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According to the explanation of Noether's theorem in Peskin & Schroeder's QFT book, pp. 17-18,

If the Lagrangian $\mathcal{L}(x)$ change to $$\mathcal{L}(x)+\alpha\partial_\mu\mathcal{J}^\mu\tag{2.10}$$ when the field $\phi(x)$ is change to $$\phi^\prime(x)=\phi(x)+\alpha\Delta\phi(x),\tag{2.9}$$ there is a current $$j^\mu=\frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi)}\Delta\phi-\mathcal{J}^\mu,\tag{2.12}$$ that is conserved.

I don't understand why use the parameter $\alpha$ though it is vanished after all. What meaning does it have? in the book, $\alpha$ refers to an infinitesimal parameter and $\Delta\phi$ is some deformation of the field. If $\Delta\phi$ is a deformation of the field, why just define the field as $$\phi^\prime=\phi+\Delta\phi~?$$

Qmechanic
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Orient
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1 Answers1

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  1. The assumption in Noether's first theorem is that there is a 1-parameter$^1$ family of field and spacetime transformations (with parameter $\epsilon\equiv\alpha\in\mathbb{R}$).

  2. We want to study the corresponding 1-parameter family of action functionals $S(\epsilon)$.

  3. The 1-parameter transformation is by definition called a quasisymmetry (QS) iff $$ \left. \frac{dS(\epsilon)}{d\epsilon}\right|_{\epsilon=0} ~=~\text{boundary integral}. \tag{QS}$$

  4. Although the 1-parameter $\epsilon$ is ultimately not needed in the formulation of the final Noether conservation law, it is useful in the definition and identification of a QS.

  5. Whether one wants to keep the 1-parameter $\epsilon$ inside or outside the $\Delta \phi$ symbol is a conventional choice. Both conventions are used in the literature.

  6. Let us mention for completeness that in the context of Noether's second theorem and gaugesymmetry, $\epsilon(x)$ is a function of spacetime.

  7. Even within the context of Noether's first theorem, sometimes the 1-parameter $\epsilon$ is promoted to a function $\epsilon(x)$ of spacetime as a trick.

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$^1$ This can be generalized to a finite number of parameters, but let's keep it simple here.

Qmechanic
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