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equations:yang_mills_equations

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Yang-Mills Equations

see also gauge_theory and Gauge Symmetry

Intuitive

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.

Concrete

\begin{equation} \partial^{\mu}F_{\mu\nu}^a-gf_{abc}A^{\mu}_bF_{\mu\nu}^c=0 \end{equation}

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. Then \begin{equation} \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}

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}


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

equations/yang_mills_equations.1522078475.txt.gz · Last modified: 2018/03/26 15:34 (external edit)