Geometric Algebra, prerequisites to studying? Anyone here study it?

  • #1
I was curious if anyone here ever studied Geometric Algebra? It seems not so mainstream and fairly new and I feel intrigued by the subject but I don't want to get in over my head. Just browsing through the table of contents of some books has a lot of unfamiliar terms to me.

Here was a couple books that I was thinking of buying:

Geometric Algebra for Physicists:

Linear and Geometric Algebra:

Any guidance would be appreciated, thanks.

Answers and Replies

  • #2
Science Advisor
Hey DrummingAtom.

One thing that I think is important when looking at geometric algebra intuitively, is to consider what it means to not only be able to multiply two vectors, but to invert the process (i.e. division).

This problems leads the consideration of multi-dimensional division algebras and the first non-commutative extension in four dimensions was the quaternion. There are of course lots of things that relate to this object, but one core idea you should be aware of is that it is a division algebra.

What happens when you consider this idea of inverting multiplication is that in the generalized form, you get a geometric product which can be used to define the inner and outer products.

From this idea, you get rotation and this is required in a geometric algebra if you want to have a division (or at least be able to invert a multiplication of vectors) since a division will end up "un-rotating" things in orientability as well as un-doing directional changes and scaling changes.

You do this in the complex numbers, and Hamilton extended these to four dimensions, and the general idea of geometric algebra is to extend this concept to any dimension that allows one to do this multiplication and inversion.

From the complex numbers on-ward you get orientation in terms of the chirality of the system and orientation is defined in an intricate way between the two products (inner and outer) as defined by the one unifying geometric product, where the two products are considered a bivector (just like a complex number has real and imaginary parts, a bivector as a scalar part and a vector part corresponding to the inner and outer products).

Now you have two kinds of ways of thinking about this: You have the whole linear algebra way which looks at vectors, matrices, determinants and all that and you also have the way that Hamilton did which is to use multiplication tables and then expand out the multiplication of two objects in the same way we do it for complex numbers (so just like we use i^2 = 1, in quaternions we get ij = k and so on which we can simplify and then collect terms to get a quantity).

There is actually a well developed theory of the 2nd way as opposed to the first way which is based on matrices and linear and tensor algebra and if I can dig up the resource for this (which I know I have) I can point it out (it is a free document on arxiv if I recall correctly).

Now you have simplifications in physics for these things for all that theoretical stuff, but the idea of having this multiplication and division in both the linear algebra, determinant, exterior algebra formulation (and other similar ways) as well as in the way that Hamilton did it (which as I said, has been extended to arbitrary cases in many ways) look at the same kind of thing.
  • #4
For prerequisites, it might be helpful to understand traditional vector/tensor algebra/calculus. Usually in physics oriented books, the authors will assume you already know the physics of the situation they're applying GA to. This makes understanding GA rather tricky when they say "let's use the Dirac equation as an example to show how GA works" when you've never seen the Dirac Equation before and don't have a clue what they're talking about.

In the early stages, be sure to take the time to learn all the arithmetic regarding the geometric, inner, and outer products. These things can seem trivial, but it can actually get quite complicated. Having a good command of how the different products work will make more advanced material easier to digest.

Be sure check out anything written by D. Hestenes of Arizona State University, he has a lot of material available on his website. He's really the guy who got people interested in GA. His book New Foundations for Classical Mechanics is a good starting point if you can find a copy. He started promoting GA back in the 1960's but it didn't really catch on until more recently, though rather slowly. The biggest community seems to be the field of computer graphics.

Doran and Lasenby is probably the best single source for a wide variety of different physics applications. Personally, I've found that books for computer graphics applications provide a very good introduction to GA and how to visually interpret the various objects and operations.

One thing you'll find when using multiple sources is that everyone pretty much uses different notation, which can get annoying and confusing when you're just starting out. Some people write vectors in regular font, some use bold font, etc. Also, since there are so many different types of objects in GA, it's really hard to figure out how to handwrite things. For instance Doran and Lasenby write linear functions in capitalized non-italicized sans-serif font and bivectors in capitalized italicized serif font. In my handwriting, these two look identical.

Check out things categorized under the name Clifford algebra too. It's essentially another name for GA, though there are subtle differences between the two. The Clifford algebra formalism is a little more mainstream than GA, so there's more written about it.
  • #5
I started learning about Clifford algebra recently. Tensors are familiar in physics; the algebra of tensors is built up from vectors using the tensor product, and it includes the vectors as a special kind of tensor. Similarly a Clifford algebra is an algebra built up from the vectors, but using a different product called the Clifford (or geometric) product. A Clifford algebra contains the vectors as well as more general elements (comparable to higher rank tensors) called multivectors.

Given any vector space we can build a tensor algebra from it. But to build a Clifford algebra we need to be given an inner (that is 'dot') product along with the vector space. Pure mathematicians (e.g. Chevalley, E. Cartan) have studied Clifford algebras in the abstract, with arbitrary vector spaces, fields and inner products. The corner of Clifford algebra known as geometric algebra deals mainly with 'ordinary space,' i.e. R^3 with the ordinary dot product, and 'Minkowski space,' i.e. R^4 with the inner product familiar from special relativity. The Clifford algebras in each case can be used much like the tensor algebras; you can use multivectors, like tensors, to represent things which vectors don't suit, like the electromagnetic field.

There are several books that I've found really helpful (I'm a beginner).

Pertti Lounesto's Clifford Algebras and Spinors starts right from the beginning, introducing vector spaces before building up the Clifford algebra in 2-D. The first half of the book introduces Clifford algebra and some physical applications, the second half covers more advanced mathematical topics (I haven't looked at the second half). It is fairly concise, and has lots of exercises with answers, which include lots of chances to get used to actually calculating things and a few proofs too.

Geometric Algebra for Physicists by Doran and Lasenby, mentioned above, is also very useful. It contains a free standing introduction to Clifford algebras, as well as 'geometric calculus,' which is (multivariable) calculus with Clifford algebra. It covers a wide variety of physical applications including point and rigid body mechanics, electromagnetism, and spinors in quantum mechanics. The applications chapters aren't stand-alone introductions, and as mentioned above they probably won't make sense unless you know the physics before.

For a summary of the maths A Survey of Geometric Algebra and Geometric Calculus by Alan Macdonald (available here ) is very clear.

The (partly unfinished) notes Clifford algebra, geometric algebra and applications of Lundholm and Svensson, found here, are much more mathematical and looked a bit intimidating to me, but they define the various products of Clifford algebra in a very simple way which allows you to prove identities essentially by verifying obvious Venn-diagram type set theory relations. They also treat duality, and how it relates the various products, very clearly (these duality relations speed up derivations a lot).

Hestenes' Spacetime Calculus, here, is a concise introduction to Clifford algebra in relativity, I found the section on electromagnetism very helpful. I haven't looked much at Hestenes' New Foundations for Classical Mechanics, and Hestenes and Sobczyk's Clifford Algebra to Geometric Calculus.

As an aside, some books on geometric algebra are written from the point of view of trying to establish geometric algebra as the 'universal mathematical language' or 'a grand unifying nexus for the whole of mathematics' (quotes from Hestenes website ). This involves reformulating everything in terms of geometric algebra and criticising other ways of doing things. As a result these books can sound a bit 'cult-like', which can be off-putting. It is not true that tensors and differential forms lack geometric interpretation; looking in the 1951 edition of 'Tensor Analysis for Physicists' by Schouten (one of the founders of tensor analysis), you find a glossy page with photos of wire and foam models of geometric representations of various kinds of tensor! Emphasising geometric interpretation is helpful, but not as revolutionary as you'd guess from reading some geometric algebra books. Also, the conviction that everything must be done using geometric algebra leads to squeezing subjects like differentiable manifolds into a formalism that doesn't seem to suit them (to me).

Geometric algebra does suit some things extremely well though, for example:

(i) Describing spinors, and understanding the Pauli and Dirac matrices as representations of basis vectors in space and spacetime respectively. This speeds up calculations and makes some Pauli and Dirac matrix identities obvious. For example the Pauli matrix identity [itex](\vec{x}\cdot\vec{\sigma})(\vec{y}\cdot\vec{\sigma}) = (\vec{x}\cdot\vec{y})I+i\vec\sigma\cdot(\vec{x} \times \vec{y})[/itex] expresses the simple GA relation [itex]\vec{x}\vec{y} = \vec{x}\cdot\vec{y}+\vec{x}\wedge\vec{y}[/itex] via a matrix representation using the 'vector of matrices' [itex]\vec\sigma[/itex].

(ii) Doing electromagnetism. Maxwell's equations reduce to a single GA equation [itex] \nabla F = J[/itex] and, more importantly, in GA [itex]\nabla[/itex] can be inverted (i.e. has a Green's function), so you can solve for the 'field tensor' [itex]F[/itex] in terms of the four-current [itex]J[/itex] directly without introducing potentials. As another example, Coulomb's law in electrostatics follows simply from [itex]\vec\nabla\cdot\vec{E}=\rho[/itex] and [itex]\vec\nabla\times\vec{E}=0[/itex], since together these can be written [itex]\vec\nabla\vec{E}=\rho[/itex], and we can invert [itex]\vec\nabla[/itex] by integrating with its Green's function.

In short, while some people get overzealous, Clifford algebra is frequently very useful.
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  • #6
Science Advisor
Homework Helper
2020 Award
this seems based on work by grassmann and clifford from the 1840's and 1870's. classic mathematical discussions include the famous book by emil artin, geometric algebra 1957, and various papers in topology from the 1960's by bott and atiyah relating clifford algebras to K theory and periodicity theorems. apparently hestenes introduced the subject to physicists somewhat later.

a nice book for undergraduates that explains clearly the geometric version of multiplication involved in its fundamental setting, is a geometric approach to differential forms, by david bachmann, a work that was read communally here a few years ago of PF.
  • #7
GA Note and Software

You may wish to check out [Broken]

The repository contains my notes on GA (following Doran and Lasenby with the blanks filled in) and python software (detailed documentation in LaTeX docs) for symbolic manipulation of multivectors.
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