Girish Mahajan (Editor)

Fuzzy sphere

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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 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 antisymmetric symbol 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 = 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

S 2 f d Ω := 2 π 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 π k Tr ( I ) = 2 π k j = 4 π r 2 j j 2 1

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

References

Fuzzy sphere Wikipedia