Comparator operator in QFT

Question

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.

Answer 1

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.