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I'm trying to prove the conserved quantities for $N$-vortices but having a difficult time doing so. I know it's something to do with Noether's theorem but not sure how to go about it. The Hamiltonian is $$ H = \frac{1}{4\pi}\sum_{\beta \neq \alpha}^N\Gamma_{\alpha}\Gamma_{\beta}logl_{\alpha\beta} $$ Where l is the distance between vortices. I need to use the fact that H is translational invariant and rotational invariant which I can show. The translational invariance gives

$$ X = \sum_{\alpha}^N\Gamma_{\alpha}z_{\alpha} $$

and the rotational invariance should give

$$ I = \sum_{\alpha}^N\Gamma_{\alpha}|z_{\alpha}|^2 $$

But again, I'm not sure how to show that X and I are conserved. Not necessarily looking a solution but just to point me in the right direction further than just 'use Noether's theorem'. Any help is appreciated!!

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Assuming no explicit time dependence, the Heisenberg equation of motion relates the time derivative of an operator to its commutator with the Hamiltonian as follows: $$\frac{d}{dt}\mathcal{O} = \frac{i}{\hbar} [H,\mathcal{O}]$$ From this we see that, in such cases, an operator that commutes with the Hamiltonian corresponds to a conserved quantity. The objects $\Gamma$ in your system probably come with some (anti)commutation relations, which you could exploit to test if $X$ and $I$ are conserved.

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  • $\begingroup$ Thanks for the reply! I have used this relation prior in a separate proof using the Poisson bracket I believe. So you are saying that with this relationship proposed, using invariance to translations and such should give the bracket = 0? $\endgroup$
    – D.Cecchi
    Apr 17, 2021 at 17:36
  • $\begingroup$ Yes, if you're dealing with a classical system, the corresponding result holds at the level of Poisson brackets. See, for example, here: en.wikipedia.org/wiki/Conserved_quantity#Hamiltonian_mechanics $\endgroup$ Apr 18, 2021 at 4:15

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