... the development of quantum field theory made the interpretation of antiparticles as holes unnecessary, even though it lingers on in many textbooks.
where the proportionality sign indicates that there might be a phase on the right hand side.
One may try to quantize an electron field without mixing the annihilation and creation operators by writing
So one has to introduce the charge conjugate antiparticle field, with its own creation and annihilation operators satisfying the relations
where the first sum is over positive energy states and the second over those of negative energy. The energy becomes