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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!
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!
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