Noether's theorem for fields is usually given as follows:

Given a field theory with action $S=\int\mathcal{L}(\phi,\partial\phi)d^4x$, and given a one-parameter variation of the fields $\phi_\epsilon$, if the variation is a symmetry of the action (which means that $\delta\mathcal{L}=\partial_\mu F^\mu$), then the current $$ \mathcal{J}^\mu=\frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi)}\delta\phi-F^\mu $$ is conserved on-shell.

Some authors (di Francesco for example) seem to make a difference between external symmetries (induced by diffeomorphisms of spacetime) and internal symmetries (changes in the fields not related to spacetime geometry), however since diffeos change the fields, I think these two can be considered being part of the same case.

Now, I have read (di Francesco, Weinberg, Tong's online QFT notes), that if a global one-parameter symmetry is made local ("gauged"), then without modifications, the action will no longer be invariant, instead, it will be proportonal to $\delta S=\int\partial_\mu\alpha\mathcal{J}^\mu d^4x$, where $\alpha$ is the local parameter function.

However, on-shell, the action's variation must vanish, therefore, $\mathcal{J}^\mu$ is a conserved current, and is implied that this current is the same as the one obtainable from the usual form of Noether's theorem.

Question 1: Is this true? I mean that this current is always the same. At least up to gauge transformation $\mathcal{J}^\mu\mapsto\mathcal{J}^\mu+\partial_\nu\Sigma^{\nu\mu}$?

Furthermore, the canonical stress-energy-momentum (SEM) tensor is defined as the Noether current associated to spacetime translations.

Now, my main source of confusion is that the equivalence of the canonical (well, the Belinfante-Rosenfeld) tensor with the Einstein-Hilbert tensor is usually given by the fact that the canonical SEM tensor is the current of rigid spacetime translations, and the EH SEM tensor is the current of "gauged" spacetime translations (infinitesimal diffeomorphisms).

BUT, as I had written above, the Noether current from "gauged" symmetries arises from the fact that (without modifications), the Lagrangian will not be symmetric with respect to the gauged symmetries, even if it is symmetric under the "ungauged" symmetries. However, when the EH SEM tensor is calculated, we use diffeomorphism-invariant actions, which are symmetric under gauged translations.

Question 2: How does the "EH SEM tensor = Belinfante SEM tensor" relationship follow from the Noether theorems? I mean, considering what I said above. As far as I am aware, when we are in flat spacetime, and use gauged translations, we do not use the generally covariant form of the action, hence it is NOT invariant under them. But GR actions are. How to reconcile these?

Question 3: Spinor fields. If we use a flat spacetime action for spinor fields, how to incorporate "gauged translations"? I mean, spinors cannot transform under the general diffeomorphism group, as their transformation is related to the double cover of the Poincaré group. In the case of a generally covariant action, spinor components are given with respect to an orthonormal vielbein, whose flat indices do not transform under diffeos, so that's taken care of, but what about "unmodified" spinor actions in flat spacetime?

Ps: I know about the Gotay/Marsden paper, but I don't understand jet bundles, so I don't fully understand the paper.


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