Fourier Modes of a Real Scalar Field

Jorge L. deLyra

Department of Mathematical Physics, Physics Institute, University of São Paulo

Version 1, June 2006

We will consider here the technical questions of how to deal with the Fourier modes of a single-component real scalar field in numerical simulations. This includes how to organize them in order to avoid over-counting the degrees of freedom, how to index them into vectors and how to build the transformations from position space to momentum space and back in an efficient way, so that they can be executed very fast, for use in stochastic simulations. It is important to note that the type of implementation presented here is not what is usually known as the Fast Fourier Transform (FFT). It may be described rather as an Indexed Fourier Transform, and is meant specifically for the relatively small lattices used in the stochastic simulations of field-theoretical models in space-time dimensions ranging from $d=2$ to $d=5$.