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Comment by westurner

24 days ago

That quaternions also solve for what we normally have 3D+time for.

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

Summarized by a model:

> 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

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westurner

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westurner  24 days ago

Shouldn't there be symmetry and unitarity given energy conservation? And quaternions express this with rotations in SO(3), but is there a better model than quaternions for EM field theory since 1861?

  -1 = i^2 = j^2 = k^2
  q = a + bi + cj + dk
  q = a + xi + yj + zk

(Edit) Quaternions and spatial rotation: https://en.wikipedia.org/wiki/Quaternions_and_spatial_rotati...

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.