When studying the Klein-Gordon equation, the introduction of creation/annihilation operators was justified by recognizing a harmonic-oscillator-like equation which we know how to quantize. Is there a similar justification when introducing these operators for the Dirac equation? Most of the resources I have looked at simply state them and move on.
In general the reason is that it was desired to find an operator solution to the equation, so that the fields could be operators in analogy to the observable operators that we have in nonrelativistic quantum mechanics.
I don't have an explicit proof of it, but a professor of mathematical physics who I spoke to told me that "putting operators in place of $a$ and $a^\dagger$ gives the most general operator solution to the equation of motion". At that point it can be shown that these operators must be annihilation and creation operators.
$\begingroup$ So it is a mathematical consequence? $\endgroup$– CBBAMMar 1 at 3:36
$\begingroup$ If you're looking for an operator solution then yes $\endgroup$ Mar 1 at 5:22
The justification is that each component of the Dirac equation is also a solution to the klein-gordon equation.