# Momentum Energy tensor and Wilson Loop in Yang-Mills Theory

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• CantorsLuck
In summary, the conversation discusses the relationship between gauge invariance and the trace of the energy-momentum tensor in Yang-Mills theory. It is shown that the Yang-Mills tensor is not gauge invariant, but the trace of the energy-momentum tensor is. The concept of holonomy and Wilson loops are also introduced, and it is explained that the curvature tensor is related to the holonomy. The physical significance of this relationship is that the curvature tensor measures the holonomy for small loops.

#### CantorsLuck

Hello Everyone. I Was Wondering how excatly the Gauge invariance of the trace of the Energy-momentum tensor in Yang-Mills theory connects with the trace of an Holonomy.
To be precise in what I'm asking:
The Yang-Mills Tensor is defined as:

$$F_{\mu \nu} (x) = \partial_{\mu} B_{\nu}(x)- \partial_{\nu} B_{\mu} (x) -ig[B_{\mu} , B_{\nu}]$$
Where ##-\frac{1}{-ig}[B_{\mu} , B_{\nu}] = F_{\mu \nu}##
If now I define a covariant derivative as follows: ##D_{\mu}=1_{2x2} \partial_{\mu} - igB_{\mu}##
where $$B_{\mu}=\frac{\sigma}{2} \beta_{\mu}$$

such that
##D_{\mu} \psi \rightarrow D'_{\mu} \psi ' = \sigma \psi \sigma^{-1}##, so ##D_{\mu}= \sigma D_{\mu} \sigma^{-1}## (1)
Applying this to ##F_{\mu \nu}## :
$$=-\frac{1}{-ig}[\sigma D_{\mu} \sigma^{-1} , \sigma D_{\nu} \sigma^{-1} ]$$
$$=-\frac{1}{-ig}(\sigma D_{\mu} \sigma^{-1} \sigma D_{\mu} \sigma^{-1} - \sigma D_{\mu} \sigma^{-1} \sigma D_{\mu} \sigma^{-1})$$
$$=-\frac{1}{-ig} \sigma [D_{\mu} , D_{\nu}] \sigma^{-1} = \sigma F_{\mu \nu} \sigma^{-1} \neq F_{\mu \nu}$$
Therefore ##F_{\mu \nu}## in not Gauge invariant.
Now, if I Consider the ##Tr(F_{\mu \nu} F^{\mu \nu})## and apply relation (1)

$$Tr(\sigma F_{\mu \nu} \sigma^{-1} \sigma F^{\mu \nu} \sigma^{-1})= Tr(\sigma F_{\mu \nu} F^{\mu \nu} \sigma^{-1}) = Tr(F_{\mu \nu} F^{\mu \nu})$$
So, the ##Tr(F_{\mu \nu} F^{\mu \nu})## is indeed gauge invariant.

Now Let's review some differential geometry concepts:

Lets consider a manifold ##M##, a Vector Bundle ##E## with a connection ##D## and a smooth path ##\gamma## that connects two points in the manifold, let's say ##p## and ##q## . We know that we can define a holonomy $$H(\gamma , D): E_{p} \rightarrow E_{q}$$

and let's remember that ##Tr(H(\gamma , D))## is gauge invariant. We think the ##Tr(H(\gamma , D))## as a Holonomy over a loop, and this loop is called the Wilson Loop. $$W(\gamma , D) = Tr(H(\gamma , D))$$

So my question is how exactly the energy-momentum tensor in Yang-Mills theory connects with the Wilson loop and what's the physical significance of this?

Thanks!

Last edited:
That's the curvature tensor, not the energy momentum tensor, just to let you know.

The relation between the holonomy and the curvature tensor is that if you expand the holonomy in the size of the curve, then the curvature tensor is the coefficient of the first term.

Physically the curvature then measures the Holonomy for small loops.

On my phone, I'll post something more detailed tomorrow.

dextercioby and nrqed