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- Thread starter arlesterc
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So for electromagnetism the symmetry is the gauge symmetry of the potential, and the resulting Noether charge is electric charge and the resulting Noether current is electric current.

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exercise ;) ;)

effect variation ##\delta A_{\mu} = \partial_{\mu} \eta## of ##A_{\mu}## where ##\eta: R^4 \rightarrow R## arbitrary function, and more, do calculate resulting variation in the lagrangian ##\mathscr{L} = -\frac{1}{4} F_{\mu \nu} F^{\mu \nu} - A_{\mu} j^{\mu}##. now proof that ##j^{\mu}## is a conserved.... :)

with that lemma, proposition follow by recalling that electric charge obtained by volume integral of the ##j^0##

effect variation ##\delta A_{\mu} = \partial_{\mu} \eta## of ##A_{\mu}## where ##\eta: R^4 \rightarrow R## arbitrary function, and more, do calculate resulting variation in the lagrangian ##\mathscr{L} = -\frac{1}{4} F_{\mu \nu} F^{\mu \nu} - A_{\mu} j^{\mu}##. now proof that ##j^{\mu}## is a conserved.... :)

with that lemma, proposition follow by recalling that electric charge obtained by volume integral of the ##j^0##

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Thanks - that's very helpful. You write:

" The Noether current and the Noether charge are related through a continuity equation which essentially defines a local conservation law. "

What is the continuity equation?

Where does the J I see so often come in? Aclaret's post as is is a few levels above my mental pay grade I am afraid. Maybe if I had a worked example it might help - actual numbers, actual case.

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It is an equation of the form $$\frac{\partial \rho}{\partial t}+\nabla \cdot \vec j = 0$$ where ##\rho## is the charge density and ##\vec j## is the current density. Basically, it says that if charge accumulates somewhere then it cannot simply magically appear there but must flow in as current that doesn’t flow out.What is the continuity equation?

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Under that variation ##\delta A_{\mu} = \partial_{\mu} \eta## that i mention (fancy-schmancy term: gauge transformation), it clear that resulting variation in lagrangian is ##\delta \mathscr{L} = - j^{\mu} \partial_{\mu} \eta##.

For theorem of noether to apply, transformed lagrangian must be same up to some divergence; i.e. ##\mathscr{L} \mapsto \mathscr{L} + \partial_{\mu} v^{\mu}## for some ##v##. That in mind, we then must assert that

##\delta \mathscr{L} = - j^{\mu} \partial_{\mu} \eta \overset{!}{=} -\partial_{\mu} (j^{\mu} \eta) \implies \eta \partial_{\mu} j^{\mu} = 0##

this hold for arbitrary ##\eta##, thus ##\partial_{\mu} j^{\mu} = 0##. this express conservation of charge :)

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To answer the question in the title of this thread: Noether's theorem goes in both directions, i.e.,

So for electromagnetism the symmetry is the gauge symmetry of the potential, and the resulting Noether charge is electric charge and the resulting Noether current is electric current.

one-parameter symmetry Lie group ##\Leftrightarrow## conserved quantity

The conserved quantity is the generator of the symmetry group. Take the Dirac equation and its action. The action is invariant under (global) U(1) transformations, i.e., the multiplication of the field with a phase factor. The corresponding Noether charge is given by

$$\hat{Q}=\int_{\mathbb{R}^3} \mathrm{d}^3 x :\hat{\bar{\psi}}(t,\vec{x}) \gamma^0 \hat{\psi}(t,\vec{x}):.$$

It's easy to show that this operator generates the symmetry.

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