Ladder operators in electron field and electron's charge

[DOTDO]

S. Weinberg says in his book, "The Quantum Theory of Fields Volume I", that

Since electrons carry a charge, we would not like to mix annihilation and creation operators, so we might try to write the field as $$\psi(x)=\sum_{k}u_k (x)e^{-i\omega_k t}a_k$$
where $u_k (x)e^{-i\omega_k t}$ are a complete set of orthonormal plane-wave solutions of the Dirac equation with $k$ labelling the 3-momentum, spin, and sign of the energy.
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At the first time, I thought it's because $\sum_{k}$ already involves the negative $\omega_k$ so that $b_k$ and $b^\dagger _k$ in $\psi(x)=\sum_{k}u_k (x)e^{-i\omega_k t}b_k+u_k (x)e^{i\omega_k t}b^\dagger _k$ can be merged into $a_k$.

But he says it's because of the electron's charge and explains no more. Can someone explain, please?

Thank you.

 

[mfb]

I don't have the book, but electrons cannot be created or destroyed in isolation - their electric charge is one reason, the lepton number is another.