The quantum many-body problem is a general name for a vast category of physical problems pertaining to deriving the behavior of multi-particle systems using fundamental quantum-mechanical principles.
The goal of many-body physics is to find new principles to describe macroscopic systems, using principles that pertain to microscopic systems.
Terminology
Many can be anywhere from three to infinity, although three- and four-body systems can be treated by specific means (respectively the Faddeev and Faddeev–Yakubovsky equations) and are thus sometimes separately classified as few-body systems.
Body, in this case, is referring to a particle (electron, nuclei, atom, etc.).
This arising complexity becomes especially clear by a comparison to classical mechanics. Imagine a single particle that can be described with <math>k</math> numbers (take for example a free particle described by its position and velocity vector, resulting in <math>k=6</math>). In classical mechanics, <math>n</math> such particles can simply be described by <math>k\cdot n</math> numbers. The dimension of the classical many-body system scales linearly with the number of particles, <math> n </math>.
In quantum mechanics, however, the dimension of the many-body wave function scales exponentially with <math> n </math>, much faster than in classical mechanics. Thus, many-body physics most often relies on a set of approximation methods, such as the variational method and perturbation theory, specific to the problem at hand. It ranks among the most computationally intensive fields of science.