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This is a follow-up on this answer, where ACuriousMind wrote

Formally, both the mass and the charge classify certain irreducible representations of the Poincaré group and the circle group, respectively.

I understand the basics of representation theory, and I know the $U(1)$ gauge transformations of the QED Lagrangian (I suppose that's the connection between electric charge and the circle group $U(1)$). I also have seen the basics of non-abelian gauge transformations.

However, I wasn't aware that there is a connection between representation theory and the conserved electric charge. Moreover, I had no idea that mass had anything to do with irreps of the Poincaré group.

What are the details of that connection? Why do mass and charge classify irreps?

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    $\begingroup$ One-particle states are irreducible representations of the Poincare group. What classifies representations of the Poincare group? $\endgroup$
    – Prahar
    Apr 1, 2016 at 14:54
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    $\begingroup$ Key words: (Quadratic) Casimir operators, and Schur's lemma which tells you that they're constant on irreps. $\endgroup$
    – Danu
    Apr 1, 2016 at 15:06
  • $\begingroup$ @Danu and Prahar Thanks for the hints, gonna follow those leads. $\endgroup$
    – Bass
    Apr 1, 2016 at 19:44

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