# Fourier transform of integration measure (Peskin and Schroeder)

1. Apr 3, 2013

### center o bass

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

$$\phi(k_n) = \frac{1}{V} \sum_n e^{-i k_n \cdot x_i} \phi(k_n)$$

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

$$D\phi = \Pi_{k_n^0 > 0} d Re \phi(k_n) d Im \phi(k_n).$$

Why is he only considering $k^0_n >0$ and how does he arrive at this conclusion? What is the relevance of the fourier transform being unitary?