Differences

This shows you the differences between two versions of the page.

--- equations:yang_mills_equations [2018/03/26 17:33]
jakobadmin ↷ Page name changed from equations:yang_mills_equation to equations:yang_mills_equations
+++ equations:yang_mills_equations [2022/02/10 16:43] (current)
2601:845:c302:8e50:4df7:1623:815b:9c99
@@ Line 1: / Line 1: @@
+<WRAP lag> $  \partial^{\mu}F_{\mu\nu}^a-gf_{abc}A^{\mu}_bF_{\mu\nu}^c=0$</WRAP>
 ====== Yang-Mills Equations ======
-//see also [[theories:classical_theories:gauge_theory]] and [[advanced_tools:gauge_symmetry|]] //
+//see also [[models:gauge_theory]] and [[advanced_tools:gauge_symmetry|]] //
-<tabbox Intuitive>
+<tabbox Intuitive> L=1/4g²∫TrF∧∗F
 <note tip>
@@ Line 11: / Line 13: @@
 <tabbox Concrete>
-<note tip>
+-->Derivation of the Yang-Mills equation from the Yang-Mills Lagrangian#
-In this section things should be explained by analogy and with pictures and, if necessary, some formulas.
-</note>
+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

User Tools

Site Tools

Differences

Page Tools