Английская Википедия:Fuzzy sphere
Шаблон:No footnotes 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 <math>j^2</math>-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 <math>J_a,~ a=1,2,3</math> that form a basis for the j dimensional irreducible representation of the Lie algebra su(2). They satisfy the relations <math>[J_a,J_b]=i\epsilon_{abc}J_c</math>, where <math>\epsilon_{abc}</math> is the totally antisymmetric symbol with <math>\epsilon_{123}=1</math>, and generate via the matrix product the algebra <math>M_j</math> of j dimensional matrices. The value of the su(2) Casimir operator in this representation is
- <math>J_1^2+J_2^2+J_3^2=\frac{1}{4}(j^2-1)I</math>
where I is the j-dimensional identity matrix. Thus, if we define the 'coordinates' <math>x_a=kr^{-1}J_a</math> where r is the radius of the sphere and k is a parameter, related to r and j by <math>4r^4=k^2(j^2-1)</math>, then the above equation concerning the Casimir operator can be rewritten as
- <math>x_1^2+x_2^2+x_3^2=r^2</math>,
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
- <math>\int_{S^2}fd\Omega:=2\pi k \, \text{Tr}(F)</math>
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
- <math>2\pi k \, \text{Tr}(I)=2\pi k j =4\pi r^2\frac{j}{\sqrt{j^2-1}}</math>
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, Шаблон:ArXiv
- 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.
