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17 Fuzzy Geometries: Sphere and Hyperboloid
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as a sort of Jacobi transform on the sphere of the function f: x|A f |x =
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(dx ) f (x )N (x )| x|x |2 2j +1 4 sin d d
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1+ r r 2
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P j ( ) r r
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(0,2
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f ( ) r (175)
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Also note that the map (173) can be extended to a class of distributions on the sphere, in the spirit of 12
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1725 Spin Coherent State Quantization of Spin Spherical Harmonics
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The quantization of an arbitrary spin harmonic Y kn yields an operator in H j whose (2 j + 1) ~ (2 j + 1) matrix elements are given by the following integral resulting from (173): A Y kn
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Y j (x) Y j
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(x) Y kn (x) (dx) (x) Y kn (x) (dx) (176)
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( 1) Y
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j (x) Y j
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1726 The Usual Spherical Harmonics as Classical Observables
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As asserted in Appendix C (Section C7), it is only when + = 0, that is, when = 0, that the integral (176) is given in terms of a product of two 3 j symbols Therefore, the matrix elements of AY m in the spin spherical harmonic basis are given in terms of the 3 j symbols by AY = ( 1) (2 j + 1) (2 + 1) 4 j j j j (177)
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This generalizes formula (27) of [217] This expression is a real quantity
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1727 Quantization in the Simplest Case: j = 1
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In the simplest case j = 1, we nd for the matrix elements (177) AY 10 AY 11 = = 3 1 m mn , 4 j ( j + 1) 3 1 4 j( j + 1) ( j n)( j + n + 1) mn+1 , 2 (178) (179)
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172 Quantizations of the 2-Sphere
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AY 1 1
3 1 4 j( j + 1)
( j + n)( j n + 1) mn 1 2
(1710)
From a comparison with the actions (675a 675c) of the spin angular momentum on the spin- spherical harmonics, we have the identi cation: AY 10 = AY 11 = AY 1 1 = 3 1 j , 4 j( j + 1) 3 3 1 j + , 8 j ( j + 1) 3 1 j 8 j ( j + 1) (1711) (1712) (1713)
The remarkable identi cation of the quantized versions of the components of r = (x i ) pointing to S 1 with the components of the spin angular momentum operator results: j (1714) Ax a = K a , with K = j( j + 1)
1728 Quantization of Functions
Any function f on the 2-sphere with reasonable properties (continuity, integrability, and so on) may be expanded in spherical harmonics as
=0 m=
mY m
(1715)
from which follows the corresponding expansion of A f However, the 3 j symbols are nonzero only when a triangular inequality is satis ed This implies that the expansion is truncated at a nite value, giving
Af =
m =
f m AY
(1716) m
This relation means that the (2 j + 1)2 observables (AY m ), 2 j, provide a second basis of O j The f m are the components of the matrix A f O j in this basis
1729 The Spin Angular Momentum Operators
17291 Action on Functions The Hermitian space H j carries a unitary irreducible representation of the group j SU (2) with generators a de ned in (671 673) The latter belong to O j Their
17 Fuzzy Geometries: Sphere and Hyperboloid
action is given in (675a 675c) and the above calculations have led to the crucial relations (1714) We see here the peculiarity of the ordinary spherical harmonics ( = 0) as an orthonormal system for the quantization procedure: they would lead to a trivial result for the quantized version of the Cartesian coordinates! On the other hand, the quantization based on the Gilmore Radcliffe spin coherent states, = j , yields the maximal value: K = 1/( j + 1) Hereafter we assume = 0 /
17292 Action on Operators The SU (2) action on H j induces the following canonical (in nitesimal) action on O j = End(H j )
La : La A = [ a , A]
(the commutator) ,
(1717)
here expressed through the generators We prove in Appendix D, (D4), that La AY
= A Ja Y
(1718)
from which we get L3 AY
= mAY
and (L j )2 AY
= ( + 1)AY
(1719)
We recall that the (AY m ) 2 j form a basis of O j The relations above make AY m appear as the unique (up to a constant) element of O j that is common eigenvector j to L3 and (L j )2 , with eigenvalues m and ( + 1) respectively This implies by linearity that, for all f such that A f makes sense, La A f = A J a f