= Bosonic species-swap generator
{title2=$C=\exp(i\pi N_-)$}
For two equal-energy bosonic species with <annihilation operators> $a,b$, set $c_-=(a-b)/\sqrt2$ and let $N_-$ be its <number operator>. Since $[N_-,c_-]=-c_-$, conjugation by $e^{i\pi N_-}$ reverses the antisymmetric combination while fixing $(a+b)/\sqrt2$, and therefore exchanges $a,b$ and their <creation operators>. For the particle and <antiparticle> modes of a <complex scalar field> this realizes <charge conjugation>, fixes the <Fock vacuum>, and squares to one. The normal-ordered generator $\int(a^\dagger-b^\dagger)(a-b)$ equals $2N_-$. Writing the annihilation factors first adds an infinite constant in the continuum, which must be subtracted or regulated before defining its exponential.
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