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A Taylor series expansion of functional

  1. Jan 4, 2017 #1
    I'm studying QFT in the path integral formalism, and got stuck in deriving the Schwinger Dyson equation for a real free scalar field,
    L=½(∂φ)^2 - m^2 φ^2
    in the equation,
    S[φ]=∫ d4x L[φ]
    ∫ Dφ e^{i S[φ]} φ(x1) φ(x2) = ∫ Dφ e^{i S[φ']} φ'(x1) φ'(x2)
    Particularly, it is in the Taylor series expansion of the functional exponential
    e^{i S[φ']}=e^{i S[φ+iα]} . Can anybody please tell me about the expansion? I have searched and haven't found anything quite helpful on the net. Thank you.
     
  2. jcsd
  3. Jan 4, 2017 #2

    Paul Colby

    User Avatar
    Gold Member

    ##e^x = \sum_{n=0}^\infty \frac{x^n}{n!}##
     
  4. Jan 4, 2017 #3
    I believe that is a power series expansion. The final answer should contain the exponential still since, we have to relate it to the path integral. I have just started learning the functional formalism and I wanted to know whether,
    ##F[\phi']=F[\phi+\epsilon]=F[\phi]+\epsilon\left.\frac{dF}{d\phi'}\right|_{\phi'=\phi}+O(\epsilon^2)##
    Which I believe is the functional analog of the Taylor series expansion, is correct and if the differential is indeed given by,
    ##\frac{dF}{d\phi'}=\int d^4y \:\varepsilon(y) \frac{\delta F[\phi(x)]}{\delta\phi(y)}##
    Should ##\epsilon## and ##\varepsilon## be the same or should one of them be omitted. Or is the formula incorrect?
     
    Last edited: Jan 4, 2017
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