✎ Yang-Mills Equations [The Physics Travel Guide]

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====== Yang-Mills Equations: $ \quad \partial^{\mu}F_{\mu\nu}^a-gf_{abc}A^{\mu}_bF_{\mu\nu}^c=0$ ======

//see also [[theories:gauge_theory]] and [[advanced_tools:gauge_symmetry|]] //

<note tip>
Explanations in this section should contain no formulas, but instead colloquial things like you would hear them during a coffee break or at a cocktail party.
</note>
  
<tabbox Concrete>

-->Derivation of the Yang-Mills equation from the Yang-Mills Lagrangian#

Let 
\begin{equation}
\mathcal{L}=-\frac 14 F^{\mu\nu}_aF_{\mu\nu}^a
\end{equation}   
and $A^{\mu}_a$ be real functions.

The [[equations:euler_lagrange_equations|Euler-Lagrange equations]] for 
$\mathcal{L}=\mathcal{L}(A^{\mu},\partial^{\nu}A^{\mu})$ are
\begin{equation}
\partial^{\nu}\left( \frac {\mathcal{L}}{\partial\left(\partial^{\nu}A_a^{\mu}\right)}\right)=\frac {\partial \mathcal{L}}{\partial A_a^{\mu}}
\end{equation}

Then

\begin{align}
&\frac {\partial F^{\mu\nu}_d}{\partial A_a^{\mu}}=-gf_{dac}A_c^{\nu}\\
&\frac {\partial F^d_{\mu\nu}}{\partial A_a^{\mu}}=-gf_{dac}A_c^{\nu}g_{\mu\mu}g_{\nu\nu}\\
&\frac {\partial \mathcal{L}}{\partial A_a^{\mu}}=\frac 12 gf_{abc}A_b^{\nu}F^c_{\mu\nu}\\
&\frac {\partial F^{\mu\nu}_d}{\partial \left(\partial^{\nu}A_a^{\mu}\right)}=-\delta_{ad}\\
&\frac {\partial F^d_{\mu\nu}}{\partial \left(\partial^{\nu}A_a^{\mu}\right)}=\delta_{ad}(g_{\mu\nu}g_{\nu\mu}-g_{\mu\mu}g_{\nu\nu})\\
&\frac {\partial \mathcal{L}}{\partial \left(\partial^{\nu}A_a^{\mu}\right)}=\frac 12 F_{\mu\nu}^a
\end{align}

Directly computing
\begin{equation}
\begin{gathered}
\frac {\partial F^{\mu\nu}_d}{\partial A_a^{\mu}}=\frac {\partial}{\partial A_a^{\mu}}\left( -gf_{dbc}A^{\mu}_bA^{\nu}_c\right)
=-g\delta_{ab}f_{dbc}A_c^{\nu}= -gf_{dac}A_c^{\nu}
\end{gathered}
\end{equation}

\begin{equation}
\begin{gathered}
\frac {\partial F^d_{\mu\nu}}{\partial A_a^{\mu}}=\frac {\partial}{\partial A_a^{\mu}}\left( -gf_{dbc}A^b_{\mu}A^c_{\nu}\right)
=\frac {\partial}{\partial A_a^{\mu}}\left( -gf_{dbc}g_{\mu\alpha}A^{\alpha}_bA^c_{\nu}\right)\\
=-gf_{dac}g_{\mu\alpha}\delta_{\alpha\mu}\delta_{ab}A^c_{\nu}= -gf_{dac}A^c_{\nu}g_{\mu\mu}\\
=-gf_{dac}g_{\nu\alpha}A^{\alpha}_cg_{\mu\mu}= -gf_{dac}A_c^{\nu}g_{\mu\mu}g_{\nu\nu}
\end{gathered}
\end{equation}

\begin{equation}
\frac {\partial \mathcal{L}}{\partial A_a^{\mu}}=-\frac 14 \left(\left(\\
\frac {\partial F^{\mu\nu}_d}{\partial A_a^{\mu}}\right)F^d_{\mu\nu}\\
+F^{\mu\nu}_d\left(\frac {\partial F^d_{\mu\nu}}{\partial A_a^{\mu}}\right)\right)
\end{equation}

\begin{gather*}
=\frac 14 gf_{dac}\left(A_c^{\nu}F^d_{\mu\nu}+F^{\mu\nu}_d A^{\nu}_cg_{\mu\mu}g_{\nu\nu}\right)\\
=\frac 14 gf_{dac}\left(A_c^{\nu}F^d_{\mu\nu}+A^{\nu}_cg_{\mu\alpha}g_{\nu\beta}F^{\alpha\beta}_d \right)\\
=\frac 14 gf_{dac}\left(A_c^{\nu}F^d_{\mu\nu}+A^{\nu}_cF^d_{\mu\nu} \right)\\
=\frac 12 gf_{dac}A_c^{\nu}F^d_{\mu\nu}=\frac 12 gf_{acd}A_c^{\nu}F^d_{\mu\nu}\\
=\frac 12 gf_{abc}A_b^{\nu}F^c_{\mu\nu}
\end{gather*}

\begin{gather*}
\frac {\partial F^{\mu\nu}_d}{\partial \left(\partial^{\nu}A_a^{\mu}\right)}=\frac {\partial}{\partial \left(\partial^{\nu}A_a^{\mu}\right)}\left(\partial^{\mu}A^{\nu}_d-\partial^{\nu}A^{\mu}_d\right)=-\delta_{ad}
\end{gather*}

\begin{gather*}
\frac {\partial F^d_{\mu\nu}}{\partial \left(\partial^{\nu}A_a^{\mu}\right)}
=\frac {\partial}{\partial \left(\partial^{\nu}A_a^{\mu}\right)}\left(\partial_{\mu}A^d_{\nu}-\partial_{\nu}A^d_{\mu}\right)\\
=\frac {\partial}{\partial \left(\partial^{\nu}A_a^{\mu}\right)}
\left(g_{\mu\alpha}g_{\nu\beta}\left(\partial^{\alpha}A^{\beta}_d-\partial^{\beta}A^{\alpha}_d\right)\right)\\
=g_{\mu\alpha}g_{\nu\beta}\delta_{ad}\delta_{\mu\beta}\delta_{\nu\alpha}
-g_{\mu\alpha}g_{\nu\beta}\delta_{ad}\delta_{\nu\beta}\delta_{\mu\alpha}\\
=-\delta_{ad}\left( g_{\mu\nu}g_{\nu\mu}-g_{\mu\mu}g_{\nu\nu}\right)
\end{gather*}

\begin{gather*}
\frac {\partial \mathcal{L}}{\partial \left(\partial^{\nu}A_a^{\mu}\right)}
=-\frac 14\left(-\delta_{ad}F_{\mu\nu}^d-g_{\mu\mu}g_{\nu\nu}\delta_{ad}F_d^{\mu\nu}\right)\\
=\frac 14\left(F_{\mu\nu}^a+g_{\mu\alpha}g_{\nu\beta}F_a^{\alpha\beta}\right)\\
=\frac 14\left(F_{\mu\nu}^a+g_{\mu\alpha}g_{\nu\beta}F_a^{\alpha\beta}\right)\\
=\frac 14\left(F_{\mu\nu}^a+F^a_{\mu\nu}\right)=\frac 12F_{\mu\nu}^a
\end{gather*}

<--

<tabbox Abstract> 
The Yang-Mills equations can be expressed with the
Hodge star operator as
\begin{equation}
0=d_A F=d_a *F \quad F=dA+A\wedge A
\end{equation} 
where $d_A$ is the gauge-covariant extension of the exterior derivative. The gauge field $A$ is a one-form
\begin{equation}
A(x)=A_{\mu}^a(x)t^adx^{\mu}
\end{equation}
with the values on the Lie algebra of a compact simple Lie group $G$. 
The curvature is a two-form
\begin{equation}
\begin{gathered}
F=dA+A\wedge A \\
F=F_{\mu\nu}^at^adx^{\mu}\wedge dx^{\nu}\\
F=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}+f^{abc}A_{\mu}^bA_{\nu}^c
\end{gathered}
\end{equation}

----

* For a nice description see [[https://arxiv.org/abs/0911.1013|Mass in Quantum Yang-Mills Theory]] by L. D. Faddeev
<tabbox Why is it interesting?>

<blockquote>
In physics, of course, Maxwell's equations of electromagnetism are linear partial differential equations. Their counterparts, the famous Yang-Mills equations, are non-linear equations which are supposed to govern the forces involved in the structure of matter. The equations are non-linear, because the Yang-Mills equations are essentially matrix versions of Maxwell's equations, and the fact that matrices do not commute is what produces the non-linear term in the equations.

<cite>[[https://books.google.de/books?id=OKGwzKKpbHIC&lpg=PA83&ots=K1BBdha4yb&dq=coset%20space%20intuitively&hl=de&pg=PA81#v=onepage&q&f=false|The Aleph of Space]] by Luciano Boi</cite>
</blockquote>

</tabbox>

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