Representations of quantum superalgebra Uq[gl(2|1)] in a coherent state basis and generalization

0.14 MB·English

by  Nguyen Cong Kien

Representations of quantum superalgebra
U [gl(2|1)] in a coherent state basis and
q
generalization
Nguyen Cong Kien1,2, Nguyen Anh Ky1,2, Le Ba Nam3,∗ and
2
1 Nguyen Thi Hong Van1
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2 1Institute of Physics, Hanoi, Vietnam

n
a 2College of Science, Vietnam National University, Hanoi, Vietnam
J
3Hanoi University of Technology, Hanoi, Vietnam
0
1
January 11, 2012

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p
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t
a Abstract
m
[ The coherent state (CS) method has proved to be useful in quantum physics

and mathematics. This method, more precisely, the vector coherent state (VCS)
2
v method, has been used by some authors to construct representations of superal-
6 gebras but almost, to our knowledge, it has not yet been extended to quantum
0
superalgebras, except U [osp(1|2)], one of the smallest quantum superalgebras.
2 q
1 In this article the method is applied to a bigger quantum superalgebra, namely
1. Uq[gl(2|1)], in constructing q–boson-fermion realizations and finite-dimensional
0 representations which, when irreducible, are classified into typical and nontypi-
2
cal representations. This construction leads to a more general class of q–boson-
1
: fermion realizations and finite-dimensional representations of Uq[gl(2|1)] and,
v
thus, at q = 1, of gl(2|1). Both gl(2|1) and U [gl(2|1)] have found different
i q
X
physics applications, therefore, it is meaningful to construct their representa-
r
tions.
a
Running title: Representations of U [gl(2|1)] in a coherent state basis
q
1 Introduction
Symmetry groups (and their infinitesimal version – algebras), including also super-
groups and quantum (super) groups (QG’s), and their representations play an ex-
tremely important role in the development of modern physics and mathematics. In
many cases, mathematical needs and physics applications require explicit construc-
tions of representations of groups or algebras. Depending on motivations and problems
∗
Present address: Department of Physics, University of South Florida, 4202 E. Fowler Avenue,
Tampa, FL 33620,USA.
1
considered, different methods of building representations of a group or an algebra can
be chosen, and the boson-fermion realization method is one of them. The first type
of boson-fermion realizations, or the so-called Jordan-Schwinger map (a boson realiza-
tion), was suggested byP. Jordan[1] andJ. Schwinger [2]. Other types were introduced
by F. Dyson [3] or by T. Holstein and H. Primakoff [4]. The Dyson’s method was later
generalized for Lie superalgebras [5, 6] as well as for quantum algebras [7] – [10]. The
Jordan-Shwinger and Holstein-Primakoff realizations of quantum algebras were also
made in [11, 12] and [13, 14], respectively. To build a group or algebra representation
via a boson-fermion realization, it is convenient to use the coherent state method which
has proved to be very efficient in investigating dynamic symmetries (mathematically
described by groups/algebras) of quantum systems [15].
The concept of coherent states (CS’s) was first introduced by E. Schro¨dinger over
80 years ago [16], and much later, in 1972, it was generalized for an arbitrary Lie group
by A. Perelomov (see [17] or more [18, 19] and their references). Further, this concept
was extended to that of the so-called vector coherent states (VCS’s) [20] – [25] and
also to that of super-coherent states (SCS’s) [26] – [32]. Hereafter the terminology
CS’s also contains VCS’s and SCS’s. The CS’s play an essential role in physics and
mathematics such as quantum optics [33], many-body physics [19], quantum gravity
and cosmology [34, 35], string theories and integrability [36, 37], path integrals [38] –
[41], quantum information (see [42] and references therein), representation theory of
Lie (super) groups and (super) algebras [17, 18, 30, 43], non-commutative geometry
[44], etc. A natural relation between CS’s and QG’s (including quantum supergroups)
was also established [11, 12, 13, 45, 46, 47]. As is well known [20, 21, 24, 30, 39, 43],
the CS theory is tightly related to the theory of induced representations of (super)
groups and (super) algebras and also of those of QG’s. However, sometimes the CS
method for construction of irreducible representations is simpler than the ordinary in-
duced representation methods (Chevalley-Harish-Chandra and Kac methods) and has
more direct physics applications. Therefore, the CS representation method sometimes
is preferable to the ordinary induced representation methods.
UsingtheCSmethod(foramoregeneralconstruction, see, forexample, [24,30,31])
Y. Zhang et al [48, 49] constructed representations of some superalgebras such as
osp(1|2), osp(2|2) and gl(2|2). However, to our knowledge, this method almost has
not yet been used for constructing representations of quantum superalgebras, except
U [osp(1|2)],oneofthesmallest quantumsuperalgebras[49]. Inthispaper(Sect. 2),we
q
deal with U [gl(2|1)], a bigger quantum super algebra. As a result, representations (q–
q
boson-fermion realizations and the corresponding finite-dimensional representations)
of U [gl(2|1)] in a CS basis and their generalization are obtained. Putting q = 1 we
q
get generalized representations of gl(2|1). Both gl(2|1) and its quantum deformation,
U [gl(2|1)], are important for physics applications (see, for example, [50] – [58] and
q
references therein), therefore, to construct their representations is meaningful. Our
conclusion is made in the final section.
2
2 Representations of U [gl(2|1)] in a CS basis
q
In this section, we will construct representations (q–boson-fermion realizations and
finite-dimensional representations) of U [gl(2|1)] in a coherent state basis (for other
q
representations of U [gl(2|1)] see, for example, [59] and references therein). Firstly, we
q
recall the definition of the quantum super algebra U [gl(2|1)].
q
2.1 Quantum superalgebra U [gl(2|1)]
q
The quantum super algebra U [gl(2|1)] as a deformation of the universal enveloping al-
q
gebra U[gl(2|1)] of the superalgebra gl(2|1) can be defined through its Weyl-Chevalley
generators E , E , E , E and E (i = 1,2,3) satisfying the following defining re-
12 21 23 32 ii
lations (see, for example, [59]):
a) the (anti-)commutation relations (1 ≤ i,j,i+1,j +1 ≤ 3):
[E ,E ] = 0, (1a)
ii jj
[E ,E ] = (δ −δ )E , (1b)
ii j,j+1 ij i,j+1 j,j+1
[E ,E ] = (δ −δ )E , (1c)
ii j+1,j i,j+1 i,j j+1,j
[E ,E ] = [H ] , (1d)
12 21 1 q
{E ,E } = [H ] , (1e)
23 32 2 q
d
i+1
H = (E − E ), H ≡ E , (1f)
i ii i+1,i+1 3 33
d
i
where d = d = −d = 1, while
1 2 3
qX −q−X
[X] := , (2)
q q −q−1
is aquantum deformation(q-deformation, for short) of anoperatoror a number X, and
b) the Serre relations:
E2 = E2 = 0, (3a)
23 32
[E ,E ] = [E ,E ] = 0, (3b)
12 13 q 21 31 q
with E and E ,
13 31
E = [E ,E ] , E = −[E ,E ] . (4)
13 12 23 q−1 31 21 32 q−1
defined as new (non-Chevalley) generators, where [A,B] := AB −rBA is a deformed
r
commutator. The generators E , i,j = 1,2,3, are q-deformations (q-analogs) of the
ij
Weyl generators e of gl(2|1) subject to the supercommutation relations
ij
[e ,e } = δ e −(−1)([i]+[j])([k]+[l])δ e , (5)
ij kl jk il il kj
with [e ,e } ≡ e e −(−1)([i]+[j])([k]+[l])e e being a super-commutator (commutator
ij kl ij kl kl ij
or anti-commutator) and [i] being a parity defined by [1] = [2] = 0, [3] = 1.
3
The subspace of U [gl(2|1)] developed on the Cartan generators H and the even
q i
Chevalley generators E and E is its even subalgebra which denoted by U [gl(2|1) ]
12 21 q ¯0
is a quantum deformation of U[gl(2|1) ]. We will work in this paper with generic q, i.e.,
¯0
when it is not a root of unity. The case of q being a root of unity will be investigated
elsewhere.
2.2 q–boson-fermion realization of U [gl(2|1)]
q
In this sub-section, we deal with q–boson-fermion realizations of U [gl(2|1)].
q
Let |Ji be a state of U [gl(2|1)] defined by
q
H |Ji = 2J |Ji, H |Ji = 2J |Ji, H |Ji = 2J |Ji,
1 1 2 2 3 3
E |Ji = E |Ji = E |Ji = 0. (6)
12 13 23
It is called the the highest weight state characterized by a set of three numbers J , J
1 2
and J called the highest weight:
3
|Ji ≡ |J ,J ,J i. (7)
1 2 3
We can construct a representation of U [gl(2|1)] induced from |Ji. When J is an
q 1
integer or a half-integer, but J and J can be arbitrary complex numbers, a U [gl(2|1)]
2 3 q
module, denoted by W, induced from |Ji has a dimension of 8J +4 and contains four
1
sub-spaces, say V , i = 1,2,3,4, which are finite-dimensional modules of the even
i
subalgebra U [gl(2|1) ] and respectively spanned on the following basis vectors:
q ¯0
|J ,J ,J ,Mi = C(1).EJ1−M|Ji ∈ V ,
1 2 3 M 21 1
|J −1/2,J +1/2,J +1/2,Pi = C(2). qJ1+P+1/2E EJ1−P+1/2|Ji
1 2 3 P 32 21
n
+[J +P +1/2] E EJ1−P−1/2|Ji ∈ V ,
1 q 31 21 2
1 o
|J +1/2,J ,J +1/2,Ri = C(3). E EJ1−R+1/2|Ji
1 2 3 R qJ1−R+1/2 32 21
(cid:26)
−[J −R+1/2] E EJ1−R−1/2|Ji ∈ V ,
1 q 31 21 3
|J ,J +1/2,J +1,Si = C(4).E E EJ1−S|Ji ∈ V , o (8)
1 2 3 S 32 31 21 4
(i)
with C being normalization coefficients which can be determined by addtional, for
K
example, Hermitian conditions, and −J ≤ M,S ≤ J , −(J −1/2) ≤ P ≤ (J −1/2),
1 1 1 1
−(J +1/2) ≤ R ≤ (J +1/2) such that (J −M), (J −P −1/2), (J −R+1/2) and
1 1 1 1 1
(J − S) are integers. These four finite-dimensional irreducible modules of the even
1
subalgebra U [gl(2|1) ] are built on the U [gl(2|1) ]-highest weight states
q ¯0 q ¯0
|J ,J ,J ;J i ≡ |J ,J ,J i = |Ji,
1 2 3 1 1 2 3
|J −1/2,J +1/2,J +1/2;J −1/2i ≡ |J −1/2,J +1/2,J +1/2i,
1 2 3 1 1 2 3
|J +1/2,J ,J +1/2;J +1/2i ≡ |J +1/2,J ,J +1/2i,
1 2 3 1 1 2 3
|J ,J +1/2,J +1;J i ≡ |J ,J +1/2,J +1i, (9)
1 2 3 1 1 2 3
4
via the formula
|L ,L ,L ;ki = C(i).(E )L1−k|L ,L ,L ;L i, i = 1,2,3,4, (10)
1 2 3 k 21 1 2 3 1
(i)
where C are normalization coefficients in (8), L = J ,J ±1/2; L = J ,J +1/2;
k 1 1 1 2 2 2
L = J ,J ±1/2,J +1and−L ≤ k ≤ L such that L −k areintegers. Forsimplicity,
3 3 3 3 1 1 1
(i)
we can choose C = 1 and work with vectors (8) non-normalized. The odd generators
k
of U [gl(2|1) intertwine these U [gl(2|1) ]-modules.
q q ¯0
Now, generalized coherent states of U [gl(2|1)] can be defined as (cf. [18, 19, 48],
q
[26]–[31])
ea12E21+α23E32+α13E31 |Ji, (11)
q
∞
where eX = Xn denotes the so-called q-exponent with [n] ! = [1] .[2] ...[n] and
q [n]q! q q q q
n=0
[n] given in P(2), while α† (α ) are q-analogs of the fermion creating (annihilating)
q ij ij
operators with the number operator N and a† (a ) is a q-analog of the boson cre-
αij 12 12
ating (annihilating) operator with the number operator N . They form the quantum
a12
Heisenberg superalgebra U [h(2|1)]:
q
† † † †
{α ,α } = 1, N = α α , [N ,α ] = α , [N ,α ] = −α
ij ij αij ij ij αij ij ij αij ij ij
a12,a†12 = q−Na12, [Na12]q = a†12a12, [Na12 +1]q = a12a†12,
q
h i
N ,a† = a† , [N ,a ] = −a .
a12 12 12 a12 12 12
h i
This quantum Heisenberg superalgebra U [h(2|1)] super-commutes with U [gl(2|1)],
q q
that is, if E is an operator of U [gl(2|1)] and X is an operator of U [h(2|1)] they
q q
super-commute with each other:
EX = (−1)deg(E).deg(X)XE, (12)
where deg(X) is the parity of X.
Let |ψi be a state vector in a (e.g., finite-dimensional) module of U [gl(2|1)]. Then
q
the mapping (cf. [48] and note the difference between the notations here and there)
|ψi → |ψi = hJ|ea†12E12+α†23E23+α†13E13|ψi|0i, (13)
J q
induces the mapping
A → Γ(A)|ψi = hJ|ea†12E12+α†23E23+α†13E13A|ψi|0i, (14)
J q
of an operator A defined in a space, which here is a U [gl(2|1)] module, containing |ψi,
q
where |0i is a vacuum state of the quantum Heisenberg superalgebra U [h(2|1):
q
a |0i = α |0i = 0. (15)
12 ij
Theoretically, a construction of a finite-dimensional representation of U [gl(2|1)]can be
q
performed step by step as in the case of non-deformed superalgebras. However, there
5
are underwater rocks preventing a smooth performance of such a construction. One of
the obstacles is to expand the expression ea†12E12+α†23E23+α†13E13 into a product of expo-
q
nents. This problem is not simple as that for a non-deformed exponent. Here, instead
of Baker-Campbell-Hausdorff formula a q-deformation of the generalized Zassenhaus
formula [60] can be used but we prefer a direct computation. As a result we obtain
eqα†13E13+α†23E23+a†12E12 = D4(Na12)α1†3E13α2†3E23 +D3(Na12)α1†3E13
n+eD2(Na12)a†12α†23E13eD1(Na12)α†23E23 ea†12E12, (16)
q
o
where,
q +1
D (N ) = ,
1 a12 qNa12+1 +1
([N ] −[N +1] +1)
D (N ) = a12 q a12 q ,
2 a12 (1−q)(1−q−1)[N +1] [N ]
a12 q a12 q
1−qNa12+1
D (N ) = ,
3 a12 [N +1] (1−q)
(cid:0)a12 q (cid:1)
D (N ) = (Na12 +2)(1−q)(1−q−1)+ 1−qNa12+2 q−1 −q−Na12−1 . (17)
4 a12 (1−q−1)(1−q)[N +2] [N +1]
(cid:0)a12 q a(cid:1)12(cid:0) q (cid:1)
Using (1), (3), (4), (14) and (16) we find an explicit boson-fermion realization of the
generators:
Γ(H ) = 2J −2N +N −N , (18a)
1 1 a12 a23 α13
Γ(H ) = 2J +N +N , (18b)
2 2 a12 α13
Γ(H ) = 2J +N +N , (18c)
3 3 α13 α23
D (N −1)
Γ(E ) = a α α† +a 1 a12 N α α†
12 12 23 23 12 D (N ) α23 13 13
(cid:18) 1 a12 (cid:19)
D (N −1) D (N )
+ a 4 a12 N N + 2 a12 [N ] α† α
12 D (N ) α13 α23 D (N ) a12 q 23 13
4 a12 3 a12
D (N ) D (N )+1
− 1 a12 2 a12 [N +1] α† α
D (N )+1 D (N ) a12 q 23 13
1 a12 3 a12
D (N −1)
+ a 3 a12 N α α† , (18d)
12 D (N ) α13 23 23
3 a12
6
2
D2(Na12) a† N α α† − D2(Na12−1) a† α† α
Γ(E21) =  D1(Na12−1) (cid:20) 12 α23 13 13 D3(Na12−2) 12 23 13(cid:21) 
 +D3(Na12)α† α 1− D2(Na12) a†(cid:16)α† (cid:17)α 
 D1(Na12) 13 23 D3(Na12−1) 12 23 13 
× −q−(2J1+1) (cid:16) (cid:17) 
 
+ (cid:0)[2J1 −Na12(cid:1)+1]qa†12α23α2†3+ α α†
[2J +2−N ] D1(Na12) a† N 13 13
( 1 a12 qD1(Na12−1) 12 α23 )
+ [2J1 −Na12 +1]qDD3(2N(Na1a21−2)2)− a† 2α† α
( [2J1 −Na12 +2]qDD1(1N(Na1a21−2)1)DD23((NNaa1122−−12)) )(cid:16) 12(cid:17) 23 13
D (N )
+ [2J −N ] 3 a12 a† N α α†
1 a12 qD (N −1) 12 α13 23 23
3 a12
D (N )
+ [2J −N +1] 4 a12 a† N N , (18e)
1 a12 qD (N −1) 12 α13 α23
4 a12
Γ(E13) = Dq(NNa12 )α13α23α2†3+qNa12DD1((NNa12))α13Nα23, (18f)
3 a12 4 a12
[2J +2J ]
1 2 q † † †
Γ(E ) = D (N )a α α α
31 q 2 a12 12 23 13 13
[2J +2J ]
+ 1 2 qD (N )α† α α†
q 3 a12 13 23 23
[2J +2J ] D (N ) D (N )
+ 1 2 q 4 a12 α† N + 2 a12 a† α† N
q D (N ) 13 α23 D (N −1) 12 23 α13
1 a12 (cid:20) 3 a12 (cid:21)
N N
+ q2J2+2J1 D (N )α† α23 +D (N ) α13 D (N )a† α†
4 a12 13D (N ) 4 a12 D (N ) 2 a12 12 23
(cid:20) 1 a12 1 a12 (cid:21)
− q2J1+2J2−Na12[Na12 −1]qD2(Na12)a†12α2†3α13α1†3
− [2J2]qq2J1−Na12+1D1(Na12)a†12α2†3α13α1†3
− q2J1+2J2−Na12−1D3(Na12)[Na12]qα1†3α23α2†3
D (N )
− [2J2 +1]qq2J1−Na12D (N4 a1−2 1)a†12α2†3Nα13
3 a12
D (N ) D (N )
− q2J1+2J2−Na12−2D4(Na12)[Na12]qα1†3 Nα23 − D (N2 a1−2 1)a†12α2†3α13
1 a12 (cid:20) 3 a12 (cid:21)
D (N )
+ q2J1+2J2−Na12D (N4 a1−2 1)a†12α2†3Nα13, (18g)
3 a12
1 1 D (N )
Γ(E ) = α α α† − 2 a12 a† α α α†
23 qNa12D1(Na12) 23 13 13 qNa12D1(Na12)D3(Na12 −1) 12 13 23 23
1 D (N ) D (N )
− 2 a12 a† α N + 3 a12 α N
qNa12−1D4(Na12 −1) 12 13 α23 qNa12D4(Na12) 23 α13
1 D (N )
+ a† α α α† + 1 a12 a† α N , (18h)
D (N −1) 12 13 23 23 D (N −1) 12 13 α23
3 a12 4 a12
7
Γ(E ) = q2J2D (N )[N ] α† α α† +[2J ] D (N )α† α α†
32 2 a12 a12 q 23 13 13 2 q 1 a12 23 13 13
D (N )
+ q2J2D (N )a α† α α† + 4 a12 [2J +1] α† N
3 a12 12 13 23 23 D (N ) 2 q 23 α13
3 a12
1 1 D (N )
+ q2J2D (N ) a α† N − 2 a12 a† α† α
4 a12 q 12 13D (N ) α23 D (N −1) 12 23 13
(cid:26) 1 a12 (cid:20) 3 a12 (cid:21)
α†
− 23 N . (18f)
D (N ) α13
3 a12 )
Applying Maclaurin’s expansion for fermion operators
qNαij = 1+qN −N (19)
αij αij
we can prove that the operators (18) really satisfy the defining relations (1) – (4) of
U [gl(2|1)]. So, the mapping (14) performs a q–boson-fermionrealization of U [gl(2|1)].
q q
2.3 Typical and nontypical representations
For the finite-dimensional representations (8), the mapping (13) gives
|J ,J ,J ,Mi = (a† )(J1−M)|0i,
1 2 3 J 12
|J −1/2,J +1/2,J +1/2,Pi = 2D (J −P +1/2)α† (a† )J1−P+1/2|0i
1 2 3 J 2 1 23 12
+2D (J −P −1/2)α† (a† )J1−P−1/2|0i,
3 1 13 12
D (J −R+1/2)[2J +1]
|J +1/2,J ,J +1/2,Ri = 2 1 1 1 q
1 2 3 J −D (J −R+1/2)[J −R+1/2] q−2J1−1
2 1 1 q
(cid:26) (cid:27)
×α† (a† )J1−R+1/2|0i
23 12
−2D (J −R−1/2)[J −R+1/2] q−2J1−1
3 1 1 q
×α† (a† )J1−R−1/2|0i,
13 12
|J ,J +1/2,J +1,Si = α† α† (a† )(J1−S)|0i. (20)
1 2 3 J 23 13 12
Using (18) we can obtain all the matrix elements of U [gl(2|1)] in the basis (20).
q
The latter constitute subspaces which being U [gl(2|1) ]-modules are invariant under
q ¯0
the action of even U [gl(2|1)] generators but shifted from one to another by the odd
q
generators:
Γ(H ) |J ,J ,J ,Mi = 2M|J ,J ,J ,Mi ,
1 1 2 3 J 1 2 3 J
Γ(H ) |J −1/2,J +1/2,J +1/2,Pi
1 1 2 3 J
= 2P|J −1/2,J +1/2,J +1/2,Pi ,
1 2 3 J
Γ(H ) |J +1/2,J ,J +1/2,Ri
1 1 2 3 J
= 2R|J −1/2,J ,J +1/2,Ri ,
1 2 3 J
Γ(H ) |J ,J +1/2,J +1,Si
1 1 2 3 J
= 2S|J ,J +1/2,J +1,Si . (21)
1 2 3 J
8
Γ(H ) |J ,J ,J ,Mi
2 1 2 3 J
= (2J +J −M)|J ,J ,J ,Mi ,
2 1 1 2 3 J
Γ(H ) |J −1/2,J +1/2,J +1/2,Pi
2 1 2 3 J
= (2J +J −P +1/2)|J −1/2,J +1/2,J +1/2,Pi ,
2 1 1 2 3 J
Γ(H ) |J +1/2,J ,J +1/2,Ri
2 1 2 3 J
= (2J +J −R+1/2)|J −1/2,J ,J +1/2,Ri ,
2 1 1 2 3 J
Γ(H ) |J ,J +1/2,J +1,Si
2 1 2 3 J
= (2J +J −S +1)|J ,J +1/2,J +1,Si . (22)
2 1 1 2 3 J
Γ(H ) |J ,J ,J ,Mi
3 1 2 3 J
= 2J |J ,J ,J ,Mi ,
3 1 2 3 J
Γ(H ) |J −1/2,J +1/2,J +1/2,Pi
3 1 2 3 J
= (2J +1)|J −1/2,J +1/2,J +1/2,Pi ,
3 1 2 3 J
Γ(H ) |J +1/2,J ,J +1/2,Ri
3 1 2 3 J
= (2J +1)|J +1/2,J ,J +1/2,Ri ,
3 1 2 3 J
Γ(H ) |J ,J +1/2,J +1,Si
3 1 2 3 J
= (2J +2)|J ,J +1/2,J +1,Si . (23)
3 1 2 3 J
Γ(E ) |J ,J ,J ,Mi
12 1 2 3 J
= [J −M] |J ,J ,J ,M +1i ,
1 q 1 2 3 J
Γ(E ) |J −1/2,J +1/2,J +1/2,Pi
12 1 2 3 J
= [J −P −1/2] |J −1/2,J +1/2,J +1/2,P +1i ,
1 q 1 2 3 J
Γ(E ) |J +1/2,J ,J +1/2,Ri
12 1 2 3 J
= [J −R+1/2] |J +1/2,J ,J +1/2,R+1i ,
1 q 1 2 3 J
Γ(E ) |J ,J +1/2,J +1,Si
12 1 2 3 J
= [J −S] |J ,J +1/2,J +1,S +1i . (24)
1 q 1 2 3 J
Γ(E ) |J ,J ,J ,Mi
21 1 2 3 J
= [J +M] |J ,J ,J ,M −1i ,
1 q 1 2 3 J
Γ(E ) |J −1/2,J +1/2,J +1/2,Pi
21 1 2 3 J
= [J +P −1/2] |J −1/2,J +1/2,J +1/2,P −1i ,
1 q 1 2 3 J
Γ(E ) |J +1/2,J ,J +1/2,Ri
21 1 2 3 J
= [J +R+1/2] |J +1/2,J ,J +1/2,R−1i ,
1 q 1 2 3 J
Γ(E ) |J ,J +1/2,J +1,Si
21 1 2 3 J
= [J +S] |J ,J +1/2,J +1,S −1i . (25)
1 q 1 2 3 J
9
Γ(E ) |J ,J ,J ,Mi = 0,
13 1 2 3 J
Γ(E ) |J −1/2,J +1/2,J +1/2,Pi
13 1 2 3 J
= 2qJ1−P−1/2|J ,J ,J ,P +1/2i ,
1 2 3 J
Γ(E ) |J +1/2,J ,J +1/2,Ri
13 1 2 3 J
= −2q−J1−R−3/2[J −R+1/2] |J ,J ,J ,R+1/2i ,
1 q 1 2 3 J
Γ(E ) |J ,J +1/2,J +1,Si
13 1 2 3 J
[J −S] q−J1−S−1
1 q
= −
2D (J −S)[2J +1]
4 1 1 q
×|J −1/2,J +1/2,J +1/2,S +1/2i
1 2 3 J
qJ1−S
− |J +1/2,J ,J +1/2,S +1/2i . (26)
1 2 3 J
2D (J −S)[2J +1]
4 1 1 q
Γ(E ) |J ,J ,J ,Mi
31 1 2 3 J
[2J +2J +1] [J +M] q−(J−M)−1
1 2 q 1 q
=
2[2J +1]
1 q
×|J −1/2,J +1/2,J +1/2,M −1/2i
1 2 3 J
[2J ] qJ1+M
2 q
− |J +1/2,J ,J +1/2,M −1/2i ,
1 2 3 J
2[2J +1]
1 q
Γ(E ) |J −1/2,J +1/2,J +1/2,Pi
31 1 2 3 J
= −2qJ1+P−3/2[2J ] D (J −P +1/2)|J ,J +1/2,J +1,P −1/2i ,
2 q 4 1 1 2 3 J
Γ(E ) |J +1/2,J ,J +1/2,Ri
31 1 2 3 J
= −2q−J1+R−3/2[2J +2J +1] [J +R+1/2] D (J −R−1/2)
1 2 q 1 q 4 1
×|J ,J +1/2,J +1,R−1/2i ,
1 2 3 J
Γ(E ) |J ,J +1/2,J +1,Si = 0. (27)
31 1 2 3 J
Γ(E ) |J ,J ,J ,Mi = 0,
23 1 2 3 J
Γ(E ) |J −1/2,J +1/2,J +1/2,Pi
23 1 2 3 J
= 2|J ,J ,J ,P −1/2i ,
1 2 3 J
Γ(E ) |J +1/2,J ,J +1/2,Ri
23 1 2 3 J
= 2[J +R+1/2] |J ,J ,J ,R−1/2i ,
1 q 1 2 3 J
Γ(E ) |J ,J +1/2,J +1,Si
23 1 2 3 J
[J +S]
1 q
=
2D (J −S)[2J +1]
4 1 1 q
×|J −1/2,J +1/2,J +1/2,S −1/2i
1 2 3 J
1
− |J +1/2,J ,J +1/2,S −1/2i . (28)
1 2 3 J
2D (J −S)[2J +1]
4 1 1 q
10