Editing Many-body problem (section)

==Explanation of the problem==
In general terms, while the underlying [[physical laws]] that govern the motion of each individual particle may (or may not) be simple, the study of the collection of particles can be extremely complex. 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 theory|information]], which usually makes exact or analytical calculations impractical or even impossible.

This 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 many-body-system is in general in a superposition of combinations of single particle states - all the <math> k^n </math> different combinations have to be accounted for. The dimension of the quantum many body system therefore scales exponentially with <math> n </math>, much faster than in classical mechanics.

Because the required numerical expense grows so quickly, simulating the dynamics of more than three quantum-mechanical particles is already infeasible for many physical systems.<ref>{{Cite journal |first1= David |last1= Hochstuhl |last2= Bonitz |first2= Michael |last3= Hinz |first3= Christopher |year= 2014 |title= Time-dependent multiconfiguration methods for the numerical simulation of photoionization processes of many-electron atoms |journal= The European Physical Journal Special Topics |volume= 223 |issue= 2 |pages= 177–336 |doi= 10.1140/epjst/e2014-02092-3 |bibcode= 2014EPJST.223..177H |s2cid= 122869981  }}</ref> Thus, many-body theoretical physics most often relies on a set of [[approximation]]s specific to the problem at hand, and ranks among the most [[High-performance computing|computationally intensive]] fields of science.

In many cases, [[Emergence|emergent phenomena]] may arise which bear little resemblance to the underlying elementary laws.

Many-body problems play a central role in [[condensed matter physics]].