Fuzzy sphere

{{No footnotes|date=April 2020}}

In mathematics, the fuzzy sphere is one of the simplest and most canonical examples of non-commutative geometry. Ordinarily, the functions defined on a sphere form a commuting algebra. A fuzzy sphere differs from an ordinary sphere because the algebra of functions on it is not commutative. It is generated by spherical harmonics whose spin l is at most equal to some j. The terms in the product of two spherical harmonics that involve spherical harmonics with spin exceeding j are simply omitted in the product. This truncation replaces an infinite-dimensional commutative algebra by a $j^2$-dimensional non-commutative algebra.

The simplest way to see this sphere is to realize this truncated algebra of functions as a matrix algebra on some finite-dimensional vector space.

Take the three j-dimensional square matrices $J\_a,~\; a=1,2,3$ that form a basis for the j dimensional irreducible representation of the Lie algebra su(2). They satisfy the relations $[J\_a,J\_b]=i\backslash epsilon\_\{abc\}J\_c$, where $\backslash epsilon\_\{abc\}$ is the totally antisymmetric symbol with $\backslash epsilon\_\{123\}=1$, and generate via the matrix product the algebra $M\_j$ of j dimensional matrices. The value of the su(2) Casimir operator in this representation is

:$J\_1^2+J\_2^2+J\_3^2=\backslash frac\{1\}\{4\}(j^2-1)I$

where $I$ is the j-dimensional identity matrix.

Thus, if we define the 'coordinates'

$x\_a=kr^\{-1\}J\_a$

where r is the radius of the sphere and k is a parameter, related to r and j by $4r^4=k^2(j^2-1)$, then the above equation concerning the Casimir operator can be rewritten as

:$x\_1^2+x\_2^2+x\_3^2=r^2$,

which is the usual relation for the coordinates on a sphere of radius r embedded in three dimensional space.

One can define an integral on this space, by

:$\backslash int\_\{S^2\}fd\backslash Omega:=2\backslash pi\; k\; \backslash ,\; \backslash text\{Tr\}(F)$

where F is the matrix corresponding to the function f.

For example, the integral of unity, which gives the surface of the sphere in the commutative case is here equal to

:$2\backslash pi\; k\; \backslash ,\; \backslash text\{Tr\}(I)=2\backslash pi\; k\; j\; =4\backslash pi\; r^2\backslash frac\{j\}\{\backslash sqrt\{j^2-1\}\}$

which converges to the value of the surface of the sphere if one takes j to infinity.