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Four Point Correlation function from Generating Functional

  1. Oct 18, 2009 #1
    Hi everyone,

    I'm working through Section 9.2 (Functional Quantization of Scalar Fields) from Peskin and Schroeder. I have trouble understanding the absence of a term in equation 9.41 which I get but the authors do not.

    Define [itex]\phi_i \equiv \phi(x_i)[/itex], [itex]J_{x} \equiv J(x)[/itex], [itex]D_{xi} \equiv D_{F}(x-x_i)[/itex] (the Feynman propagator). Repeated subscripts are integrated over implicitly.

    Equation 9.41 in the book reads

    [tex]\langle 0|T\phi_1\phi_2\phi_3\phi_4|0\rangle = \frac{\delta}{\delta J_1}\frac{\delta}{\delta J_2}\frac{\delta}{\delta J_3}\frac{\delta}{\delta J_4}e^{-\frac{1}{2}J_x D_{xy} J_{y}}[/tex]

    [tex]= \frac{\delta}{\delta J_1}\frac{\delta}{\delta J_2}\frac{\delta}{\delta J_3}\left[-J_x D_{x4}\right]e^{-\frac{1}{2}J_x D_{xy} J_{y}}[/tex]
    [tex]= \frac{\delta}{\delta J_1}\frac{\delta}{\delta J_2}\left[-D_{34}+J_{x}D_{x4}J_{y}D_{y3}\right]e^{-\frac{1}{2}J_x D_{xy} J_{y}}[/tex]
    [tex]= \frac{\delta}{\delta J_1}\left[D_{34}J_{x}D_{x2}+D_{24}J_{y}D_{y3} +J_{x}D_{x4}D_{23}\right]e^{-\frac{1}{2}J_x D_{xy} J_{y}}[/tex]
    [tex]= D_{34}D_{12} + D_{24}D_{13} + D_{14}D_{23}[/tex]

    where J is set equal to zero after all the 4 functional derivatives have been evaluated.

    When I do this by hand, I get (in the second last step), an extra term:

    [tex]= \frac{\delta}{\delta J_1}\left[D_{34}J_{x}D_{x2}+D_{24}J_{y}D_{y3} +J_{x}D_{x4}D_{23}-J_x D_{x4} J_{y}D_{y3} J_z D_{z2}\right]e^{-\frac{1}{2}J_x D_{xy} J_{y}}[/tex]

    which isn't given in the book. I just follow the prescription of differentiating with proper order and bringing down a -J*D type of term every time the exponent is differentiated. What happened to this term? Please help..

    Thanks in advance.
     
  2. jcsd
  3. Oct 18, 2009 #2
    Wait...I think I got it...the terms that are O(J^2) survive the derivatives but go to zero when J is set equal to zero, whereas the O(J) terms are robbed of their J dependence by the last functional derivative. The authors just don't show the steps.

    I should've read what I was writing :uhh:
     
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