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Fourier transform of integration measure (Peskin and Schroeder)

  1. Apr 3, 2013 #1
    At page 285 in Peskin and Schroeder's Introduction to quantum field theory the author defines the integration measure [itex]D\phi = \Pi_i d\phi(x_i)[/itex] where space-time is being discretised into a square lattice of volume L^4. He proceeds by Fourier-transforming

    [tex] \phi(k_n) = \frac{1}{V} \sum_n e^{-i k_n \cdot x_i} \phi(k_n)[/tex]

    and considering only [itex]k^0_n >0[/itex] as independent variables he concludes that since the fourier transformation is unitary, the measure [itex]D\phi = \Pi_i d\phi(x_i)[/itex] can equivalently be expressed as (equation 9.22)

    [tex] D\phi = \Pi_{k_n^0 > 0} d Re \phi(k_n) d Im \phi(k_n).[/tex]

    Why is he only considering [itex]k^0_n >0[/itex] and how does he arrive at this conclusion? What is the relevance of the fourier transform being unitary?
  2. jcsd
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