Foundations of Wilson Lines and Loops From Basic Concepts to Gauge Field Reconstruction

This review presents a concise and unified treatment of Wilson lines and Wilson loops in continuum gauge theory, emphasizing their interpretation as parallel transport and their status as gauge covariant transporters and gauge invariant loop observables. Starting from the problem of comparing charged fields at separated points, we derive the path ordered exponential formulation and collect the key kinematical identities: gauge covariance, composition under path concatenation, inversion, and base point dependence. We then explain how local information is encoded in suitable small segment and small loop limits, and how these limits enter reconstruction questions for the underlying gauge connection. Seen as an inverse problem, this perspective clarifies how much of the gauge connection is encoded in loop observables and why traced Wilson loops, while gauge invariant, are intrinsically lossy without additional input. Special attention is given to clarifying common sources of confusion, in particular the distinction between untraced holonomies and traced Wilson loops, and the role of reference path choices when relating open transports to based loop data. The presentation is intended to serve both as an entry point for newcomers and as a compact reference for readers in high energy theory, mathematical physics, and related fields.