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In Peskin and Shroeder, for a local $U(1)$ transformation, the comparator operator is expanded as: \begin{equation} U(x+\epsilon n, x) = 1 -ie\epsilon n^{\mu}A_{\mu} + \mathcal{O}(\epsilon^2) \tag{15.5} \end{equation} for $\epsilon\rightarrow0$. I am not sure how one arrives at this expression, apart from "feeling" that it should depend on the distance $\epsilon n^{\mu}$, thus needing some other vector quantity.

Later, when talking about $SU(2)$, they expand $U$ as: \begin{equation} U(x+\epsilon n,x) = 1 + ig\epsilon n^{\mu}A^i_{\mu}\frac{\sigma^i}{2}+\mathcal{O}(\epsilon^2). \tag{15.23} \end{equation} Where did this sign change come from? The authors also say that $U$ can be consistently restricted to be a unitary matrix and because of that, can be expanded as above. How do we know it is possible? This question was already asked here, but the only answer is cyclical.

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A concept you're going to want to look in to is parallel transport. Long story short - you have some underlying manifold (your space) that you want to move locally defined quantities around in, and that movement process has to be continuous/smooth, and preserve inner products. Everything else follows from those requirements and how the vector space being transported relates to the manifold and its symmetries: producing notions of covariant derivatives, expansion at linear order being sufficient to build the transformations up from infinitesimal steps, etc.

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