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On the symmetry of Wigner (3-j) and Racah (6-j) coefficients
M Hage-Hassan
To cite this version:
M Hage-Hassan. On the symmetry of Wigner (3-j) and Racah (6-j) coefficients. 2020. �hal-02630475�
On the symmetry of Wigner (3-j) and Racah (6-j) coefficients
M. Hage-Hassan
Université Libanaise, Faculté des Sciences Section (1) Hadath-Beyrouth
Abstract
The Clebsh-Gordan coefficient formulas are determined using the integration of the product of three Wigner D-matrix. So we determine the generating function of these coefficients and the explicit expression of Van der Waerden formula for Wigner 3-j coefficient is derived simply. We also determine a new generating function of Wigner 6-j, by the coupling of three angular moments, and we derive without any difficulty Racah's formula for 6-j. We also propose a new method for the determination of the symmetries of the symbols 3-j and 6-j only from the expressions of these formulas. We also prove the connection of the coupling coefficients in bosons basis (Schwinger) and the analytic Fock space.
1. Introduction
The angular momentum has been studied for long time ago by many scientists and by different methods [1-11]. Many of the formulas of the Clebsh-Gordan and 6-j coefficients have been found by several authors [7-13] and by different methods but the calculation methods are generally difficult [9, 10]. In addition, the study of symmetries has been studied mainly by Racah, Wigner, Regge and others [5, 9-13].
This work is the continuation of my paper on the angular momentum for the
determination of formulas for coefficients 3-j and 6-j symbols [14-15]. Using the triple integral of Wigner D-matrix we determine the generating functions (GF) of 3-j and the Van der Waerden formula [9]. We determine by the coupling of three angular moments in the analytic Fock space, the GF of 6-j, and the Racah formula for 6-j [9].
We propose a new simple method for the study of the symmetries of these coefficients only using the Van der Waerden formula for the symbols 3-j and the Racah formula for the study of the 6-j symmetry.
We also summarize the method for the determination of other expressions of the 3-j symbols from the triple integral of Wigner D-matrix.
The connection between the boson space and the Fock space has been studied for a long time [16] for the study of “Anharmonic vibrations in nuclei “[16-18]. We will apply our method for the connection of the expression of the coupling coefficient in bosons (Schwinger) and in Fock-Bargmann space.
The plan of this paper is as follows: We begin, in sections 2 and 3, by reviewing the angular momentum and the coupling of angular momentum. In section 4, we derive the Van der Waerden formula for 3-j symbols. In sections 5, we study the symmetries of Clebsh-Gordan. In section 6 we show the derivation of some other formulas for the three symbols. In section 7 and 8 we derive the Racah formula for the 6-j and we study the
symmetries of these symbols. In the appendices we deal with the connection between the boson spaces and the analytic Fock space.
2. The representation of SU(2) and the generating function of Wigner's D-matrix
We starting from the representation of SU (2) in the analytic Fock's space [7, 14-15]
For the determination of Wigner's D-matrix and its generating function.
2.1- The Analytic Hilbert space
The Analytic Oscillator space (or Fock space) Is defined by:
! ) (
n z z u
n
n (2.1) With scalar product is:
( ) ,!
! ' ,
( n' n n' n d z n,n' n
z n u z
u
(2.2) And d
z is the measure of integration:d(z)
1 exp
zz
dxdyi, zxi iyi (2.3) Furthermore, it is easy to check the useful relationship:
z z d z e
exp[ ] ( ) (2.4) It is important to note that the functions
un(z)ezz/2
are the wave functions ofsubspaces of the cylindrical basis of the Harmonics oscillators [19].
2.2 The representation of SU(2) in the analytic Hilbert space
We define the generators of SU(2) in the analytic Hilbert space, ui,i1,2, by:
)
1 ( 1
2 , 1 1 1
, 1 1 1
2 2 1 1 3
1 2 2 1 2
1 2 2 1
1 u u
u u u J
u u i u u J
u u u
J
(2.5)
And
2 2 1 1
2 1 1
2 , 1 ) 1
( u u
u u N
N N J
(2.6) We find that:
J2jm(u) j(j1)jm(u), J3jm(u)mjm(u)
(2.7)
With jm(u) is the representation of SU(2) in the analytic Hilbert space with:
)!
m j ( )!
m j (
u jm u
u ) u (
) m j ( 2 ) m j ( 1
j m
(2.8)
It is important to note that the space of the bosons was introduced by Schwinger for the study of the angular moment with:
00 )!
m j ( )!
m j (
a jm a
) m j ( 2 ) m j ( 1
(2.9)
2.3- The Wigner’s D-matrix
The Wigner’s D-matrix [1-10] of the rotation R()eiJzeiJyeiJz is given in Dirac notations by:
D(jm,'m}()
jmR()jm
eim'd(jm,'m)()eim (2.10) With (,,) are the angles of Euler.The four dimensions analytic Hilbert space is:
D(jm,'m}(z)2jD(jm,'m}(), () (2.11) With:
)!
m j ( )!
m j (
) u z u z ( ) u z u z ) ( u ( ) ( R
m j 2 1 1 2 m j 2 2 1 1 j m
j 2
(2.12)
And 1 1 1 ( )/2 2 2 2 ( )/2
sin2 2 ,
cos
x iy ei z x iy ei
z (2.13)
From formula (2.12) we deduce the expression:
2 s
s
s
m ' m j
2 m ' m j
) m ,' m (
)) 2 / ( tg )!( s m ' m ( )!
s m j ( )!
s ' m j (
! s
) 1 (
) 2 / sin(
( )
2 / (cos(
)!
m j ( )!
m j ( )!
' m j ( )!
' m j ( ) ( d
(2.14) We can also write this expression in terms of hypergeometric functions by:
:
(j m'), (j m);m' m 1; tan /2
,.F
)!
m ' m (
) 2 / (cos )
2 / (cos )!
' m j ( )!
m j (
)!
' m j ( )!
m j ) ( ( d
2 1
2
m ' m '
m m j 2 2
1 j
m ,' m
(2.15) We emphasize that we have identified using Wigner d-matrix symmetries twenty-four formulas [22] which are solutions of the Gauss equation.
2.4- The generating function of Wigner's D-matrix
Multiplying (2.12) by jm'(v)and summing with respect to j,m,m'we find the generating function of Wigner's D-matrix
(v)D (z) (u)u u z z
z v z
v exp ) u , z , (
G j m
' m , m , j
j } m ,' m ( ' j m 2
1 1 2
2 1 2
1
(2.16)
3. The coupling of angular momentum and the Clebsch-Gordan coefficient
We will determine the coupling of two angular moments and the coefficients of the Clebsh-Gordan and 3-j symbols. And we write the triple integral formula of the Wigner D-matrix, in terms of the symbols 3-j, and which is very useful for the calculation of these coefficients.
3.1 The coupling of two angular momentums and the coefficient de clebsh-Gordan The eigenfunctions of total angular momentum J J1 J2
is (jj )jm (u1,u2)
3 3 2
1 , with:
(u ,u ) jm j m (j j )j m jm (u1) jm (u2).
m m
3 3 2 1 2 2 1 1 2
1 m j ) j j
( 1 1 2 2
2 1 3
3 2
1
(3.1)With m3 m1m2, j3
m3 , and j3
j1j2
.j1m1j2m2 (j1j2)j3m3 or j1m1j2m2 (j1j2)j3m3 (3.2) The coefficient (3.2) is the clebsh-Gordan and jm (u1) jm (u2)
2 2 1
1
is the two uncoupled
representation of SU(2).
3.2 The Wigner 3-j symbols
The study of symmetries of these coefficients is facilitated by the use of the Wigner 3-j symbols [9].
j m j m (j j )j ( m )
1 j 2
) 1 ( m m m
j j j
3 3 2 1 2 2 1 1 3
m j j
3 2 1
3 2
1 1 2 3
(3.3) 3.3 The triple integral formula for Wigner D-matrix
We deduce from the orthogonality of matrices D (Ω) the relations which are very important for the calculation of the symbols of 3-j.
D ( )D ( )D ( )d ,d d sin d d8
1 2
0 0
2 0
j ) m , ' m ( j
) m , ' m ( j
) m , ' m 2 (
3 3 3 2
2 2 1
1 1
3 2 1
3 2 1 3 2 1
3 2 1
m m m
j j j ' m ' m ' m
j j
j (3.4)
4- The generating function of Wigner 3-j symbol and the Van der Waerden's formula.
There are several formulas for the Clebsh-Gordan [8-12] but we will determine the Van der Waerden formula by developing the generating function of these coefficients.
4.1 The generating function of Wigner 3-j symbol Using (2.16) and (3.4) we find:
3i1 1 2 ii,z,v )]d (z )d (z ) u
( G
[
exp
v2,v3
u2,u3
v3,v1
u3,u1
[v1,v2][u1,u2]
(4.1) or
exp
det(t,v)
exp
det(t,u)
d(t)
G3j(t,v)G3j(t,u)d(t) (4.2) With
3 2 2 2 1 2
3 1 2 1 1 1
3 2 1
u u u
u u u
t t t ) u , t det(
exp (4.3)
C- The function G3j(t,u)is the generating function of the 3-j symbols [6, 7].
G (t,u) exp
det(t,u)
exp
t
u ,u
t u ,u
t3[u1,u2]
1 3 2 3 2 1 j
3
3 2 1
3 2 1 3 2 j m 1
3 m j 2 m j 1 m
j m m m
j j ) j
t t t ( ) u ( ) u ( ) u
( 2 2 3 3
1
1 (4.4)
And
1 2 3
1/2j 2 J 3 j 2 J 2 j 2 J 2 1
/ 1 3
2
1 (J 2j )!(J 2j )!(J 2j )!
t t )! t
1 J ( ) t t t (
3 2 1
(4.5)
With J=j1+j2+j3,
1j i 2 j 2 i 1 ji,u u u u u
u and
2ji 1 j 2 i 1 j
i.u u u u u
u (4.6) The invariance of the determinant by permutation of the columns and the rows of (4.3) allowed Regge [4] to deduce the symmetries of the symbols 3-j, as we write below.
4.2 The first derivation of the expression of Clebsh-Gordan coefficients
The development of the first member of the expression (4.4) and its identification with the second member gives us:
u ,u
u ,u
[u ,u ] (J1)!(j1,j2,j3) j2 J 2 j 1
2 1 J j 3 2 3 J
2 1 2 3
j m
3 2 1
3 2 3 1
m j 2 m j 1 m
j m m m
j j ) j
u ( ) u ( ) u
( 2 2 3 3
1
1 (4.7)
jm 3
q j 2 J 2 1 1 2 q 2 2 1 1 2
p j 2 J 1 1 3 2 p 1 2 3 1 1
t j 2 J 2 2 3 1 t 3 2 2 1
)!
q j 2 J (
! q
) u u ( ) u u ( )!
p j 2 J (
! p
) u u ( ) u u ( )!
t j 2 J (
! t
) u u ( ) u u
( 2 3
Identification with the first term gives:
p j1j2 tm3,q j1j3 tm2.
(4.8)
Therefore:
3 2 1
3 2 1 3 2
1 m m m
j j ) j
j , j , j ( )!
1 J (
(4.9)
t 1 2 3 1 3 2 3 3 2 2 3 2 1
q p J
)!
t j j j ( )!
t m j ( )!
t m j ( )!
t m j j ( )!
t m j j (
! t ) 1 1 (
2 1
! 1 c b a
)!
c b a ( )!
c b a ( )!
c b a
abc (
(4.10)
On putting s j2 m2 tand m3 par –m3 we derive the well known:
Van der Waerden’s formula of the Clebsh-Gordan:
21
3 2 1
3 2 1 3
3 2 1 3
3 2 1 2 2 1
1 (j j j 1)!
) j , j , j ( ) 1 j 2 m ( , m m m
j ) j j ( m j m
j
(j1m1)!(j1m1)!(j2 m2)!(j2 m2)!(j3 m3)!(j3 m3)!
(4.11)
s 1 2 1 1 2 2 3 2 1 3 1 2
s
)!
s m j j ( )!
s m j j ( )!
s m j ( )!
s m j ( )!
s j j j (
! s ) 1 1 (
It is quite clear that our method of deriving the Van der Waerden's formula is very simple by comparison with the complexity of the other methods [9].
5- The symmetries of the 3-j symbols
We can determine the symmetries of the Clebsh-Gordan coefficients or the symbols 3-j from all the formulas of these coefficients [4-9] but we will limit ourselves to Van der Waerden's formula (4.11). So we look transformations that leave invariant the product of the summation formula (4.11).
For this we put: AXB
(5.1)
0 m j j
m j j
m j
m j
j j j
u u u c b a
1 1 1 0 0 0
0 1 0 1 0 1
0 0 1 1 1 0
0 1 0 0 1 0
0 0 1 0 0 1
0 0 0 1 1 1
2 1 3
1 2 3
2 2
1 1
3 2 1
3 2 1
(5.2)
With
3 2 1
3 2 1 3
2
1 m m m
j j j u
u u
c b a
(5.3) 5.1 The symmetries of the three symbols
a- It is simple to verify from (4.9) and (4.11) that:
3 2
1
3 2
1 j j j 3
2 1
3 2 1
m m
m
j j
) j 1 m (
m m
j j
j 1 2 3 (5.4)
b- Symmetries by permutation
The symmetries by permutation of the components of X = (j1, j2, j3, m1, m2, m3) give:
3 1 2
3 1 j 2
j j 3
2 1
3 2 1
m m m
j j ) j
1 m (
m m
j j
j 1 2 3
1 1 3
1 1 3 j j j 2
3 1
2 3 1 j j j
m m m
j j ) j
1 m (
m m
j j ) j
1
( 1 2 3 1 2 3 (5.5)
5.2 Regge symmetries
We also find (a, b, c, u1, u2, u3) from the permutations of the components of the column B which leave the coefficients 3-j invariant.
For exemple:
a- tB
j1m1
j1 j2 j3
j2 m2
j3 j2 m1
j3 j1m2
0
(5.6) We derive the formula.
3 1 3 2 3 1 3 2 2 3
1 3 2 1
3 2 1
3 2 1
3 2 1
2 m m j m j
2 m j j j
j
2 m j j 2
m j j j
m m m
j j j
(5.7) b- tB
j1m1
j2m2
j1j2j3
j3 j2m1
j3j1m2
0
(5.8) We find also:
2 m j j j
2 m j j j
2 m j j j
2 m j j 2
m j j 2
m j j )
1 m (
m m
j j j
3 2 1 3 2 3 1 2 1 3 2 1
3 2 1 2
3 1 1
3 2 j j j 3
2 1
3 2
1 1 2 3
(5.9)
Note that Regge has studied the symmetries of the 3-j symbols starting from the invariance of the determinant [4].
6- Triple integral of the Wigner D-Matrix and The expressions of the Clebsh-Gordan coefficients
In this part, we present the calculation of the Clebsh-Gordan coefficients by a direct calculation using the triple integral formula of the Wigner D-Matrix and its symmetries.
By posing in expression (3.4) m'1 j1, m'2 j2,m'1m'2m'30 we find:
d ()d ( )d ( )d 81 2
0 0
2 0
3 j
) m , 2 j 1 j ( 2
j ) m , 2 j ( 1
j ) m , 1 j
2 ( 3 3
2 2 1
3 2 1
3 2 1 3 1 2 1
3 2
1
m m m
j j j j j j j
j j
j (6.1) From expression (2.15) we find that:
(jj,m) (cos( /2))j m(sin( /2))j m )!
m j ( )!
m j (
)!
j 2 ) (
(
D1 1
(6.2)
And (j j,m) (cos( /2))j m(sin( /2))j m )!
m j ( )!
m j (
)!
j 2 m (
j ) 1 ( ) ( D1
1
(6.3) By setting m1 = j1, m2 = -j2 and m3 = -m1-m2 and using (4.8) and (6.2), (6.3) we find that:
(j j j )!(j j j 1)!
)!
j 2 ( )!
j 2 ( j
j j j
j j
j
3 2 1 3 2 1
2 1 2
1 2 1
3 2
1
(6.4) As we have already pointed out that the d-matrix can be expressed using several formulas [22]. Then we get several expressions of the symbols 3-j.
In addition the calculation of (6.1) is carried out using a Beta functions [21-22]:
2
0
1 y 2 1 x
2 sin d
sin ) 2
y x (
) y ( ) x ) ( y , x ( B
(6.5) Note that the symmetries of the 3-j symbols can also be deduced using these formulas.
7- The Racah's formula for 6-j symbols
We summarize the coupling of three angular moments [6, 14, 15], and the derivation of the Racah coefficients and its compact form in terms of 6-j.
7.1 The coupling of three angular momentums and Racah coefficients
The coupling of three angular moments is carried out by two different ways and we arrive at two different states. And the Racah coefficients are the coefficient of passage from one representation to another.
The coupling of three angular moments can be done in two ways:
J
J1 J2
J3 J1
J2 J3
, WithJ12 J1 J2, AndJ23 J2 J3
(7.1)
1- The base of the first representation is: (j1j2)j12j3;j 2- The basis of the second representation is: j1(j2j3)j23;j The Racah coefficient is defined by:
(j1j2)j12j3;j j1(j2j3)j23;j (2j121)(2j231)W(j1j2jj3;j12j23)
(7.2) 7.2 The generating function of the non-coupled representation
The generating function of the first representation is deduced from the non-coupled representation
exp
t1u1 t2u2 t3u3
(7.3) By application on
3 1 2 2 x.u1 1 x.t2 , t
exp t (7.4) And
3 3 2 y.x 1 y.t3 , t
exp x (7.5) So we get the generating function of the base (jajb)jabjc;j :
u
(2j1) j j j
j j j()
z (j j )j j ;j
u j
3 12 2 1 jm
3 12 12
2 1 2 / 1 1
31
2 3
3 1 2 3 2 1
3 u ,u u ,u u ,u
exp (7.6)
22
y.u1 21
y.u2 1 y.u3
exp
Q1(u)
Likewise the generating function of the representation ja(jbjc)jbc;j is:
j
23 3 2 1 jm 23
1 23
3 2 2 / 1
2 (2j 1) j j j ' j j j( ') z j ,(j j )j ;j
3 1
3 1
1 2 2 3 3 2
3 u ,u ' ' u ,u ' ' u ,u
'
exp
' '
y.u ' '
y.u '
y.u
exp
Q2(u)
1 1 3 1 2 2 2
2
(7.7)
7.3 The generating function of the coupled representation The generating function of the 6-j symbols is:
3
2 , 1 j , 1 i
i j 1
2 1
2 j
6 d (y) du
G (7.8)
we make the change of the variables u2iby u2i j1..3and we denote the column (u) by )
u u u u u u
( 11 12 13 12 22 32 then we perform the integration with respect to y,
We find: G 1 3 du exp( (u)tM
u )2 , 1 j , 1 i
i j n
j
6
The calculation of this expression can be done using the Gauss formula:
n t 1
1
i i i
n
)) M (det(
) Mz z ( exp dy
1 dx
(7.9)We find:
3 t
12 , 1 j , 1 i
i j n
j
6 1 du exp( (u) M u ) (det(M))
G
With:
vt 1 ut xt
0
vy uy
1 xy 0
vp up
xp 1 0
0 vt
1 vy vp
0 ut
uy 1 up
0 xt xy
xp 1
M
23 13
23 12
13 12
23 13
23 12
13 12
(7.10)
And
1 2 2
2 1
1 1
2 2
2 3
23 1 3 13 2 3 12
1 3 23 2 3 13 3
12
' ' v ' ' u ' x
t y
, p ' '
' '
' (7.11)
7.4 The generating function of the 6-j symbols
The determinant of the coupling matrix of angular momentum (M) is:
det(M)
1xpuyvt121213132323
p23t12 y13
x23v12u13
2 (7.12) Using (7, 11) we find the generator function of the symbols 6-j:j1j2j12
j12j3j( ) j2j3j23
' j1j23j( ')W(j1j2jj3;j12j23)j j j j j j1231223
(7.13)
1'122'2'221'2'113'3'232'3'131'3'1'331
2
It is important to note that we can also make the coupling in terms of bosons, Schwinger [6] (appendiceI)
7.5 The formula of 6-j
We can develop G6j in the form:
( )!(m)!(n)!(p)!
) ' ' ( ) ' ' ( ) '
' ( ) '
( )!
1 z G (
p 2 3 3 n 1 1 2 m 1 2 2 2 2 2 1 j
6
(q)!(r)!(s)!
) '
' ( ) ' ( ) ' '
(3231 q 313 r 1331 s
(7.14) With z mnpqrsmore
1- The powers of iare : P(1)mr, P(2)q and P(3)ps
We deduce that: 1 2 12
3
1 i
i) z n j j j
(
P
then nz
j1j2 j12
2- The powers of iare: P(1)ns,P(1)lmetP(1)qr P(3)ps We deduce that: P( ) z p j1 j12 j3
1 i
i
then pz
j1 j12j
3- The powers of iare. P('1)nq, P('2)mp and P('3)rs
We deduce that: 1 2 12
3
1 i
i) z l j j j
' (
P
then lz
j2 j3 j23
4 The powers of iare: P('1)s,P('1)mnand P('3)pq We find that: P( ' ) z r z
j1 j23 j
3
1 i
1
donc rz
j1j23j
And q j1 j23 j12 j3 z m j12 j j2 j23z s j1 j2 j3 jz Finally we find the Racah formula for the coefficients 6-j that we will write in
the usual notations [4,9].
) ef
; abcd ( W )
1 f (
e d
c b a j
j j
j j
j a b c d
23 3
12 2
1
(7.15) 7.6 Racah's formula
Racah's formula is:
z z
! f e a z
! c b a z
)!
1 z ) (
1 ( dec dbc aef
f abc e d
c b
a (7.16)
zbdf
! zdec
abdez
! bcef z
! aedf z
!1
With
abc is given in (4.10).
8- The symmetries of 6-j coefficients
We are looking for transformations which leave the summation invariant (7.16) 8.1 The symmetries of 6-j coefficients
We put again: AXB
