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The complex scalar field admits the conserved quantity $$Q=i \int{d^3x (\Pi \psi - \Pi^* \psi^*)}$$ Is it the consequence of a symmetry of the system and the Noether theorem? If yes is every conserved quantity the consequence of a symmetry?

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    $\begingroup$ Looks like the charge for the $U(1)$ symmetry (even if without knowing the Lagrangian is difficult to tell). Usually the conservation laws are a consequence of symmetries, at least in Lagrangian systems. $\endgroup$
    – yuggib
    May 11, 2018 at 6:08
  • $\begingroup$ You can find the symmetry transformation related to a conserved quantity by taking $$\delta \phi = i [Q,\phi].$$ This is discussed in detail in every introductory QFT book, e.g. Weinberg. See also here: physics.stackexchange.com/q/137499 $\endgroup$ May 11, 2018 at 6:50
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    $\begingroup$ Last subquestion is a duplicate of physics.stackexchange.com/q/24596/2451 $\endgroup$
    – Qmechanic
    May 11, 2018 at 6:57

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