The most general solution to the Klein-Gordon equation is written as
\begin{equation} \Phi(x)= \int \mathrm{d }k^3 \frac{1}{(2\pi)^3 2\omega_k} \left( a(k){\mathrm{e }}^{ -i(k x)} + a^\dagger(k) {\mathrm{e }}^{ i(kx)}\right) \end{equation}
where I guess the second part was added to make the solution real, i.e. $c+c^\dagger= 2Re(c)$, is this correct?
The general solution to the Dirac equation is written \begin{equation} \Psi = \sum_r \sqrt{\frac{m}{(2\pi)^3}} \int \frac{ d^3p}{\sqrt{w_p}} \left(c_r(p) u_r(p) {\mathrm{e }}^{-ipx}+ d_r^\dagger (p) v_r(p) {\mathrm{e }}^{+ipx} \right) \end{equation}
and to the Dirac-adjoint equation \begin{equation} \bar \Psi = \sum_r \sqrt{\frac{m}{(2\pi)^3}} \int \frac{ d^3p}{\sqrt{w_p}} \left(c_r^\dagger(p) \bar u_r(p) {\mathrm{e }}^{+ipx}+ d_r (p) \bar v_r(p) {\mathrm{e }}^{-ipx} \right) \end{equation} and I would be interested in how the naming of the Fourier coefficents is justified in the first place. I know that they are interepreted in terms of creating and annihilating particles and anti-particles in QFT, but why do we name the coefficents here $c$ and $d^\dagger$ and not for example $c$ and $c^\dagger$ for $\Psi$, just as for the scalar case?