# The Conservation Laws in Peskin and Schroeder (page 309)

I'm working on the Conservation Laws in Peskin (page 309), but I was confused for it.

In last section, I know that

Classical: the action is stationary.i.e. $$\delta S =0$$ when $$\phi(x)\rightarrow \phi(x)+\epsilon(x)$$, so we obtain Euler-Lagrange equation

Quantum: the generating functional is invariant.i.e. $$\delta Z[J]=0$$ when $$\phi(x)\rightarrow \phi(x)+\epsilon(x)$$, so we obtain Dyson-Schwinger equations

However, when I started to derive Noether's theorem, I got confused.

When $$\phi(x)\rightarrow \phi(x)+\epsilon(x)\Delta\phi_a(x)$$ (Eq.9.93), in the classical case and in the quantum case, which is stationary?

What's the difference between $$\phi(x)\rightarrow \phi(x)+\epsilon(x)$$ and $$\phi(x)\rightarrow \phi(x)+\epsilon(x)\Delta\phi(x)$$, they're both infinitesimal transformations. Why would one give DS equations and one give Noether theorem? What are the differences and connections between them？