Thanks, MAA, for a really cool exposition.
Numbers are really good at describing, well, anything at all -- be it geometrical (like the number line) or weird (like, er, rotations in 5-space) or even logical (like, er, mathematical proofs). Godel's theorem
(which among geeks is so cliche and trite that
we forget how powerful and interesting it is) is the famous proposition that every interesting logical system must have unprovable tautologies -- universal truths that cannot be proven within the logical system. "Interesting" here means that the system is powerful enough to contain statements about itself. He proved it, in part, by pointing out (in another revolutionary argument that is now so trite it seems hokey) that it's possible to represent any statement in any logical system with a number; and it's possible to construct mathematical operations that correspond to the laws of derivation for any logical system.
On another topic, non-commutativity has a lot of really interesting applications in the real world. For example, you can use matrices (another weird, multidimensional form of number) to represent rotations in three dimensions. There are special (but simple) rules for multiplying matrices. If you have two matrices that represent rotations, you can find out the result of making both rotations by multiplying the two matrices together. But it matters which order you do rotations (for example, turn your head 90 degrees right, then 90 degrees up. Now you're looking straight up, with a crick in your neck. If instead you turn your head 90 degrees up then 90 degrees right, you end up looking to the right with your head bent sideways). So of course matrices have to break commutativity too, to match that behavior.
Another really deep, strange, and interesting application of non-commutativity comes with quantum mechanics. It's so interesting that I'll take a few minutes to
go off on a tangent and describe it.
There are two very interesting formulations of classical mechanics that use energy, rather than forces, to represent the system (Remember, in classical mechanics you're trying to describe the behavior of, basically, systems of springs, levers, ropes, wheels, brakes, and such -- macroscopic objects). Those two formulations are are Hamiltonian and Lagrangian mechanics, and they're taught in upper-division college physics curricula everywhere. In both systems you write down all the possible free parameters of the system as "generalized dimensions" that aren't necessarily spatial -- they could represent, for example, how far you have bent your little finger, or how electrically charged you've become, or whatever -- so long as each one describes some aspect of the system.
So far, so good. Lagrange mechanics is really neat, because it turns the subtle art of figuring out where things will move into a mechanical process: just write down all possible parameters of the system, figure out how the potential and kinetic energy of the system depends on each one, and plug in a simple formula -- and out pops the answer for how the system behaves.
But that's not the cool bit. Enter a mathematician named Emma Noether, from around the beginning of the 20th century. She made an incredible, brilliant discovery that tied together two bits of physical theory that most folks don't even think about. Allow me to digress for a couple more paragraphs, to set the stage.
One important concept to any aspiring physicist is the conserved quantity. You've probably heard of the conservation of energy, in the sense that there's something called "energy" that is associated with every physical system. You can compute how much there is, and then come back later and account for it. The energy isn't created or destroyed, only moved around the system. Likewise, Newton's second law ("for every action there's an equal and opposite reaction") can instead be expressed as a conservation of momentum. There are perhaps half a dozen of these important conserved quantities that you learn in first-year physics.
There's another aspect of physical law that people wonder at, which is invariance, and many people have spent time trying to figure out how you can tweak physical systems and leave the physics intact. By that, I mean that the experiments come out the same. For example, it doesn't matter what time you do an experiment -- we expect that physics experiments will get the same outcome whether they're done Monday morning or Friday afternoon, or 100 years in the future. Likewise, we don't expect the experiments to change just because we move the apparatus across the room or around the world. (Unless, of course, the experiment depends on the stuff around it -- you have to move the whole apparatus, or at least the bits that matter -- so it's no fair pointing out, for example, that moving a boat from the ocean to to Death Valley at sea level will make it fall 300 feet to the ground.)
Noether's great discovery is that symmetries in physical law (like invariance under motion or time of day or angle) are intimately connected to conserved quantities. In face, she constructed a mechanical apparatus -- an algorithm -- for converting physical symmetries into conserved quantities, and vice versa. You can start with a physical symmetry like invariance
under displacement in time, plug it into the
Noetherian apparatus, and get out a conservation law (in this case, conservation of energy). Plug in displacement in position, and you get out momentum. Plug in
change in angle, and you get out angular momentum. Every symmetry has an associated conservation law, and vice versa. Wow.
even curiouser, if you multiply together any conserved quantity and its associated symmetric variable (like energy and time, or momentum and position), you get out something
with units of "action", energy * time. That's the same units as Planck's Constant, the fundamental quantum mechanical constant.
In fact, if you start with ordinary Lagrangian classical mechanics, and then take away commutativity (in a limited way), you get back out quantum mechanics for free! That's very strange. The rule change is that you keep commutativity (ab=ba) for most things, but that any quantity and its associated ``Noetherian J-form'' (the thing that's conserved if physics is invariant under
changes in your quantity) fail to commute. In particular, you just have to set (for example)
(momentum) (position) = (position)(momentum) - i h/ 2 / \pi, where 'i' is sqrt(-1) and h is a measurable physical constant with units of action (6.6e-34 J s). When you've done that, all of a sudden you can use exactly the same theorems and techniques to solve your problems, except that now you get quantum mechanics instead of classical mechanics.
That's bizarre, but it shows how important non-commutativity can be!
Yow -- that was a lot longer than I intended.
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