Fermion sign problem: Decoupling transformation and simulation algorithm

We discuss the fermion sign problem and, by examining a very general Hubbard-Stratonovich transformation, argue that the sign problem cannot be solved with such methods. We propose a different kind of transformation which, while not solving the sign problem, shows more detailed information about the system. With our transformation it is trivial to tell which auxiliary field configurations give a positive sign and which give a negative sign. We then discuss briefly various properties of this transformation and construct an algorithm which with one simulation gives results for a whole range of particle densities and Hubbard U values, positive and negative. Our approach is in excellent agreement with exact calculations.