# Many-body problem

The **many-body problem** is a general name for a vast category of physical problems pertaining to the properties of microscopic systems made of a large number of interacting particles. *Microscopic* here implies that quantum mechanics has to be used to provide an accurate description of the system. A *large number* can be anywhere from 3 to infinity (in the case of a practically infinite, homogeneous or periodic system, such as a crystal), 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. In such a quantum system, the repeated *interactions* between particles create quantum correlations, or entanglement. As a consequence, the wave function of the system is a complicated object holding a large amount of information, which usually makes exact or analytical calculations impractical or even impossible. Thus, many-body theoretical physics most often relies on a set of approximations specific to the problem at hand, and ranks among the most computationally intensive fields of science.

## Examples

- Condensed matter physics (solid-state physics, nanoscience, superconductivity)
- Bose–Einstein condensation and Superfluids
- Quantum chemistry (computational chemistry, molecular physics)
- Atomic physics
- Molecular physics
- Nuclear physics (Nuclear structure, nuclear reactions, nuclear matter)
- Quantum chromodynamics (Lattice QCD, hadron spectroscopy, QCD matter, quark–gluon plasma)

## Approaches

- Mean-field theory and extensions (e.g. Hartree–Fock, Random phase approximation)
- Dynamical mean field theory
- Many-body perturbation theory and Green's function-based methods
- Configuration interaction
- Coupled cluster
- Various Monte-Carlo approaches
- Density functional theory
- Lattice gauge theory
- Matrix product state

## Quotes

"It would indeed be remarkable if Nature fortified herself against further advances in knowledge behind the analytical difficulties of the many-body problem."— Max Born, 1960

## Further reading

- Jenkins, Stephen. "The Many Body Problem and Density Functional Theory".
- Thouless, D. J. (1972).
*The quantum mechanics of many-body systems*. New York: Academic Press. ISBN 0-12-691560-1. - Fetter, A. L.; Walecka, J. D. (2003).
*Quantum Theory of Many-Particle Systems*. New York: Dover. ISBN 0-486-42827-3. - Nozières, P. (1997).
*Theory of Interacting Fermi Systems*. Addison-Wesley. ISBN 0-201-32824-0. - Mattuck, R. D. (1976).
*A guide to Feynman diagrams in the many-body problem*. New York: McGraw-Hill. ISBN 0-07-040954-4.