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Multiple functional derivatives

  1. Jul 17, 2010 #1
    Hi all,

    Long time stalker, first time poster. I've finally got stumped by something not already answered (as far as I can tell) around here. I'm trying to make sense of double functional derivatives: specifically, I would like to understand expressions like

    [tex]\int dx \frac{\delta^2}{\delta \phi(x) \delta \phi(x)} \Psi[\phi] [/tex].

    What can happen is that taking one derivative gives me an expression with a [tex]\phi(x) [/tex] sitting in front of [tex]\Psi[/tex], and then I'm not sure how to act with the second functional, since I now have something like a function times a functional; the naive approach gives me a bunch of delta functions. For example, for a Gaussian

    [tex]\Psi[\phi] = exp \left[ -\frac{1}{2} \int dx' \phi(x') \phi(x') \right] [/tex]

    the first derivative is

    [tex]\frac{\delta \Psi}{\delta \phi(x)} = \left[ -\int dx' \delta(x-x') \phi(x') \right] \Psi = - \phi(x) \Psi [/tex].

    Now naively

    [tex]\frac{\delta^2 \Psi}{\delta \phi(x) \delta \phi(x)} = \frac{\delta \phi(x)}{\delta \phi(x)} \Psi + \phi(x) \frac{\delta \Psi}{\delta \phi(x)} [/tex]

    but the first term is just [tex]\delta(0)[/tex]! Which is even worse when I then try to integrate this over dx.

    So, my guess is that I'm supposed to instead treat the functional derivative as a partial derivative when there's some function of [tex]\phi(x)[/tex] sitting in front of the functional. But I

    a. don't know if this is true
    b. don't know why it should be true.

    Any help appreciated!

    Thanks,
    Dan
     
  2. jcsd
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