Suppose a fermion being a composite of two other fermions $$\psi=\varphi\cos\theta+\chi\sin\theta.$$ If $\varphi$ and $\chi$ satisfy the Dirac equations $$i\!\!\not\!\partial\varphi=e_{\varphi}\!\!\not\!\!A\varphi \\ i\!\!\not\!\partial\chi=e_{\chi}\!\!\not\!\!A\chi,$$ then $\psi$ should satisfy some Dirac equation $$i\!\!\not\!\partial\psi=e_\psi\!\!\not\!\!A\psi.$$ However, summing the first two equations, I get $$i\!\!\not\!\partial\psi=\,\not\!\!A[e_{\varphi}\cos(\theta)\,\varphi+e_{\chi} \sin(\theta)\,\chi].$$ How do I get the charge $e_{\psi}$?
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$\begingroup$ An example in nature? $\endgroup$– anna vCommented Dec 30, 2022 at 4:24
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$\begingroup$ @annav for example, generation mixing with CKM-matrix, the only difference is that charges are the same $\endgroup$– JavaGamesJARCommented Dec 30, 2022 at 10:14
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$\begingroup$ that is a mathematical example. by nature I mean an existing particle, atom or molecule $\endgroup$– anna vCommented Dec 30, 2022 at 10:35
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1 Answer
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$\psi = \varphi \cos \theta+\chi \sin \theta$ does not describe a "composite" field of the two (massless) fermions $\varphi$ and $\chi$ but a "mixture". However, fields with different charges $e_\varphi \ne e_\chi$ cannot be mixed (superselection rule). Only fields of the same charge sector can be mixed, which is usually done to identify the mass eigenfields.
