This shows you the differences between two versions of the page.
Both sides previous revision Previous revision Next revision | Previous revision | ||
equations:yang_mills_equations [2018/03/26 17:34] jakobadmin [Concrete] |
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 ====== | ====== 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> | <note tip> | ||
Line 10: | Line 12: | ||
| | ||
<tabbox Concrete> | <tabbox 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# | -->Derivation of the Yang-Mills equation from the Yang-Mills Lagrangian# | ||
Line 22: | Line 20: | ||
\end{equation} | \end{equation} | ||
and $A^{\mu}_a$ be real functions. | 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 | Then | ||
Line 32: | Line 39: | ||
&\frac {\partial \mathcal{L}}{\partial \left(\partial^{\nu}A_a^{\mu}\right)}=\frac 12 F_{\mu\nu}^a | &\frac {\partial \mathcal{L}}{\partial \left(\partial^{\nu}A_a^{\mu}\right)}=\frac 12 F_{\mu\nu}^a | ||
\end{align} | \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*} | ||
+ | |||
<-- | <-- | ||