Many-body problem

# Many-body problem

Discussion

Encyclopedia
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
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

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
Periodic function
In mathematics, a periodic function is a function that repeats its values in regular intervals or periods. The most important examples are the trigonometric functions, which repeat over intervals of length 2π radians. Periodic functions are used throughout science to describe oscillations,...

system, such as a crystal
Crystal
A crystal or crystalline solid is a solid material whose constituent atoms, molecules, or ions are arranged in an orderly repeating pattern extending in all three spatial dimensions. The scientific study of crystals and crystal formation is known as crystallography...

), although three- and four-body systems can be treated by specific means (respectively the Faddeev
The Faddeev equations, named after their inventor Ludvig Faddeev, are equations that describe, at once, all the possible exchanges/interactions in a system of three particles in a fully quantum mechanical formulation. They can be solved iteratively....

and Faddeev-Yakubovsky equations) and are thus sometimes separately classified as few-body systems
Few-body systems
In quantum mechanics and classical mechanics, a few-body system consists of a small number of well-defined structures or point particles.In quantum mechanics, examples of few-body systems include light nuclear systems , small molecules, light atoms , atomic collisions, and quantum...

. In such a quantum system, the repeated interactions between particles create quantum correlations, or entanglement
Quantum entanglement
Quantum entanglement occurs when electrons, molecules even as large as "buckyballs", photons, etc., interact physically and then become separated; the type of interaction is such that each resulting member of a pair is properly described by the same quantum mechanical description , which is...

. As a consequence, the wave function of the system is a complicated object holding a large amount of information
Information
Information in its most restricted technical sense is a message or collection of messages that consists of an ordered sequence of symbols, or it is the meaning that can be interpreted from such a message or collection of messages. Information can be recorded or transmitted. It can be recorded as...

, which usually makes exact and/or analytical calculations impractical. 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
Condensed matter physics
Condensed matter physics deals with the physical properties of condensed phases of matter. These properties appear when a number of atoms at the supramolecular and macromolecular scale interact strongly and adhere to each other or are otherwise highly concentrated in a system. The most familiar...

(solid-state physics
Solid-state physics
Solid-state physics is the study of rigid matter, or solids, through methods such as quantum mechanics, crystallography, electromagnetism, and metallurgy. It is the largest branch of condensed matter physics. Solid-state physics studies how the large-scale properties of solid materials result from...

, nanoscience, superconductivity
Superconductivity
Superconductivity is a phenomenon of exactly zero electrical resistance occurring in certain materials below a characteristic temperature. It was discovered by Heike Kamerlingh Onnes on April 8, 1911 in Leiden. Like ferromagnetism and atomic spectral lines, superconductivity is a quantum...

)
• Bose-Einstein condensation and Superfluids
• Quantum chemistry
Quantum chemistry
Quantum chemistry is a branch of chemistry whose primary focus is the application of quantum mechanics in physical models and experiments of chemical systems...

(computational chemistry
Computational chemistry
Computational chemistry is a branch of chemistry that uses principles of computer science to assist in solving chemical problems. It uses the results of theoretical chemistry, incorporated into efficient computer programs, to calculate the structures and properties of molecules and solids...

, molecular physics
Molecular physics
Molecular physics is the study of the physical properties of molecules, the chemical bonds between atoms as well as the molecular dynamics. Its most important experimental techniques are the various types of spectroscopy...

)
• Atomic physics
Atomic physics
Atomic physics is the field of physics that studies atoms as an isolated system of electrons and an atomic nucleus. It is primarily concerned with the arrangement of electrons around the nucleus and...

• Molecular physics
Molecular physics
Molecular physics is the study of the physical properties of molecules, the chemical bonds between atoms as well as the molecular dynamics. Its most important experimental techniques are the various types of spectroscopy...

• Nuclear physics
Nuclear physics
Nuclear physics is the field of physics that studies the building blocks and interactions of atomic nuclei. The most commonly known applications of nuclear physics are nuclear power generation and nuclear weapons technology, but the research has provided application in many fields, including those...

(Nuclear structure
Nuclear structure
Understanding the structure of the atomic nucleus is one of the central challenges in nuclear physics. This article is written from a nuclear physics perspective; as such, it is suggested that a casual reader first read the main nuclear physics article....

, nuclear reactions, nuclear matter
Nuclear matter
Nuclear matter is an idealized system of interacting nucleons that exists in several phases that as yet are not fully established...

)
• Quantum chromodynamics
Quantum chromodynamics
In theoretical physics, quantum chromodynamics is a theory of the strong interaction , a fundamental force describing the interactions of the quarks and gluons making up hadrons . It is the study of the SU Yang–Mills theory of color-charged fermions...

(Lattice QCD
Lattice QCD
Lattice QCD is a well-established non-perturbative approach to solving the quantum chromodynamics theory of quarks and gluons. It is a lattice gauge theory formulated on a grid or lattice of points in space and time....

In particle physics, a hadron is a composite particle made of quarks held together by the strong force...

spectroscopy, QCD matter
QCD matter
Quark matter or QCD matter refers to any of a number of theorized phases of matter whose degrees of freedom include quarks and gluons. These theoretical phases would occur at extremely high temperatures and densities, billions of times higher than can be produced in equilibrium in laboratories...

, quark-gluon plasma
Quark-gluon plasma
A quark–gluon plasma or quark soup is a phase of quantum chromodynamics which exists at extremely high temperature and/or density. This phase consists of asymptotically free quarks and gluons, which are several of the basic building blocks of matter...

)

## Approaches

• Mean-field theory and extensions (e.g. Hartree-Fock
Hartree-Fock
In computational physics and chemistry, the Hartree–Fock method is an approximate method for the determination of the ground-state wave function and ground-state energy of a quantum many-body system....

, Random phase approximation
Random phase approximation
The random phase approximation is an approximation method in condensed matter physics and in nuclear physics. It was first introduced by David Bohm and David Pines as an important result in a series of seminal papers. For decades physicists had been trying to incorporate the effect of microscopic...

)
• Dynamical Mean Field Theory
Dynamical mean field theory
Dynamical Mean Field Theory is a method to determine the electronic structure of strongly correlated materials. In such materials, the approximation of independent electrons, which is used in Density Functional Theory and usual band structure calculations, breaks down...

• Many-body perturbation theory
Perturbation theory
Perturbation theory comprises mathematical methods that are used to find an approximate solution to a problem which cannot be solved exactly, by starting from the exact solution of a related problem...

and Green's function
Green's function (many-body theory)
In many-body theory, the term Green's function is sometimes used interchangeably with correlation function, but refers specifically to correlators of field operators or creation and annihilation operators....

-based methods
• Configuration interaction
Configuration interaction
Configuration interaction is a post-Hartree–Fock linear variational method for solving the nonrelativistic Schrödinger equation within the Born–Oppenheimer approximation for a quantum chemical multi-electron system. Mathematically, configuration simply describes the linear combination...

• Coupled cluster
Coupled cluster
Coupled cluster is a numerical technique used for describing many-body systems. Its most common use is as one of several quantum chemical post-Hartree–Fock ab initio quantum chemistry methods in the field of computational chemistry...

• Various Monte-Carlo
Quantum Monte Carlo
Quantum Monte Carlo is a large class of computer algorithms that simulate quantum systems with the idea of solving the quantum many-body problem. They use, in one way or another, the Monte Carlo method to handle the many-dimensional integrals that arise...

approaches
• Density functional theory
Density functional theory
Density functional theory is a quantum mechanical modelling method used in physics and chemistry to investigate the electronic structure of many-body systems, in particular atoms, molecules, and the condensed phases. With this theory, the properties of a many-electron system can be determined by...

• Lattice gauge theory
Lattice gauge theory
In physics, lattice gauge theory is the study of gauge theories on a spacetime that has been discretized into a lattice. Gauge theories are important in particle physics, and include the prevailing theories of elementary particles: quantum electrodynamics, quantum chromodynamics and the Standard...