I have been studying the mathematics of gauge theories for the past several months and now that I am beginning to understand, I find myself struggling to put everything in context historically. I have two main questions:

1) In physics textbooks they introduce gauge theories non-rigorously by "promoting global gauge invariance to local gauge invariance". What is the physical motivation of this? Why is this necessary?

2) More mathematically, what physically motivated physicists to consider the principal bundle construction? In GR the idea of an affine connection arises as a necessity but I don't see the necessity for an Ehresmann connection in gauge theories. So often in the literature the author will simply say "introducing a connection on the principal bundle..." without motivating this construction whatsoever.

If anyone knows about the history of how these theories were developed I would greatly appreciate your insight! It is unsatisfying to know how all this machinery works without understanding how it was assembled.


I can help you with the first question.

In general, we know that Minkowski space has Poincare invariance as a global symmetry. In the other hand, General relativity is based on a general manifold, but by the equivalence principle, locally we can consider a Minkowski space-time who has Poincare invariance, i.e. we have Poincaré symmetry locally.

So, to have the same symmetry we apply the Gauge principle.

I hope that it is clear.


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