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 many 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 three 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.

This article is about the many-body problem in quantum mechanics. For the n-body problem in classical mechanics, see n-body problem.

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

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.