# Problem with expansion of normal ordering

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 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.

## 1 Answer

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.

• I have edited my question and added my solution. Is that correct..I am unable to grasp that first term where G and phi commute and zero..and also I have highlighted in red...that is my assumption but don't know why. – Radha Krishna May 4 at 16:32
• I updated the answer. – Qmechanic May 4 at 16:41
• Thanks for help..is my answer correct which I have attempted? – Radha Krishna May 4 at 16:42
• It looks OK apart from some curly bracket typos (v6). – Qmechanic May 4 at 16:45