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equations:yang_mills_equations [2018/03/26 17:29] jakobadmin [Abstract] |
equations:yang_mills_equations [2022/02/10 16:43] (current) 2601:845:c302:8e50:4df7:1623:815b:9c99 |
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- | ====== Yang-Mills Equation ====== | + | <WRAP lag> $ \partial^{\mu}F_{\mu\nu}^a-gf_{abc}A^{\mu}_bF_{\mu\nu}^c=0$</WRAP> |
- | //see also [[theories:classical_theories:gauge_theory]] and [[advanced_tools:gauge_symmetry|]] | + | ====== Yang-Mills Equations ====== |
- | <tabbox Intuitive> | + | //see also [[models:gauge_theory]] and [[advanced_tools:gauge_symmetry|]] // |
+ | |||
+ | <tabbox Intuitive> L=1/4g²∫TrF∧∗F | ||
<note tip> | <note tip> | ||
Line 11: | Line 13: | ||
<tabbox Concrete> | <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> | <tabbox Abstract> | ||
- | The Yang-Mills equations (2.22) can be expressed with the | + | The Yang-Mills equations can be expressed with the |
Hodge star operator as | Hodge star operator as | ||
\begin{equation} | \begin{equation} | ||
0=d_A F=d_a *F \quad F=dA+A\wedge A | 0=d_A F=d_a *F \quad F=dA+A\wedge A | ||
\end{equation} | \end{equation} | ||
- | where $d_A$ is the gauge-covariant extension of the exterior derivative. | + | where $d_A$ is the gauge-covariant extension of the exterior derivative. The gauge field $A$ is a one-form |
- | This is described in a clearer way in [2]. The gauge field $A$ is a one-form | + | |
\begin{equation} | \begin{equation} | ||
A(x)=A_{\mu}^a(x)t^adx^{\mu} | A(x)=A_{\mu}^a(x)t^adx^{\mu} | ||
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\end{gathered} | \end{gathered} | ||
\end{equation} | \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?> | <tabbox Why is it interesting?> | ||