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In the book Condensed Matter Field Theory by Altland, on page 32, it is given while explaining Noether's theorem that

To understand the impact of a symmetry transformation, it is fully sufficient to consider its infinitesimal version. (Any finite transformation can be generated by successive application of infinitesimal ones.) Consider, thus, the two mappings $$ \begin{aligned} x_{\mu} \rightarrow x_{\mu}^{\prime} &=x_{\mu}+\left.\frac{\partial x_{\mu}}{\partial \omega_{a}}\right|_{\omega=0} \omega_{a}(x) \\ \phi^{i}(x) \rightarrow \phi^{\prime i}\left(x^{\prime}\right) &=\phi^{i}(x)+\omega_{a}(x) F_{a}^{i}[\phi] \end{aligned} $$ expressing the change of both fields and coordinates to linear order in a set of parameter functions $\left\{\omega_{a}\right\}$ characterizing the transformation. (For a three-dimensional rotation, $\left(\omega_{1}, \omega_{2}, \omega_{3}\right)=(\phi, \theta, \psi)$ would be the rotation angles, etc.) The functionals $\left\{F_{a}^{i}\right\}-$ which need not depend linearly on the field $\phi$, and may explicitly depend on the coordinate $x-$ define the incremental change $\phi^{\prime}\left(x^{\prime}\right)-\phi(x)$.

We now ask how the action Eq. (1.16) changes under the transformation Eq. (1.42), i.e. we wish to compute the difference $$ \Delta S=\int d^{m} x^{\prime} \mathcal{L}\left(\phi^{\prime i}\left(x^{\prime}\right), \partial_{x_{\mu}^{\prime}} \phi^{\prime i}\left(x^{\prime}\right)\right)-\int d^{m} x \mathcal{L}\left(\phi^{i}(x), \partial_{x_{\mu}} \phi^{i}(x)\right) $$

However, $\frac{\partial x_{\mu}}{\partial \omega_{a}}$ in the first equation doesn't make any sense; it means changing the mapped point if the transformation at that point were to change.

Could someone help me to decipher what kind of transformation the author is talking about in here?

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  • $\begingroup$ This cannot be answered without a copy of the book. Consider adding the definition of the transformation, and the role of $\omega_a$ in it, into the text of the question. $\endgroup$ – Emilio Pisanty Apr 18 at 6:34
  • $\begingroup$ @EmilioPisanty there is no definition for $w_a$ but see my edit; I have added more of the text that proceeds it. $\endgroup$ – Our Apr 18 at 6:43
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Consider the example that Altland suggests: let $\theta,\phi,\psi$ be the Euler angles specifying a rotation. Then $x,y,z \to x(\theta,\phi,\psi),y(\theta,\phi,\psi), z(\theta,\phi,\psi)$ and the derivatives $\partial x/\partial \theta$ etc. give the infinitesimal form of the rotation.

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