Wilson Loop in Yang-Mills Gauge Invariant

Wilson Loop in Yang-Mills Theory is Gauge Invariant Johar M. Ashfaque We wish to show that the Wilson loop Tr P exp ig Z s 0 ds 0 dx μ (s 0 ) ds 0 A a μ (x(s 0 ))t a in Yang-Mills theory is gauge invariant. Define U P (z,y)= P exp ig Z s 0 ds 0 dx μ (s 0 ) ds 0 A a μ (x(s 0 ))t a where z = x(s) is the final point and y = x(0) is the initial point of the path P . We imagine that path P is a subpath of the full path from y = x(0) to x(1). Here P exp is the path-ordered exponential prescribing that higher values of s stand to the left. Define the matrix in representation space M (s 0 )= ig dx μ (s 0 ) ds 0 A a μ (x(s 0 ))t a so that U P (z,y)= P exp Z s 0 ds 0 M (s 0 ) . We know d ds U P (z,y)= M (s)U P (z,y) ⇒ M (s)U P (z,y)= ig dx μ (s) ds A a μ (x(s))t a U P (z,y) which can be rewritten as dx μ (s) ds D μ U P (x, y)=0 where D μ contains a x μ -derivative which acts on the x of U P (x, y). A first-order equation like this along with an initial condition at s = 0 uniquely determines the solution. Here, the initial condition at s =0 translates to U P (y,y)= . Now we consider (A α ) a μ to be the α-gauge transformation of A a μ . From A α we naturally form U α P as well as the covariant derivative D α μ . Therefore, we automatically have dx μ (s) ds D α μ U α P (x, y)=0 as well as U α P (y,y)= . We know that the covariant derivative guarantees that D μ ψ transforms exactly like ψ with ψ α = e iα a t a ψ, we have D α μ ψ α = e iα a t a D μ ψ. This is valid for any function ψ, hence it implies D α μ = e iα a t a D μ e -iα a t a as a differential-operator equation. That is to say dx μ (s) ds D μ e -iα a t a U α P (x, y)=0. Clearly, e -iα a t a U α P (y,y)e iα a t a = and from the above ˜ U P (x, y) := e -iα a t a U α P (x, y)e iα a t a satisfies the equation dx μ (s) ds D μ ˜ U P (x, y)=0. 1

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Transcript of Wilson Loop in Yang-Mills Gauge Invariant

Page 1: Wilson Loop in Yang-Mills Gauge Invariant

Wilson Loop in Yang-Mills Theory is Gauge Invariant

Johar M. Ashfaque

We wish to show that the Wilson loop

(P exp

[ig

∫ s

ds′dxµ(s′)

ds′Aaµ(x(s′))ta

])in Yang-Mills theory is gauge invariant.

Define

UP (z, y) = P exp

[ig

∫ s

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

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.

Page 2: Wilson Loop in Yang-Mills Gauge Invariant

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).