It seems that Einstein once remarked of a colleague that "he could calculate, but he couldn't think". I find myself in an analogous situation here; I have a degree of understanding of the mathematics used in gauge theories, but I am struggling with their physical interpretation.
Here's what I've got so far; I would like you phys jocks to help me out.
We are talking spacetime, a manifold, for now. Let's say that a group of coordinate transformation is "rigid" if, for any element of the group at one point in the manifold, I must have the same transformation (group element) at all points.
Example: the Lorentz group of coordinate transformations. The allowed transformations are rotations, translations and boosts. But the Special Theory does not allow me to apply a boost here, a rotation there and a translation elsewhere. It seems that physics likes to work in a coordinate-independent manner, which is tantamount to asking that any transformation group is not rigid in the above sense.
Do I fully understand this last statement? No, so please help.
Anyway, it seems that the way to relax rigidity in the above sense is to adjoin to each point in spacetime a "copy" of the relevant transformation group. This is more like it, from my point of view.
The structure that results from the adjunction of a
continuous group of transformations, ie. a Lie group to each point of a manifold is called a "principal bundle". Let's call this
for now, and accept (it's not at all hard to show) that this is a manifold [it's essentially the product of 2 manifolds - the Lie group and spacetime]; let's call spacetime as the "base manifold"
.
So, since a continuous group describes infinitesimal transformations, then for any
I may have an infinity of maps
. And further, for any open curve
, an uncountable number of possible curves
.
This is over-long. How am I doing so far, tough guys?
