Science Forums 2

[Jay-qu]

Originally Posted by Ben

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.

This was a wise statement, as a teacher to often do I see kids rushing to find a formula that relates quantities rather than thinking up the desired relationship from first principles. Note that at this level of physics it can be somewhat impossible to do without much trial and error.

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.

This is a condition of a rigid transformation, but more importantly the space must remain undistorted under the transformation. Imagine transforming the position of your arm, from straight up to straight in front of you, it is only rigid if you dont bend or twist your arm.

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.

I dont follow the logic of your final sentence. Physics does like to be coordinate independent, this is because we believe there is no preferred position or inertial reference frame - the laws of physics should be the same in every frame.

For the Lorentz transformations, the rotations and translations are both rigid so long as it is a global transformation - applies the same everywhere in space. The boost however is not a rigid transformation. Boosting induces the effects of length contraction and time dilation, these are non-rigid effects.
Do I fully understand this last statement? No, so please help.

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 .

I am not very familiar with the fibre bundle approach (a very mathematical approach) to gauge theories, that said I see what you are saying above but Im afraid I dont see the point of this argument. Did you have a question or comment about this argument?

J