Wick's Theorem Proof (Peskins and Schroder)

1. Jan 23, 2012

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.

2. Jan 23, 2012

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.