# Some basic concepts in quantum field theory part 2

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.

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
$$[\psi (x), i \psi^\dagger (y)] = i \delta (x-y)$$