Because SU(2) we get a lot of interesting phenomena, including that there are two types of particles, bosons and fermions. We get some interesting phenomena that only rotating by 720deg (two full rotations) bring back to the initial state. And I am not talking only about USB-A, but about spinors (https://en.wikipedia.org/wiki/Spinor) - there are some party tricks around that (vide https://www.reddit.com/r/physicsmemes/comments/181oldw/a_ger...).
That's one of those things I've vaguely learned but really want to spend the time to learn academically so I can actually do the math instead of just hearing about it.
Also lots of HN readers are actual physicists or mathematicians. It's not all techies.
Also, lots of engineers have at some point learned some computer graphics and so been exposed to quaternions in that setting. Since they're mysterious and hard to wrap your head around most people don't really 'get it', leaving a sort of standing curiosity that articles like this tap into.
sl(2, R) ≅ so(2,1)
sl(2, C) ≅ so(3,1)
sl(2, H) ≅ so(5,1)
sl(2, O) ≅ so(9,1)
Dirac equation is the C case, the other cases have their uses.
I mean, there are practical reasons too (which are mostly just isomorphic to the stuff in the paper). But really that's why. It's part of our cultural history in ways that more esoteric math isn't.
And Lewis Carroll (Oxford (Math)), preferred Euclidean geometry over quaternions, for "Alice's Adventures in Wonderland" (1865).
Quaternions:
q = a + bi + cj + dk
-1 = i^2 = j^2 = k^2
> In a quaternion, if you lose the scalar (a) — the "real" or "time" component — you are left with only the three imaginary components (i, j, k) rotating endlessly in a circle.
(An exercise for learning about Lorentzian mechanics, then undefined: Rotate a cube about a point other than its origin. Then, rotate the camera about the origin.)
4D Quaternions (a + bi + cj + dk) are more efficient for computers than 3D+t Euclideans. Quaternions do not have the Gimbal Lock problem that Euclidean vectors have. Quaternions interpolate more smoothly and efficiently, which is valuable for interpolating between keyframes in a physical simulation.
Why are rotations and a scalar a better fit?
Quaternions were published by William Rowan Hamilton (Trinity,) in 1843, in application to classical mechanics and Lagrangian mechanics.
Maxwell's (1861,1862) original ~20 equations are also quaternionic; things are related with complex rotations in EM field theory too. Oliver Heaviside then "simplified" those quaternionic expressions into accessible vectors.
Is there Gimbal Lock in the Heaviside-Hertz vector field reinterpretation of Maxwell's quaternionic EM field theory? Maxwell's has U(1) gauge symmetry.
And then quantum has complex vectors and some unitarity, too
History of quaternions: https://en.wikipedia.org/wiki/History_of_quaternions
-1 = i^2 = j^2 = k^2
q = a + bi + cj + dk
q = a + xi + yj + zk
QED: Quantum electrodynamics: https://en.wikipedia.org/wiki/Quantum_electrodynamics :
> Mathematically, QED is an abelian gauge theory with the symmetry group U(1), defined on Minkowski space (flat spacetime). The gauge field, which mediates the interaction between the charged spin-1/2 fields, is the electromagnetic field. The QED Lagrangian for a spin-1/2 field interacting with the electromagnetic field in natural units gives rise to the action [...]
And QED is the basis for the Standard Model of particle physics and for some theories of n-body quantum gravity.
There's a 1-1 mapping between complex numbers and 2D rotation matrices that only do rotation and scaling. The benefit is that the complex number only has two coefficients, not four like the matrix. Multiplying these complex numbers is the same as multiplying the equivalent matrices. Quaternions are the same idea just in 3 dimensions (so with 3 imaginary units i j k, not just i, one per plane).
I justify quaternions to myself with the intuition from [1]. In essence quaternions represent rotations in 4D, where multiplying by a "unit" (i,j,k), rotates two distinct planes by 90 degrees. The reason introducing a single unit j doesn't work is the same reason this rotation-is-multiplication trick doesn't work in 1D (or really any odd-number of dimensions). Anyways if we call this 4th axis w and pick a simple rule like ij = k then we get some nice properties like
- multiplying by i rotates xy + zw planes by 90 degrees
- j rotates xz + yw
- k rotates xw + yz
- 1 rotates nothing
[1] https://www.reedbeta.com/blog/why-quaternions-double-cover/
funny thing is quaternions had that exact same energy in the computer graphics community for years. after ken shoemake introduced them to CG in 1985, there was a long period of "why are we using euler angles like cavemen when this exists??". now quaternions are well known tooling for people in graphics and the mystique has worn off at least in that community.
scaling -> real numbers
1d rotations and scaling -> complex numbers
2d rotations and scaling -> quaternions
In the case of quaternions, there is called double-covering, which turns out (rather than being an artefact), play fundamental role in particle physics.
A 2d quaternion just has no j or k term and works for 2d math.