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I am reading normal ordering..and far now I'm able to understand. I am stuck in third line from second expression in the book Lectures On Quantum Field Theory By Ashok Das in page no. 237.

It is given that

$$ \begin{aligned} \,&[\;:\!\phi(x)\phi(y)\!:,\phi^{(-)}(z)]\\ =& [\phi(x)\phi(y) + i G^{(+)}(x-y), \phi^{(-)}(z)] \\ = & [\phi^{(+)}(x),\phi^{(-)}(z)]\phi(y) + \phi(x) [\phi^{(+)}(y),\phi^{(-)}(z)] \end{aligned}\tag{6.89} $$

I understand it is derived as enter image description here

I understand that Term highlighted in red commute and zero. Is this correct or am I doing anything wrong. Does G term commutes with field as I have highlighted in first.

Radha Krishna
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Between the 2nd & 3rd line in eq. (6.89) Ashok Das is using that:

  • The contraction $\langle 0|\phi(x)\phi(y)|0\rangle=-iG^{(+)}(x-y)$ commutes with anything, cf. eq. (6.82). For more details, see my related Phys.SE answer here.

  • $\phi = \phi^{(+)} + \phi^{(-)}$, cf. eq. (6.83).

  • The creation parts $\phi^{(-)}(x)$ and $\phi^{(-)}(y)$ commute.

Qmechanic
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