Fuzzy sphere explained

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.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\epsilon} _{abc} J _{c} , where \epsilon _{abc} is the totally antisymmetric symbol with \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 ^{2= (1)/(4)} _{3} (j ^{2-1)I}

where I is the j-dimensional identity matrix.Thus, if we define the 'coordinates'

x ^{-1} _{a = kr} 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 ^{2= r} _{3} ^{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 (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,
• 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.