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, $[psi(x),pi(y)]=idelta^{(3)}(x-y)$, where $pi(x)$ is momenta. However, this part seems missing in the Dirac field in Peskin's book.
Could anyone please help?

Michele Grosso · Answer

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

Some basic concepts in quantum field theory part 2

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