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Grassmann variables and functional derivatives

  1. Jun 29, 2012 #1
    Hi all! I'm sorry if this question has been already asked in another post...
    I'm studying the path integrals formalism in QED. I'm dealing with the functional generator for fermionic fields. My question is:

    The generating functional is:

    $$Z_0=e^{-i\int{d^4xd^4y \bar{J}(x)S(x-y)J(y)}}$$

    Where $$J(x)$$ and $$\bar{J}(x)$$ are Grassmann numbers.
    When I have to extract Green function from the generating functional I have to perform, for example, a functional derivative rispect to J(z). Does the functional derivative follow the same rule as the ordinary derivative? Do I have to anticommutate $$\bar{J}$$ and $$J$$ before deriving??

    Thaks to all
  2. jcsd
  3. Jun 30, 2012 #2


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    Normally yes, because you'd have to use either the left functional derivative or the right one constantly throughout. For example, if you use the right derivative and the Jbar is to the left, you'd have to shift it past J (that is bring it to the right) and then differentiate. So the answer is yes.
  4. Jun 30, 2012 #3
    Thank you very much! You have been great!

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