How can I identify gauge transformations of fields with gauge transformations on a principal bundle?
Answers
So indeed if we start with a section (thus also a Yang-Mills field) on the principal bundle and with a section on the associated bundle, at a gauge transformation Ω:M→G the section on the associated bundle will also change with ρ(Ω−1)..
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I start from the following construction which I hope is ok: with a Lie group G and a smooth manifold M I can construct a locally trivial principal bundle. If M is Minkowski which can be covered by a single chart, the bundle is trivial, so the total space is P=M×G. I choose a representation of G on a carrier space V and I construct an associated bundle with fibre V. It will have the same base space M and the total space PV=P×V/∼ where the equivalence relation is the usual one for associated bundles. (the left action of G on V is given by the representation) Fields are then sections in this associated bundle. My problem is related to (local) gauge transformations. As far that I know, they are maps S:M→G which are in one-to-one correspondence with automorphisms on P. On the principal bundle I can choose a connection and then a smooth section in order to pullback the connection to the base space. I can make a gauge transformation to change my section and look at the new components of the connection on base space.