Yang-Mills Equations

Intuitive

L=1/4g²∫TrF∧∗F

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 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*}

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 Mass in Quantum Yang-Mills Theory by L. D. Faddeev

Why is it interesting?

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.

The Aleph of Space by Luciano Boi