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I'm trying to understand Wikipedia's proof of Noether's theorem for a field theory on Minkowski space. Link. Their proof is clearly just the one from Goldstein (starting on page 588 in the second edition) with details omitted, but I can't understand Goldstein either. I'm going to ask a couple of basic questions first, and see if the answers will help me figure out what they're doing. If that doesn't work, I'll post more specific questions about the derivation.
1. Isn't the theorem supposed to be about what happens when the action is invariant under a transformation of the fields? Then why are Goldstein and Wikipedia messing with the set that the integration is performed over? And why would Goldstein postulate that the form of the Lagrangian is invariant?
2. Is it correct to say that the assumption we start with is that there's an n-parameter family of fields [itex]\epsilon\mapsto\phi_a[/itex] with [itex]\phi_0=\phi[/itex] and such that each [itex]\phi_a[/itex] satisfies both the Euler-Lagrange equations and the boundary conditions that we have imposed on [itex]\phi[/itex]?
3. If the answers to 1 and 2 are "no" and "yes" respectively, then why doesn't the derivation (and the final result) look like this?
[tex]0=\frac{d}{d\epsilon^r}\Big|_0 S[\phi_\epsilon]=\dots=\int d^4x \partial_\mu\bigg(\frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi)}\frac{d}{d\epsilon^r}\Big|_0\phi_\epsilon \bigg)[/tex]
The conserved currents would be
[tex]j_r^\mu=\frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi)}\frac{d}{d\epsilon^r}\Big|_0\phi_\epsilon[/tex]
4. If the answers to 1 and 2 are "yes" and "no" respectively, then what kind of variation is causing the [itex]\mathcal L[/itex] term to appear in the conserved current? (It can't be a variation of the fields, can it?)
1. Isn't the theorem supposed to be about what happens when the action is invariant under a transformation of the fields? Then why are Goldstein and Wikipedia messing with the set that the integration is performed over? And why would Goldstein postulate that the form of the Lagrangian is invariant?
2. Is it correct to say that the assumption we start with is that there's an n-parameter family of fields [itex]\epsilon\mapsto\phi_a[/itex] with [itex]\phi_0=\phi[/itex] and such that each [itex]\phi_a[/itex] satisfies both the Euler-Lagrange equations and the boundary conditions that we have imposed on [itex]\phi[/itex]?
3. If the answers to 1 and 2 are "no" and "yes" respectively, then why doesn't the derivation (and the final result) look like this?
[tex]0=\frac{d}{d\epsilon^r}\Big|_0 S[\phi_\epsilon]=\dots=\int d^4x \partial_\mu\bigg(\frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi)}\frac{d}{d\epsilon^r}\Big|_0\phi_\epsilon \bigg)[/tex]
The conserved currents would be
[tex]j_r^\mu=\frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi)}\frac{d}{d\epsilon^r}\Big|_0\phi_\epsilon[/tex]
4. If the answers to 1 and 2 are "yes" and "no" respectively, then what kind of variation is causing the [itex]\mathcal L[/itex] term to appear in the conserved current? (It can't be a variation of the fields, can it?)