Baez week 61



The group of rotations in n-dimensional Euclidean space. This group is called SO(n). It is not simply connected if n > 1, meaning that there are loops in it which cannot be continuously shrunk to a point. This is easy to see for SO(2), which is just the circle - or, if you prefer, the unit complex numbers. It's a bit trickier to see for SO(3), but it is easy enough to demonstrate - either mathematically or via the famous "belt trick" - that the loop consisting of a 360 degree rotation around an axis cannot be continuously shrunk to a point, while the loop consisting of a 720 degree rotation around an axis can.

This "doubly connected" property of SO(3) implies that it has an interesting "double cover", a group G in which all loops *can* be contracted to a point, together with a two-to-one function F: G -> SO(3) with F(gh) = F(g)F(h). (This sort of function, the nice kind of function between groups, is called a "homomorphism".) And this double cover G is just SU(2), the group of 2x2 complex matrices which are unitary and have determinant 1. Better yet - if we are warming up for the octonions - we can think of SU(2) as the unit quaternions!

Now elements of SO(n) are just nxn real matrices which are orthogonal and have determinant 1, so given an element g of SO(n) and a vector v in R^n, we can do matrix multiplication to get a new vector gv in R^n, which of course is just the result of rotating v by the rotation g. This makes R^n into a "representation" of SO(n), meaning simply that

(gh)v = g(hv)


1v = v.

We call R^n the "vector" representation of the rotation group SO(n), for obvious reasons.

Now SO(n) has lots of other representations, too. If we consider SO(3), for example, there is in addition to the vector representation (which is 3-dimensional) also the trivial 1-dimensional representation (where the group element g acts on a complex number v by leaving it alone!) and also interesting representations of dimensions 5, 7, 9, etc.. The interesting representation of dimension 2j+1 is called the "spin-j" representation by physicists. All representations of SO(3) can be built up from these representations, and none of these representations can be broken down into smaller ones - one says they are irreducible.

But the double cover of SO(3), namely SU(2), has more representations! Using the two-to-one homomorphism F: SU(2) -> SO(3) we can convert any representation of SO(3) into one of SU(2), but not vice versa. For example, since SU(2) consists of 2x2 complex matrices, it has a representation on C^2, given by the obvious matrix multiplication. This is called the "spinor" or "spin-1/2" representation of SU(2). It doesn't come from a representation of SO(3).

To digress a bit, the reason physicists got so interested in SU(2) is that to describe what happens when you rotate a particle (in the framework of quantum theory) it turns out you need, not just the representations of SO(3), but of its double cover, SU(2). E.g., an electron, proton or neutron is described by the spin-1/2 representation. This implies that when you turn an electron around 360 degrees about an axis, its wavefunction changes sign, but when you rotate it another 360 degrees, its wavefunction is back to where it started. You can't describe this behavior using representations of SO(3), but you can using SU(2). In general, for any j = 0, 1/2, 1, 3/2, 2, etc., there is an irreducible representation of SU(2), called the "spin-j" representation, which is (2j+1)-dimensional. Only when the spin is an integer does the representation come from one of SO(3).