Joined: Apr 2002
Sat May 25, 02 12:50 AM
This is a truly lovely little book and is fun to read:
FLATLAND: A Romance of Many Dimensions
by E. Abbott
The concept of holonomies is nicely illustrated by thinking about living in "Sphereland"
Read a bit of Flatland, and then start thinking about what happens if they lived on "Sphereland" instead.
Sphereland is like Flatland, except if you were to continue walking in any given direction in a "straight" line, you'd end up right back where you started from.
If you were to walk around a loop on Sphereland holding a spear in your hand trying desparately to keep the spear pointed in some fixed direction as you move around, i.e. as you turn left you compensate by rotating the spear to the right so it remains in the original direction, then when you return to your starting point, you will have found that the spear has actually rotated by a total amount proportional to the area of the loop you covered.
The exact amount is given by
delta(theta) = Area/a^2
where a is the radius of the sphere. For example, if you walk around a great circle, then you split the sphere into two equal halves of area 2*pi*a^2 each so that
delta(theta) = 2*pi
and the spear is not rotated at all when you return. If you trace out an octant of the sphere, i.e. a spherical triangle with each angle pi/2, then the area of the loop is 1/8*(4*pi*a^2) so that the amount rotated is
delta(theta) = pi/2.
There is a nice java applet demonstrating this at:
Parallel Transport on a Sphere
Another thing to note is that in the limit that a -> infinity, then delta(theta) -> 0 for any loop of finite area.
In other words, in the limit that Sphereland becomes Flatland, then you will not observe any of this "rotation" business.
This means (among other things) that when you transport something around a closed loop, if it returns "rotated", then that means there was some "curvature" floating around somewhere.
The curvature is obtained from a connection. The connection is basically a rule that tells you how to "transport" stuff.
In the case of electromagnetic theory, there is a U(1)-connection. The group U(1) is just the group of elements of the form exp(i*t) for some real t, i.e. U(1) is the group of "phases". Hence, EM theory is invariant under changes of "phase", i.e. gauge transformations.
If you tilt your head sideways a bit and think really hard, you can picture the AB effect as transporting a SINGLE electron around a closed loop. When the electron gets back to its point of origin, it is "rotated." In this case the rotation group is U(1) so that a rotation means a "change of phase."
Now, like a said, if transporting around a closed loop results in a "rotation" that means there is some curvature somewhere. In the case of EM, the magnetic flux density B plays the role of the curvature. So if you turn off the B-field, you have no curvature -> no rotation. If you turn on the B-field you have curvature -> you have rotation = change in phase.
Now, in a feeble attempt to relate this to math finance, arbitrage is like transporting a portfolio around a closed loop (as in my very first post in this thread). So if there is no arbitrage opportunity (curvature), then no matter what you do, i.e. no matter how you carry your portfolio around in loops, you will never make any money (i.e. rotate your portfolio). So speculators and arbitaguers have the effect of eliminating curvature in the market
The analogies to geometry are pretty striking. I'm looking forward to learning more about it.
Edited: Sat May 25, 02 at 05:41 AM by IAmEric