Question

A path deformation byϵμ(s)ϵμ(s)induces a variation of the connectionA′(s)=A(s)+ΔA(s)A′(s)=A(s)+ΔA(s). I'm trying to obtain the first-order expansion of the holonomyHγ(A)=Pei∫γAHγ(A)=Pei∫γA: ΔHγ=i∫10dt[Pei∫t0dsA(s)]ΔA(t)[Pei∫1tdsA(s)]ΔHγ=i∫01dt[Pei∫0tdsA(s)]ΔA(t)[Pei∫t1dsA(s)]. How does the path-ordering operator work during the expansion?

Answer 1

In the expansion of the holonomy, the path-ordering operator ensures that the operators representing the connection at different points along the path are properly ordered.

Step-by-step explanation:

In the expansion of the holonomy, the path-ordering operator is denoted by the symbol P and it ensures that the operators representing the connection at different points along the path are properly ordered.

During the expansion, the path-ordering operator works by rearranging the connection operators so that they are ordered according to their positions along the path. This is important because the connection operators do not commute, meaning that their order matters when calculating the holonomy.

By applying the path-ordering operator, the first-order expansion of the holonomy can be expressed as an integral over the path with the connection operators properly ordered.