Examples
If an operator is not simply expressed as a product, but as a function of another operator, we must first perform Taylor expansion of this function. This is the case of the Wilson loop that is defined as a path-ordered exponential; this guarantees that the Wilson loop encodes the holonomy of the gauge connection. The parameter that determines the ordering is a parameter describing the contour, and because the contour is closed, the Wilson loop must be defined as a trace in order to become gauge-invariant.
Read more about this topic: Path-ordering
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