This talk will present the basic ideas involved in integrating the non-relativistic Schroedinger equation for many-body systems. In imaginary time, it is a many-dimensional diffusion equation in which the physical potential has the effect of creating or destroying Brownian walkers. Monte Carlo methods, often modified by importance-sampling and other variance reduction methods can be readily applied. The outcome is a numerical procedure that can give estimates of quantities of physical interest for the fundamental mode that have no uncontrolled approximations.
Unfortunately, for many important systems, the fundamental mode is of no interest: what is required is the lowest “fermionic” mode, in which the solution is “antisymmetric,” i.e. where the solution changes sign when the coordinates of like-spin particles are interchanged. This presents a deep challenge to the numerical method.
The body of the talk will be devoted to the exposition of this challenge, and a class of proposals for its solution.