Consider a positive-energy solution with charge . The equivalent Klein-Gordon equation satisfied by the positive-energy solution is
The negative energy solution satisfies
Taking the complex conjugate of the negative energy equation gives
Comparing equation 4.77 with equation 4.79 shows
Thus is the charge conjugate solution and represents the charge conjugate state of . Similarly, is the charge conjugate state of . If we (arbitrarily) call the particle described by ``the particle'', then we call the particle described by the antiparticle. For example, if we call the meson the particle, then the meson is the antiparticle. The undesirable negative-energy solutions have now been interpreted as antiparticles.
Neutral particles fit into this picture too, in that the charge-conjugate state is the state itself. In other words, neutral particles are their own antiparticles. Let and so that we can write
where is a proportionality constant which has to be real. can be deduced since for neutral particles both and are real. Realizing that
it follows that
so that and . Accordingly there exist two different kinds of neutral particles, namely
Neutral particles are thus eigenfunctions of the charge conjugation operator, while charged particles are not.