Configuration space (mathematics)

In mathematics, a configuration space is a construction closely related to state spaces or phase spaces in physics. In physics, these are used to describe the state of a whole system as a single point in a high-dimensional space. In mathematics, they are used to describe assignments of a collection of points to positions in a topological space. More specifically, configuration spaces in mathematics are particular examples of configuration spaces in physics in the particular case of several non-colliding particles.

For a topological space ${\displaystyle X}$ and a positive integer ${\displaystyle n}$, let ${\displaystyle X^{n}}$ be the Cartesian product of ${\displaystyle n}$ copies of ${\displaystyle X}$, equipped with the product topology. The n^th (ordered) configuration space of ${\displaystyle X}$ is the set of n-tuples of pairwise distinct points in ${\displaystyle X}$:

This space is generally endowed with the subspace topology from the inclusion of ${\displaystyle \operatorname {Conf} _{n}(X)}$ into ${\displaystyle X^{n}}$. It is also sometimes denoted ${\displaystyle F(X,n)}$, ${\displaystyle F^{n}(X)}$, or ${\displaystyle {\mathcal {C}}^{n}(X)}$.

There is a natural action of the symmetric group ${\displaystyle S_{n}}$ on the points in ${\displaystyle \operatorname {Conf} _{n}(X)}$ given by

This action gives rise to the n^th unordered configuration space of X,

which is the orbit space of that action. The intuition is that this action "forgets the names of the points". The unordered configuration space is sometimes denoted ${\displaystyle {\mathcal {UC}}^{n}(X)}$, ${\displaystyle B_{n}(X)}$, or ${\displaystyle C_{n}(X)}$. The collection of unordered configuration spaces over all ${\displaystyle n}$ is the Ran space, and comes with a natural topology.

For a topological space ${\displaystyle X}$ and a finite set ${\displaystyle S}$, the configuration space of X with particles labeled by S is

For ${\displaystyle n\in \mathbb {N} }$, define ${\displaystyle \mathbf {n} :=\{1,2,\ldots ,n\}}$. Then the n^th configuration space of X is denoted simply ${\displaystyle \operatorname {Conf} _{n}(X)}$.