# Why does Peskin and Schroeder move normal ordering move outside a commutator?

The equation trying to prove that Wick's theorem by induction in P&S on page 90 implies that normal ordering can be moved outside a commutator (at least with a positive frequency field), which I just don't understand. It implies

$$[\phi_1^+,N(\phi_2\ldots\phi_m)] = N([\phi_1^+, \phi_2\ldots\phi_m]).$$

What am I missing here?

• Did you find an answer to your question elsewhere? Commented Jun 6, 2021 at 22:09

$$\phi_1^+N(\phi_2\cdots\phi_m) = N(\phi_2\cdots\phi_m)\phi_1^+ + [\phi_1^+,N(\phi_2\cdots\phi_m)]$$ $$=N(\phi_1^+\phi_2\cdots\phi_m) + N\big([\phi_1^+,\phi_2^-]\phi_3\cdots \phi_m + \phi _2[\phi_1^+,\phi_3^-]\phi_4\cdots\phi_m + \cdots\big)$$
The implication here is that $$N(\phi_2\cdots\phi_m)\phi_1^+=N(\phi_1^+ \phi_2\cdots\phi_m)$$. This is simply because $$\phi_1^+$$ contains only annihilation operators, and so normal ordering puts it on the right hand side.
• I'm referring to the second term, not the first. The first term absorbing $\phi_1^+$ makes sense. Commented May 16, 2020 at 16:55