Fuzzy sphere

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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 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 $[J a, J b] = i ε a b c J c$ , where $ε a b c$ is the totally anti-commuting tensor with $ε 123 = 1$ , 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

$J_1^2+J_2^2+J_3^2=\frac{1}{4}(j^2-1)I$

where I is the j-dimensional identity matrix. Thus, if we define the 'coordinates' $x a = k r - 1 J a$ where r is the radius of the sphere and k is a parameter, related to r and j by $4 r 4 = k 2 (j 2 - 1)$ , 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

$\int_{S^2}fd\Omega:=2\pi k Tr(F)$

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

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

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

[edit] See also

Fuzzy torus

[edit] Notes

John Madore, An introduction to Noncommutative Differential Geometry and its Physical Applications, London Mathematical Society Lecture Note Series. 257, Cambridge University Press 2002