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.