General Relativity has four dimensions.
Quarternions have four dimensions.
Quarternions are useful in calculating the rotation of an object.
If quantum spin is indeed the geometric rotation of an object, can the rules of GR and quarternions be used to describe spin?
Spin is not geometric rotation. Check out hyperphysics’s entry.
…and the reason we know it’s not “geometric” spin is because if it were, the electron would be spinning at roughly twice the speed of light.
And for non-physicists ? 
I thought ‘spin’ was just a made-up attribute used to label a concept that otherwise has no comparible description in the macro-world. Just like quarks have ‘colors’.
Well, the quaternions can be represented as complex 2x2 matrices; scroll down a bit in this page. The quaternions I, J, and K are simply i times the Pauli spin matrices, which are used an awful lot when studying the quantum mehcanics of spin-1/2 particles. The reason that physicists like to use the Pauli matrices instead of quaternions for this purpose is that (a) matrices have a natural interpretation as operators on vector spaces, which is really the essential feature of quantum mechanics from a mathematical standpoint; and (b) the Pauli matrices generalize to higher-spin particles, while the quaternions don’t.
As for the GR angle: I won’t get into too many of the details, but the quaternion/dimension of spacetime thing is just a coincidence. If the Universe had n spatial dimensions instead of three, we would describe rotations in space with an algebra containing n(n-1)/2 + 1 objects (more precisely, the algebra of SO(n) has dimensions n(n-1)/2, and we’ll throw in the identity for good measure.) The fact that n(n-1)/2 + 1 = n + 1 (the spatial dimensions plus time) only holds when n = 3.
Oooh, so close!
Pauli matrices are a basis of su[sub]2/sub, which is the Lie algebra of SU[sub]2/sub, which is isomorphic to Spin(3), which is the double-cover of SO(3), which is the group of rotations in 3-dimensional space. Now, guess what the group of unit quaternions turns out to be…
The Pauli matrices “generalize” because they are also the matrices of the defining (2-complex-dimensional) representation of su[sub]2/sub. In general, there is one irreducible representation of su[sub]2/sub for each dimension (called the “spin-j representation” V[sub]j[/sub] for dimension 2j+1) and a basis for the action of su[sub]2/sub on V[sub]j[/sub] is the “generalization of the Pauli matrices”.
Now, representations of a Lie group correspond with those of its Lie algebra, so there are similarly actions of SU[sub]2/sub on V[sub]j[/sub], and the matrices representing the action of the group of unit quaternions are identical to the “generalized Pauli matrices”.
Nope. Spin comes directly from quantum mechanics. Quarks really do have colors, electrons really do have spin. It’s just not the same macro-world spin/color that you’re used to.
Would you rather they have called it arpgip and guingo instead of spin and color? Physicists just reused old words for new concepts, that’s all.
More on spin:
Spin of the electron comes from the Dirac equation, which is the relativistic form of the Schrodinger equation, which describes the energy of an electron orbiting an atomic nucleus (more or less). There are 4 solutions to the Dirac equation, coupled into two pairs, one pair having positive energy and the other negative energy. The positive energy pair are electrons. The negative pair are positrons. Dirac predicted the existence of positrons years before they were experimentally observed. The fact that there are two positive energy solutions is the reason there is ‘spin’, one electron is spin up, the other spin down.
Absolutely, yes.
That would prevent huge amounts of confusion and needless explanations.
It is.
But “spin” isn’t a completely crazy name for such a property, since it is certainly an angular momentum, and (like classical spin), it doesn’t depend on the center of mass of the particle moving.
To nudge this back in the direction of the original question, it’s pretty common for mathematical physicists to come up with weird and wonderful models using the quaternion algebra, so all sorts of theories involving quaternions are out there. Personally, I’ve never been convinced that there’s anything profound about this tendency. Indeed I suspect that it’s just everybody’s somehow convinced that “quaternion” looks sexier in titles and abstracts than a reference to some everyday group.
Which, given how clumsy quaternions seem to have been when they were intended as cutting-edge technology, is really rather odd.
Back to the OP, then:
Quantum spin is not the geometric rotation of an object, therefore I can’t see how quarternions would be useful. General relativity usally doesn’t play well with quantum mechanics, so it would be of little use.
Well, from a mathematician’s perspective, there really is something interesting about quaternions.
First, some definitions. A “real associative algebra” is a real vector space with an associative multiplication. That is, it’s a collection of objects with a way of adding two of them, an additive identity (0), an additive inverse for every object (-x), a “scalar multiplication” (a way of multiplying by real numbers with certain properties like distributivity), and a way of multiplying two objects that satisfies the associative law (as well as distributivity and other basic axioms). A “real division algebra” is a real associative algebra with a left multiplicative inverse for every object.
Now, the amazing thing is that the only real division algebras are R itself, the complex numbers C, and the quaternions H. In the field of representation theory, if you have two real vector spaces V and W carrying an “irreducible” group action (basically, no vector subspace is left invariant by the action but the 0 space and the whole space), the collection of “intertwinors” (maps f from V to W such that gf(v) = f(gv)) between V and W must form a real division algebra; R, C, or H.
Since quantum field theory says the space of field values for any particle species is a representation of some symmetry group (notably, the Poincare group which contains SO(3)), and that fundamental particles correspond to irreducible representations, it’s not at all surprising that quaternionic actions start playing a heavy role.