In trying to understand precisely how fiber bundle theory maps to physical models, I came across this quotation:
We can think of the elements of the principal bundle as generalized frames for the original fiber bundle. This means they correspond with different ways that we can convert the intrinsic dynamics described abstractly by a section of the fiber bundle to something concrete that we observe.
These generalized frames are often called 'gauges'. The structure group of the principal bundle is called the 'gauge group' and the automorphism of the principal bundle that fixes the base is called a 'gauge transformation.'
In the case of a vector bundle, this means that, by choosing a gauge, or an orthonormal frame for the fiber, we obtain a set of numbers: the coordinates of the section with respect to the frame.
So please tell me if I understand this correctly: we have two logically distinct fiber bundles here: the principal bundle and the associated vector bundle. The associated bundle is the 'matter field' whose sections represent the observable quantities, e.g. phase, and the principal bundle is the bundle of 'generalized frames' whose sections represent the bases we use to describe the sections of the associated bundle numerically?
Is this correct?
Sorry if this is vague. I am trying to get an intuitive 'handle' on how the technical math of fiber bundles maps to what I know from physics.