For most of the usecases associated with the examples you mention - Pauli and Dirac algebras - I don't think it matters very much. For instance, I haven't seen a decent QM course that doesn't devote at least a lecture or two to discussing the origin of Pauli matrices and the associated abstract algebra and group theory - as they should - but I don't think the course would be any more concise or useful (in terms of the physical content) if it then proceeded to treat QM in the language of geometric algebra. There's nothing wrong with the matrix representations (especially in a more computational context), though obviously one should know where the representation comes from and how to derive it.
I would argue that E&M and anything involving rotations (or Lorenzt boosts!) are two great examples where GA is miles ahead of anything other than differential forms but still has modest advantages of differential forms as well.
The main advantage I see in geometric algebra is that its a unifying framework. The same langauge used to understand E&M in GA is also useful in quantum mechanics, general relativity, complex analysis, linear algebra, etc. I'd argue that physicists don't mind this mish-mash of notations because they've already put in the work to learn them all separtely but I think there are real advantages (especially pedagogical) to unifying them into one framework.
Your first point got me curious, though. I found E&M in differential forms quite satisfying from an intellectual point of view (although of little practical benefit to the problems I was interested in compared to the classical approach, which I was plenty familiar with already). What advantages do you see for GA vs. differential forms in this context? This might make for a few interesting seminars with our math folks...
One advantage I see in the GA treatment of E&M is in radiation. If one takes the Faraday bivector for a stationary charge (ie. the Coulumb field) and then just 'rotates' it into a moving frame using the GA rotors you can find the standard radiation fields of E&M quite easily. This is cool because it's not only a neat proceedure mathematically, but its computationally quite efficient, say if you need to find the fields produced by N moving charged particles.
I'm sure there are more examples of things like this but it's been a while since I last looked at E&M in GA so none are coming to mind right now.
To get radiation fields, you need Maxwell's equations and boundary conditions. When you have those, what is the use of GA?
1) Start with the Coulomb potential A(r) = qγ₀/(4πr)
2) make it covariant by replacing r -> r -> X⋅γ₀ where X = r(γ₀ + γᵣ). This is a trivial transformation in the static frame as one can easily show X⋅γ₀ == r, but it's important as it encodes the causal information you astutely pointed out is vital to obtaining radiation solutions.
This gives A(r) = qγ₀/(4π X⋅γ₀)
3) Now, one can 'rotate' this to a frame with (not necessarily constant) velocity v via A'(r) = R A(R̃ x R) R̃ ie. evaluate the rotated A at inverse rotate spacetime position. Then
A'(r) = q v/(4π X⋅v)
This is the Liénard–Wiechert potential (this can also be obtained by solving ∇²A = J using the retarded Green's function G(r, t) = δ(|r|-t)/(4π|r|).
4) Now once can get the radiation fields by taking the derivative of this and doing some algebra:
F = ∇A = (q/4π) (X ∧ v + ½ X(v̇ ∧ v)X)/(X⋅v)³
The term proportional to X ∧ v is just the Coulomb field, but the term with the X(v̇ ∧ v)X is the radiation field which one can show propagates to infinity.
This is unfortunately much less elegant and straightforward than I let on initially so I apologize.
I still think this has advantages over standard treatments but your milage may vary.
Preparing for the additional requirements of the lectures of the coming semester is whar the semester break is for.
Especially important in high school is to dive into not only the “complex numbers” but also the “split complex numbers”.
I'm a math professor. To me "geometric algebra" refers to any use of geometry in mathematics to illuminate algebraic principles. There are very very many of these -- this article illustrates one good example. I try to incorporate intuition from geometry into my teaching whenever possible.
It seems that the two of you are referring to some particular unifying framework, and/or set of definitions and notation. Could I ask you to elaborate on what you mean?
Thanks.
https://en.m.wikipedia.org/wiki/Geometric_algebra
https://en.m.wikipedia.org/wiki/Clifford_algebra
For teaching geometric algebra, there are curricula and textbooks available...
http://geometry.mrao.cam.ac.uk/home/introduction-to-ga/
https://arxiv.org/abs/1205.5935v1
http://www.faculty.luther.edu/~macdonal/GA&GC.pdf
http://assets.cambridge.org/052148/0221/sample/0521480221WS....
This looks a bit too high-level to inform my undergraduate teaching, but certainly I could learn something from this.
http://geocalc.clas.asu.edu/html/Evolution.html#References
under "Recent Books". Notably "Clifford Algebra to Geometric Calculus" and "New Foundations for Classical Mechanics" by Hestenes.
PD: if you ever attempt to read any of those two, I'd advise to go to the latest (corrected) reprint.
Less ambiguously called "Clifford algebras", though Clifford himself prefered the term geometric algebra.
Now consider relativity, which was originally motivated by Maxwell’s equations. We can describe flat spacetime as a 4D space with the Minkowski metric, which we can extend into a geometric algebra Cl_1,3(R). We can then describe field and current as multivectors, and we end up with the equation (singular!)
∇ F = µ_0 c J
This has the additional property that it captures how observations of electromagnetism change under Lorentz transforms. If you think of a Lorentz transform as just a change of basis in spacetime algebra, and if you think of electricity and magnetism as together being the basis for electromagnetism, then it’s obvious that (for example) a moving observer would see a magnetic field generated by a stationary charge.
This is obvious because the basis change from a stationary to moving observer will directly correspond to a basis change from an electric field to a magnetic field. This is simplifying a bit but I find it easier to remember and reason about the geometric algebra version of many of these formulas.
The real shame is that the best opportunity to introduce physicists to geometric algebra was a hundred years ago when we were first discovering spinors. First the Pauli algebra and then the Dirac algebra. If physicsts at the time had been made aware of the geometric algebra approach to these things, I think it could have been an appropriate motivation to switch to geometric algebra. Instead, people came up with matrix representations of the Pauli and Dirac algebras and the field is stuck using those and saying "meh. Good enough."