Abstract
A proof of the nonexistence of self-conjugate isospin multiplets with half-integral isospin but with any spin is presented. The proof is based on a Lorentz- and isospin-covariant field equation, describing a self-conjugate isospin multiplet of fields of isospin $I$ and of spin $s$, together with the usual representation of its solutions in terms of particle creation and annihilation operators. No assumptions are made with regard to local commutativity, the usual connection between spin and statistics, or the $\mathrm{CPT}$ theorem.
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