|This kind of thing is decpetively useful for dealing with angular momentum in Quantum Mechanics. If you look at the main write up there, the algebra works out exactly the same if you let:
...and then all the commutation relations come out analagously. In fact, there is another operator useful in representing sl(2,C), called the Cazimir operator; Ϊ = ef + fe + (1/2)h2. Then given this, it's not hard to show that Ϊ = J2/(h-bar)2.
- h = 2J3/(h-bar)
- e = J+/(h-bar)
- f = J-/(h-bar)
Of course, all of this is no coincidence: the angular momentum theory is derived from the Lie algebra of the rotation group of 3-d space, which is sl(2,C). It's still quite cool, though.