Wick's Theorem Proof (Peskins and Schroder)

[jameson2]

I'm having a bit of trouble working through the induction proof they give in the book.
The step I don't understand is: (page 90 in the book, halfway down)
$$N(\phi_2...\phi_m)\phi_1^+ + [\phi_1^+,N(\phi_2...\phi_m)] = N(\phi_1^+\phi_2...\phi_m) + N([\phi_1^+,\phi_2^-]\phi_3...\phi_m + \phi_2[\phi_1^+,\phi_3^-]\phi_4...\phi_m + ...)$$

I've gone through the m=2 case in the book, and I did m=3 myself. But I just can't see how they get between the two lines above, even though I've convinced myself it should work.

If someone could explain it's be great, thanks.

 

[kof9595995]

First of all, automatically $N(\phi_2...\phi_m)\phi_1^{+}= N(\phi_1^{+}\phi_2...\phi_m)$ since $\phi_1^{+}$ is purely made up of annihilation operators.
Second, you need to prove $[\phi_1^{+}, N(\phi_2...\phi_m)]=N([\phi_1^{+},(\phi_2...\phi_m)])$. To prove this you need induction again and the relation $[A,BC]=[A,B]C+B[A,C]$. After these you just commute the $\phi_1^{+}$ through the string of operators $\phi_2...\phi_m$ then you should get it.