, and Doran and Lasenby's book, but unable to actually calculate anything. Dorst, Fontijne, & Mann have very good exercises, and build up the general framework slowly and clearly.

When I took a graduate class in General Relativity under Robert Brandenberger, one of the more unconventional (and brilliant) approaches he took with his students was to introduce the formalism of geometric algebra quite early on, along with tetrads and spinors. While confusing at first, it proved to be extremely enlightening when Maxwell's equations were brought into play.

Would you mind expanding on this? With as much detail as you feel like writing? It's incredibly interesting.

Were Maxwell's equations still condensed to 4 equations (Heaviside's equations in Gibb's vector notation), or something less concise, as I believe they were with quanternions?

I believe they become Maxwell's Equation, and it's super simple.

Expressing Maxwell's equations using the six-component electromagnetic tensor F, they become

  dF  = μJ
  d*F = 0

where μ is the magnetic permeability of the vacuum and J is the electromagnetic 4-current. The operator 'd' is the differential operator from exterior algebra, and the '*' is the Hodge dual.

Using the bivector field F = E + iB from geometric algebra, they become

  DF = μJ

where μ and J are as before, and D is the covector derivative.

For comparison, using the traditional Gibbs/Heaviside notation Maxwell's equations are

  ∇ . E = ρ/ε
  ∇ . B = 0
  ∇ x E = -∂B/∂t
  ∇ x B = μ(J + ε ∂E/∂t)

I don't think it's that uncommon. The advanced GR course I took used tetrads and spinors. It seems to me that this is rather standard for a graduate GR course.

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.

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.

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?

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.

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

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...

This 1996 paper is an introduction to Geometric Algebra, which is in effect a coordinate system where some manipulations are greatly simplified.

My resident core-stability expert (who might now have to learn about GA) alerted me to this thread. Can't say how happy I am that people are looking at it -- am happy even with the useful criticisms (have just emailed the link to the other Lasenbys, Doran, Dorst, Hestenes). Another similar semi-popular IEEE article (without equations) will appear soon.

I really enjoyed reading it but I got stuck on the equation at the bottom of page 2. Why is the condition satisfied equal to -1? Why not 1 or 0?

Thanks :-)

The equation is a definition; the letters i, j and k do not represent variables :)

Edit: If you want a more detailed explanation, the Wikipedia entry on Quaternions might be helpful (specifically the multiplication table at the top).

Grassmann/Exterior Algebra is really neat. It also gives a natural deduction of a lot of basic multidimensional calculus concepts and has a nice lead-in to differential geometry.

Gauss-Bonet all the way! :)

The survey of historical background here is nicely done. Accessible and quite concise given the amount of material it surveys, but with substance.

I remember being exposed to Geometric Algebra by Geomerics who were bragging of their new approach to global illumination when they were still a relatively new company.

Their demos were quite impressive and they sparked a lot of interest in GA in the graphics programming community.

EDIT: actually, I just realized that the founder of Geomerics ( Chris Doran ) is quoted as one of authors in this paper.

For those who want to learn geometric algebra I strongly recommend Linear And Geometric Algebra by Macdonald. I taught myself this subject entirely from this book (making sure I did all the exercises) and even enjoyed the process! Slightly weirdly an application of the algebra became useful at work shortly after finishing the book.

At the bottom of page 15 they have a reference from 1999 -- after the publication of this paper!

The wonderful thing about standards is that there are so many from which to choose!

Edit: With regard for dang's call for substantive comments (are koans not substantive?):

In physics, we emerged from a forest of units to a standardization on CGS units, then we changed gears to SI. CGS remains no less elegant than the day it was invented; electrodynamics is beautiful when viewed through the lens of CGS.

Languages are tools, we shouldn't expect one of them to suit all situations.

Changing conventions for units and moving toward standardizing across all scientific fields to the SI units (meters, etc.) is useful, but this article isn't about changing units. This idea of a new set of abstractions for representation and exploration of physics is a much much more powerful idea.

When I first learned special relativity, I remember struggling with my intuition and having to rely on unfamiliar formula to find the answer to even simple problems. The farther one goes in physics the more one has to trust the equations (electo-magnetics, special and general relativity, quantum physics, etc.) and the math itself starts to obscure whats happening. Having simpler representations and more powerful abstracts is an exciting possibility. I saw this again when trying to solve some simply stated problems (e.g. a particle falling off of a frictionless sphere), mathematics like Lagrangians which I didn't know when I first attempted this problem make the solution so much easier.

As an analogy for those that aren't really interested in the mathematics, these ideas are a bit like the jump from algebra and infinite series math to integral calculus. Although, in theory, one could solve many problems of physics without calculus (see for example [1]), the use of calculus immediately opens up a better understanding and the ability to describe and solve more realistic problems (like cars that don't travel at a constant speed).

[1] "Feynman's Lost Lecture: The Motion of Planets Around the Sun" by David Goodstein

Please.. Do the same for machine learning and statistics...

What do you feel is the division between them? I feel like probability is pretty close to the "unified language of ml and stat"

Reply on news.ycombinator.com