Wilson Loop in Yang-Mills Theory is Gauge Invariant

Johar M. Ashfaque

We wish to show that the Wilson loop

Tr

(P exp

[ig

∫ s

0

ds′dxµ(s′)

ds′Aaµ(x(s′))ta

])in Yang-Mills theory is gauge invariant.

Define

UP (z, y) = P exp

[ig

∫ s

0

ds′dxµ(s′)

ds′Aaµ(x(s′))ta

]where z = x(s) is the final point and y = x(0) is the initial point of the path P . We imagine thatpath P is a subpath of the full path from y = x(0) to x(1). Here P exp is the path-ordered exponentialprescribing that higher values of s stand to the left. Define the matrix in representation space

M(s′) = igdxµ(s′)

ds′Aaµ(x(s′))ta

so that

UP (z, y) = P exp

[ ∫ s

0

ds′M(s′)

].

We knowd

dsUP (z, y) = M(s)UP (z, y) ⇒M(s)UP (z, y) = ig

dxµ(s)

dsAaµ(x(s))taUP (z, y)

which can be rewritten asdxµ(s)

dsDµUP (x, y) = 0

where Dµ contains a xµ-derivative which acts on the x of UP (x, y). A first-order equation like this alongwith an initial condition at s = 0 uniquely determines the solution. Here, the initial condition at s = 0translates to UP (y, y) = 1.

Now we consider (Aα)aµ to be the α-gauge transformation of Aaµ. From Aα we naturally form UαP as wellas the covariant derivative Dα

µ . Therefore, we automatically have

dxµ(s)

dsDαµU

αP (x, y) = 0

as well as UαP (y, y) = 1. We know that the covariant derivative guarantees that Dµψ transforms exactlylike ψ with ψα = eiα

ataψ, we haveDαµψ

α = eiαataDµψ.

This is valid for any function ψ, hence it implies

Dαµ = eiα

ataDµe−iαata

as a differential-operator equation. That is to say

dxµ(s)

dsDµe

−iαataUαP (x, y) = 0.

Clearly, e−iαataUαP (y, y)eiα

ata = 1 and from the above UP (x, y) := e−iαataUαP (x, y)eiα

ata satisfies theequation

dxµ(s)

dsDµUP (x, y) = 0.

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By the uniqueness of the solution, we must have

UP (x, y) = UP (x, y).

ThusUαP (x, y) = eiα

ataUP (x, y)e−iαata .

This is the transformation property of UP (x, y). It is not invariant under gauge transformations. However,taking the trace, we obtain a gauge-invariant quantity

TrUαP (x, y) = TrUP (x, y).

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