Equations of motion with replacing the Lagrangian by irrep diagrams generating functional

+ 1 like - 0 dislike
2490 views

I have read that equations of motion of ghosts is equal to $$\tag 1 \frac{\delta \Gamma}{\delta \bar{c}^{a}(x)} = -\partial^{\mu}_{x}\frac{\delta \Gamma}{\delta K^{\mu , a}(x)},$$ where $\Gamma = W - \int d^{4}x \sum_{i}(J_{i} \cdot \varphi^{i})$ is generating functional for irreducible Feynman diagramms of nonabelian gauge theory given by $$S = \int \mathrm{d}^{4}x\left( -\frac{1}{4}F_{\mu \nu}^{a}F^{\mu \nu}_{a} - B_{a}f^{a} + \frac{1}{2 \varepsilon}B_{a}B^{a} - \bar{\omega}^{a}M_{ab}\omega^{b} + K_{\mu}^{a}\delta A_{a}^{\mu} + L_{a}\delta \omega^{a} \right),$$ and $K_{\mu}^{a}$ is the source for $\delta A_{\mu}^{a}$, where $\delta \varphi$ is BRST transformations of $\varphi$ field (for other descriptions see here).

Why $(1)$ is correct (i.e. why we can replace $L$ to $\Gamma$ in equations of motion)?

This post imported from StackExchange Physics at 2014-09-01 11:14 (UCT), posted by SE-user Andrew McAddams
 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysicsOverfl$\varnothing$wThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.