twoform
Jul17-10, 03:05 PM
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
\int dx \frac{\delta^2}{\delta \phi(x) \delta \phi(x)} \Psi[\phi] .
What can happen is that taking one derivative gives me an expression with a \phi(x) sitting in front of \Psi, 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
\Psi[\phi] = exp \left[ -\frac{1}{2} \int dx' \phi(x') \phi(x') \right]
the first derivative is
\frac{\delta \Psi}{\delta \phi(x)} = \left[ -\int dx' \delta(x-x') \phi(x') \right] \Psi = - \phi(x) \Psi .
Now naively
\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)}
but the first term is just \delta(0)! 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 \phi(x) 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
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
\int dx \frac{\delta^2}{\delta \phi(x) \delta \phi(x)} \Psi[\phi] .
What can happen is that taking one derivative gives me an expression with a \phi(x) sitting in front of \Psi, 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
\Psi[\phi] = exp \left[ -\frac{1}{2} \int dx' \phi(x') \phi(x') \right]
the first derivative is
\frac{\delta \Psi}{\delta \phi(x)} = \left[ -\int dx' \delta(x-x') \phi(x') \right] \Psi = - \phi(x) \Psi .
Now naively
\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)}
but the first term is just \delta(0)! 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 \phi(x) 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