Noether current for translational invariance in QFT In Weinberg's QFT book, Vol-I (page 311), he derives the conserved current for translational invariance.
He considers the transformations of the form
$$\psi^l(x)\to \psi^l(x+\epsilon (x)).\tag{7.3.30}$$
Then he writes down the variation of the action as
$$\delta I=\int d^4 x \left(\frac{\partial \cal{L}}{\partial \psi^l}\epsilon^\mu\partial_\mu \psi^l+\frac{\partial \cal{L}}{\partial (\partial_\nu \psi^l)} \partial_\nu\left[\epsilon^\mu \partial_\mu \psi^l \right]\right)\tag{7.3.31}$$
which, as I understand, comes form the replacement (7.3.30). Now what I am confused about is that why he did not consider the change in the integration measure $\int d^4 x$. I thought we should also replace $\int d^4 x \to \int J d^4 x $, where $J$ is the Jacobian for the transformation $x^\mu \to x^\mu +\epsilon^\mu (x)$. If someone can clarify why the Jacobian is not considered it will be very helpful.
 A: Conceptually, changing the measure $\int d^4 x$ would defeat the point. The action is a functional of an input field,
$$
S = S[\psi]
$$
and a symmetry transformation $\delta \psi$ is a transformation which satisfies
$$
S[\psi + \delta \psi] = S[\psi] + \mathrm{boundary \; term}.
$$
This means that if $\psi$ extremizes the action, then $\psi + \delta \psi$ must also extremize the action, meaning that our symmetry transformation maps one solution to the equations of motion, $\psi$, to another solution to the equations of motion, $\psi + \delta \psi$.
Now, note that the symmetry transformation always acts on the field, and not spacetime! You just plug $\psi$ into the action and get out a number. There is no other input to the action. In other words, if $\delta \psi = \epsilon^\mu \partial_\mu \psi$, for $\epsilon^\mu$ a constant infinitesimal translation, then this symmetry transformation translates the field in spacetime, say $0.1$ cm to the left. The points of space don't vary, it's just the field that is moving.
A: OP asks a good question. In eq. (7.3.30) it may at first seems like Weinberg considers

*

*(i) an infinitesimal spacetime (=horizontal) transformation $\delta x$.

However in eq. (7.3.31) he changes perspective and think of it as

*

*(ii) an infinitesimal vertical transformation $\delta_0\psi=\epsilon^{\mu}\partial_{\mu}\psi$ in the $\psi$ target space.

If the action $I_V[\psi]=\int_V \! d^4x {\cal L}$ is integrated over a spacetime region $V\subseteq \mathbb{R}^4$, it turns out that a purely horizontal transformation (i) only produces boundary terms along the boundary $\partial V$ whose contributions get cancelled in the corresponding full Noether current.
In contrast, to successfully apply Noether's theorem, it is important that the infinitesimal transformation contains a vertical component (ii).
In the second method (ii), Weinberg considers a purely vertical transformation $\delta_0\psi=\epsilon^{\mu}\partial_{\mu}\psi$ with no horizontal component $\delta x=0$. This answers OP's main question about why the spacetime integration measure $d^4x$ is not changed in eq. (7.3.31).
Note that Weinberg's transformation (ii) is only a quasi-symmetry rather than a strict symmetry.
All this is explained in more details in my Phys.SE answer here for the case of point mechanics.
