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say_cheese

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I have read suggested threads and I am still at a loss. I am an experimental physicist and know basics of electrodynamics and QM. I am reading the Primer for Gauge Theory by Moriyasu. Though the book is reviewed well and is suggested for relative beginner physicists, the terminologies are not explained fully. In particular, while broaching the topic,

It gives the introduction to Weyl's scale or gauge invariance. The book states "Weyl proposed that the absolute magnitude or the norm of a physical vector should not be an absolute quantity, but should depend on the location in space time. A new connection would then be necessary in order to relate the lengths of vectors at different positions. This idea became known as scale or gauge invariance. It is important to note here that the true significance of Weyl's proposal lies in the local property of gauge symmetry ..."

I understand the connection is similar to the GTR connection. But that aside, why the terminology "local" and why the word symmetry here? There is actually a variation. Perhaps I need an analogy from classical physics. My present understanding is that certain transformations like the one for the EM vector potential leave the fields unchanged, but I don't understand that how is this or the one of Weyl is local. I do understand that in the EM case, the fact that the fields do not change with the transformation, is a form of symmetry, but Weyl's??

I also don't understand why the relationship between the phase of a wavefunction and the vector potential (Aharanov-Bohm) is a gauge symmetry and local.

I broadly know that an objects properties can be known by the way it changes or does not change under a transformation, while revealing the symmetry. But I am not seeing the context of the terminologies.

Thanks

Jay