Fuzzy sphere

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 ${\displaystyle j^{2}}$-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 matrices ${\displaystyle J_{a},~a=1,2,3}$ that form a basis for the j dimensional irreducible representation of the Lie algebra su(2). They satisfy the relations ${\displaystyle [J_{a},J_{b}]=i\epsilon _{abc}J_{c}}$, where ${\displaystyle \epsilon _{abc}}$ is the totally antisymmetric symbol with ${\displaystyle \epsilon _{123}=1}$, and generate via the matrix product the algebra ${\displaystyle M_{j}}$ of j dimensional matrices. The value of the su(2) Casimir operator in this representation is

${\displaystyle J_{1}^{2}+J_{2}^{2}+J_{3}^{2}={\frac {1}{4}}(j^{2}-1)I}$

where I is the j-dimensional identity matrix. Thus, if we define the 'coordinates' ${\displaystyle x_{a}=kr^{-1}J_{a}}$ where r is the radius of the sphere and k is a parameter, related to r and j by ${\displaystyle 4r^{4}=k^{2}(j^{2}-1)}$, then the above equation concerning the Casimir operator can be rewritten as

${\displaystyle x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=r^{2}}$,

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

${\displaystyle \int _{S^{2}}fd\Omega$ :=2\pi k\,{\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

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

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