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 {\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 square matrices J a , a = 1 , 2 , 3 {\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 [ J a , J b ] = i ϵ a b c J c {\displaystyle [J_{a},J_{b}]=i\epsilon _{abc}J_{c}} , where ϵ a b c {\displaystyle \epsilon _{abc}} is the totally antisymmetric symbol with ϵ 123 = 1 {\displaystyle \epsilon _{123}=1} , and generate via the matrix product the algebra M j {\displaystyle 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 = 1 4 ( j 2 − 1 ) I {\displaystyle J_{1}^{2}+J_{2}^{2}+J_{3}^{2}={\frac {1}{4}}(j^{2}-1)I}
where I {\displaystyle I} is the j-dimensional identity matrix. Thus, if we define the 'coordinates'
x a = k r − 1 J a {\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 4 r 4 = k 2 ( j 2 − 1 ) {\displaystyle 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 {\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
∫ S 2 f d Ω := 2 π k Tr ( F ) {\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
2 π k Tr ( I ) = 2 π k j = 4 π r 2 j j 2 − 1 {\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.
Notes Jens Hoppe, "Membranes and Matrix Models", lectures presented during the summer school on ‘Quantum Field Theory – from a Hamiltonian Point of View’, August 2–9, 2000, arXiv:hep-th/0206192 John Madore, An introduction to Noncommutative Differential Geometry and its Physical Applications, London Mathematical Society Lecture Note Series. 257, Cambridge University Press 2002
References J. Hoppe, Quantum Theory of a Massless Relativistic Surface and a Two dimensional Bound State Problem. PhD thesis, Massachusetts Institute of Technology, 1982.
