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173 Link with the Madore Fuzzy Sphere 1731 The Construction of the Fuzzy Sphere la Madore
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Let us rst recall the Madore construction of the fuzzy sphere as it was originally presented in [178] (p 148), which we slightly modify to make the correspondence with the coherent state quantization It starts from the expansion of any smooth function f C (S 2 ) in terms of spherical harmonics,
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Let us denote by V the (2 + 1)-dimensional vector space generated by the Y xed
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Through the embedding of S 2 into R3 any function in S 2 can be considered as the restriction of a function on R3 (which we write with the same notation), and, under some mild conditions, such functions are generated by the homogeneous polynomials in R3 This allows us to express (1721) in a polynomial form in R3 : f (x) = f (0) +
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where each subsum is restricted to a V and involves all symmetric combinations of the i k indices, each one varying from 1 to 3 This gives, for each xed value of , 2 + 1 coef cients f (i 1 i 2 i ) ( xed), which are those of a symmetric traceless 3 ~ 3 ~ ~ 3 ( times) tensor The fuzzy sphere with 2 j +1 cells is usually written as S fuzzy , j , with j an integer or semi-integer We here extend the Madore procedure and this leads to a -indexed family of fuzzy spheres S fuzzy , j Let us list the steps of this construction: 1 We consider a (2 j + 1)-dimensional irreducible unitary representation of SU (2) The standard construction considers the vector space V j of dimension 2 j + 1, on which the three generators of SU (2) are expressed as the usual (2 j + 1) ~ (2 j + 1) Hermitian matrices J a We make instead a different j choice, namely, the three a , which correspond to the choice of the rep j resentation space H (instead of V j in the usual construction) Since they obey the commutation relations of su(2), [ a , b ] = i abc c ,
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the usual procedure may be applied As we have seen, H j can be realized as the Hilbert space spanned by the orthonormal basis of spin spherical harmonics { Y j } = j j , with the usual inner product Since the standard derivation of all properties of the fuzzy sphere rests only upon the abstract commutation rules (1723), nothing but the representation space changes if we adopt the representation space H instead of V j 2 The operators a belong to O j and have a Lie algebra structure through the skew products de ned by the commutators But the symmetrized products of operators provide a second algebra structure, which we write as O j , at the basis of the construction of the fuzzy sphere: these symmetrized prod j ucts of the a , up to power 2 j, generate the algebra O j (of dimension 2 (2 j + 1) ) of all linear endomorphisms of H j , exactly like the ordinary J a s do in the original Madore construction This is the analog of the standard j construction of the fuzzy sphere, with the J a and V j replaced by a and j H 3 The construction of the fuzzy sphere of radius R is de ned by associating an operator in O j with any function f Explicitly, this is done by rst f replacing each coordinate x i by the operator
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