Some basic concepts in quantum field theory part 2

Question

This question is a sequel of

Some basic concepts in quantum field theory part 1

In Peskin's book, the part How Not to Quantize the Dirac Field, the writer says,"...in analogy with Klein-Gordon field...impose the canonical comutation relations $[\psi_{a}(x),\psi^{+}_{b}(y)]=\delta_{x-y}\delta_{ab}$, where $a$ and $b$ denote the spinor components of $\psi$...turn out to be a blind alley"

Though leading to a blind alley for the Dirac field, the contents seem to hint that $[\psi(x),\psi^{+}(y)]=\delta_{x-y}$ for complex scalar field. However, equ(1) would give $[\psi(x),\psi^{+}(y)]=0$.

Also, in Klein-Gordon field, $[\psi(x),\pi(y)]=i\delta^{(3)}(x-y)$, where $\pi(x)$ is momenta. However, this part seems missing in the Dirac field in Peskin's book.

Could anyone please help?

Answer 1

To quantize a classical field you promote the field and its conjugate momentum to operators and impose the canonical commutation relations, e.g. for a Klein-Gordon field $\phi (x)$
$[\phi (x), \pi (y)] = i \delta (x-y)$
where $\pi (x) = \frac{\partial \mathcal L}{\partial \dot \phi (x)} = \dot \phi (x)$ is the conjugate momentum.

As for a Dirac field $\psi$, in analogy with the Klein-Gordon, however eventually turning to a blind alley, starting from the Lagrangian
$\mathcal L = \bar \psi (i \gamma^\mu \partial_\mu - m) \psi$
where $\bar \psi = \psi^\dagger \gamma^0$, we have as conjugate momentum $\frac{\partial \mathcal L}{\partial \dot \psi (x)} = i \psi^\dagger$, hence
$[\psi (x), i \psi^\dagger (y)] = i \delta (x-y)$