Other Spin Coherent States from Spin Spherical Harmonics in .NET framework

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68 Other Spin Coherent States from Spin Spherical Harmonics
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682 Orthogonality Relations
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Let us equip the SU (2) group with its invariant (Haar) measure: (d ) = sin 2 d d 1 d 2 , (653)
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in terms of the bicomplex angular parameterization Note that the volume of SU (2) with this choice of normalization is 8 2 The orthogonality relations satis ed by the j matrix elements D m1 m2 ( ) read as D m1 m2 ( ) D m
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j j m2
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( ) (d ) =
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SU (2)
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8 2 j j m 1 m 1 m 2 m 2 2j +1
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(654)
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683 Spin Spherical Harmonics
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The spin spherical harmonics, as functions on the 2-sphere S 2 , are de ned as follows: r Y j ( ) = = 2j +1 j D ( (R )) = ( 1) r 4 2j +1 j D 4
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2j +1 j D ( (R )) r 4 (655)
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(R ) , r
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where (R ) is a (nonunique) element of SU (2) that corresponds to the space rotar tion R introduced in (623) and that brings the unit vector e 3 = k to the unit vector r r with spherical coordinates ( , ): We immediately infer from the de nition (655) the following properties: r Y j ( )
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= j
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= ( 1) Y
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(656) (657)
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r Y j ( )
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2j +1 4
Now, from the quaternionic description of 3-space rotations, we have the group ~ homomorphism = (R) SU (2) R SO(3) = SU (2)/Z2 : = r ix 3 x 2 + ix 1 x 2 + ix 1 ix 3 =R = r ix 3 x 2 + ix 1 x 2 + ix 1 ix 3
(658)
In the particular case of (655) the angular coordinates , 1 , 2 of the SU (2) element (R ) are constrained by r cos 2 = cos , e i( 1 + 2 ) = ie i , sin 2 = sin , so so 2 2 = , (659) (660)
1 + 2 = +
6 The Spin Coherent States
Here we should pay special attention to the range of values for the angle , depending on whether j and consequently and m are half-integer or not If j is halfinteger, then the angle should be de ned mod (4 ) whereas if j is integer, it should be de ned mod (2 ) We still have one degree of freedom concerning the pair of angles 1 , 2 We leave open the option concerning the -dependent phase factor by putting i e i ( 1 2 ) = e i ,
(661)
where is arbitrary With this choice and considering (651) and (652) we get the expression for the spin spherical harmonics in terms of , /2, and , and of Jacobi polynomials, valid in the case in which > 1 8): r Y j ( ) = ( 1) e i ~ 2j +1 4 ( j )!( j + )! ( j )!( j + )! (662)
+ 1 ( , + ) (1 + cos ) 2 (1 cos ) 2 P j (cos ) e i 2
For other cases, it is necessary to use alternative expressions based on the relations [18] Pn
( l )
(x) =
n+ l n l
x 1 2
P n l (x) ,
(l )
( )
(x) = 1
(663)
Note that with = 0 we recover the expression for the normalized spherical harmonics (see Appendix C) Finally, introducing the single complex variable = z 2 /z 1 , one should retain the following expression, directly emerging from (651), for the generating function of the spin spherical harmonics: 2j +1 4 + sin e i 2 2
j j+
sin
i e + cos 2 2
m= j
( j + )!( j )! ( j + m)!( j m)!
( 1) e i Y
r jm ( )
(664)
In particularizing the above equation to the case = 0, one recovers the kernel (640)
8) This expression is not exactly in agreement with the de nitions of Newman and Penrose [77], Campbell [76] (note that there is a mistake in the expression given by Campbell, in which a cos should read 2 cot ), and Hu and White [78] Besides the 2 presence of different phase factors, the disagreement is certainly due to a different relation between the polar angle and the Euler angle