1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Zee question I.2.2

  1. Nov 16, 2007 #1
    I've decided to slowly work my way through Zee's quantum field theory in a nutshell over the vacation. I'm confused by the second question of the book which deals with matrix differentiation.

    1. The problem statement, all variables and given/known data

    Derive the equation [itex]\langle x_i x_j \cdots x_k x_l \rangle = \sum_{\textrm{wick}} (A^{-1})_{ab}\cdot (A^{-1})_{cd}[/itex] where {a,b,...,c,d} is a permutation of {i,j,...,k,l}

    2. Relevant equations

    [itex]\langle x_i x_j \cdots x_k x_l \rangle = \frac{\int d\mathbf{x} e^{1/2 \mathbf{x}^TA \mathbf{x}}x_i x_j \cdots x_k x_l}{\int d\mathbf{x} e^{1/2 \mathbf{x}^TA \mathbf{x}}}[/itex]

    3. The attempt at a solution

    I have no trouble deriving this equation for the 1-variable case. For the multi-variable case Zee suggests repeatedly differentiating the equation

    [itex]\int d\mathbf{x} e^{-1/2 \mathbf{x}^TA\mathbf{x} + J\mathbf{x}} = [(2\pi)^N /\det A]e^{(1/2) J A^{-1}J}[/itex]

    with respect to J. How does one differentiate wrt a matrix anyway?
    Last edited: Nov 16, 2007
  2. jcsd
  3. Nov 16, 2007 #2
    Doh! It just occured to me that J is a vector and therefore we can differentiate componentwise. I still don't quite get how to prove this, however. I can show that the claim is true for small n. Is some kind of inductive argument required here??
  4. Nov 16, 2007 #3
    I figured it out: For future knowledge, the trick is to use that the matrix A must be symmetric, and thus the derivative of the [itex]\frac{1}{2}\mathbf{x}^t A^{-1} \mathbf{x}[/itex] with respect to [itex]x_i[/itex], say can be written

    [itex]\sum_n (A^{-1})_{in} x_n[/itex].
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook