describe the unitary Cayley-Klein groups and construct their Jordan-Schwinger representations. In Sec. Ill we regard in detail SU2(j1) and SU3(j1,j2) groups for ...
J. Math. Phys., 1990, 31(5), pp. 1054–1059
The Jordan-Schwinger representations of Cayley-Klein groups. II. The unitary groups. N. A. Gromov Komi Scientific Centre, Academy of Sciences of the USSR, Syktyikar, 167610, USSR V. I. Man’ko P. S. Lebedev Physical Institute, Academy of Sciences of the USSR, Moscow, 117333, USSR (Received 12 May 1989; accepted for publication 25 October 1989) The unitary Cayley-Klein groups are defined as the groups that are obtained by the contractions and analytical continuations of the special unitary groups. The Jordan-Schwinger representations of the groups under consideration are discussed based on the mixed sets of creation and annihilation operators of boson or fermion type. The matrix elements of finite group transformations are obtained in the bases of coherent states.
1
Introduction
The purpose of this series of papers is to present the Jordan-Schwinger representations [1]–[3] of the groups that are obtained from the classical one’s by the contractions and analytical continuations of the group parameters. The Inonu-Wigner contractions [4] and analytical continuations of the groups are regarded on the base of the unified description of the groups [5], [6]. The main feature of such an approach is the transformations of the well-known algebraic constructions (generators, commutators, etc.) of the classical groups into the algebraic constructions of the groups under consideration. The Jordan-Schwinger representations of the groups are built by both boson and fermion creation and annihilation operators. The matrix elements of the finite group transformations in the Glauber coherent state basis [7] are calculated with the help of the methods of the well-developed theory of the quantum systems that are quadratic in creation and annihilation operators Hamiltonians [8]–[10]. The first paper in this series [11] (hereafter referred to as I and whose equations will be subsequently quoted by their number preceded by I) was devoted to a consideration of the Jordan-Schwinger representations of the orthogonal Cayley-Klein groups that were obtained from the special orthogonal groups by the contractions and analytical continuations of the group parameters. In this paper we discuss the case of the special unitary groups. In Sec. II we describe the unitary Cayley-Klein groups and construct their Jordan-Schwinger representations. In Sec. Ill we regard in detail SU2 (j1 ) and SU3 (j1 , j2 ) groups for which the calculations are made in explicit form.
2
The special unitary groups in complex Cayley-Klein spaces Let us define the map of the (n + 1)-dimensional complex space Cn+1 into the complex space 1
Cn+1 (j) as follows: Cn+1 → Cn+1 (j),
ψ:
ψz0 = z0 ,
ψzk = zk
k Y
j,
(1)
m=1
where k = 1, 2, . . . , n; z0 , zk are the complex Cartesian coordinates, j = (j1 , j2 , . . . , jn ) and each of the parameters jk may be equal to the real unit 1, the Clifford dual unit ιk , or the imaginary unit i. The dual numbers are not often used and we briefly review their algebraic properties. Each of the dual units is not equal to zero ιk 6= 0; different dual units obey the commutative law of multiplication ιk ιm = ιm ιk 6= 0, k 6= m; the square of a dual unit is always equal to zero ι2k = 0. Division of a real or complex number by a dual unit is not defined, but division of a dual unit by itself is equal to a real unit ιk /ιk = 1. A function of a dual argument is defined by its P Taylor expansion. The quadratic form (z, z) = nm=0 |zm |2 of Cn+1 transforms under the map (1) into the following quadratic form of the complex Cayley-Klein space Cn+1 (j) (Ref. [12]): 2
(z, z) = |z0 | +
n X
2
|zk |
k=1
k Y
2 jm ,
(2)
m=1
where zk = (x2k + yk2 )1/2 is the absolute value of the complex number zk = xk + iyk , and z = (z0 , z1 , . . . , zn ) is the complex vector. The unitary Cayley-Klein group SUn+1 (j) is defined as the group that keeps invariant the quadratic form (2) transformations in the space Cn+1 (j). The map (1) induces the transformation of the special unitary group SUn+1 into the group SUn+1 (j). All (n + 1)2 − 1 generators of SUn+1 are Hermite matrices. The commutation relations for these Hermite generators are very complicated and usually the matrix generators Y˜km , k, m = 0, 1, . . . , n of the general linear group GLn+1 (R) are used [13]. The only nonzero matrix element of Y˜km is (Y˜km )km = 1. The generators Y˜ of GLn+1 (R) satisfy the commutation relations [Y˜km , Y˜pq ] = δmp Y˜kq − δkq Y˜mp .
(3)
The independent Hermite generators of SUn+1 are defined by the equations ˜ µν = (i/2)(Y˜µν + Y˜νµ ), Q
˜ µν = (1/2))Y˜νµ − Y˜µν ), L
P˜k = (i/2)(Y˜k−1,k−1 − Y˜kk ),
(4)
where µ = 0, 1, . . . , n − 1, ν = µ + 1, µ + 2, . . . , n. The matrix generators Y˜ are transformed under the map (1) as follows:
Y˜µν (j) =
ν Y
jm Y˜µν (→) =
m=µ+1
Y˜νµ (j) =
ν Y
2 ˜ jm Yµν ,
m=µ+1
ν Y
jm Y˜νµ (→) = Y˜νµ ,
µ < ν,
m=µ+1
Ykk (j) = Y˜kk ,
2
(5)
where Y˜µν (→) and Y˜νµ (→) denote the transformed generators Y˜µν , Y˜νµ with the following nonzero matrix elements:
ν Y
(Y˜µν (→))µν =
m=µ+1
ν Y
(Y˜νµ (→))νµ =
ν Y
jm (Y˜µν )µν =
jm ,
m=µ+1
ν Y
−1 ˜ jm (Yνµ )νµ =
m=µ+1
−1 jm ,
(6)
m=µ+1
The generators (5) satisfy the commutation relations [Ykm , Ypq ] =
l2 Y l=l1
jl
l4 Y
jl δmp Ykq
l6 Y
jl−1 − δkq Ymp
jl−1 ,
(7)
l=l7
l=l5
l=l3
l8 Y
where l1 = 1 + min(k, m), l2 = max(k, m), l3 = 1 + min(p, q), l4 = max(p, q), l5 = 1 + min(k, q), l6 = max(k, q), l7 = 1 + min(m, p), l8 = max(m, p). The same laws of transformations as in Eq. (5) are held for the Hermite generators (4). Then we obtain the matrix generators of the unitary Cayley-Klein group SUn+1 (j) in the form
ν Y
ν Y
ν Y 2 ˜ µν (→) = i (Yνµ (j) + Yµν (j)) = i Y˜νµ + Y˜µν jm Q Qµν (j) = jm , 2 2 m=µ+1 m=µ+1 ν Y 2 ˜ µν (→) = 1 (Yνµ (j) − Yµν (j)) = 1 Y˜νµ − Y˜µν jm L jm , Lµν (j) = 2 2 m=µ+1 m=µ+1
i ˜ Pk (j) = P˜k = Yk−1,k−1 − Y˜kk , k = 1, 2, . . . , n (8) 2 The commutation relations of these generators may be derived with the help of Eq. (7), but they are very cumbersome and we do not write them here. The finite group transformation
Ξ(r, s, w, j) = exp Z(r, s, w, j)
(9)
of SUn+1 (j) correspond to the general element n(n+1)/2
Z(r, s, w, j) =
X
(rλ Qλ (j) + sλ Lλ (j)) +
λ=1
n X
ωk Pk (j)
(10)
k=1
of the algebra sun+1 (j), where rλ , sλ , ωk are real group parameters and index λ is connected with the indices µ, ν by Eq. ([11] (19)). The explicit form of the finite transformation Ξ(r, s, w, j) may be obtained for the groups SU2 (j1 ) and SU3 (j1 , j2 ). The Jordan-Schwinger representations of SUn+1 (j) are provided by the operators ˆ µν (j) = ψˆ Q a+ Qµν (j)ψˆ a,
ˆ µν (j) = ψˆ L a+ Lµν (j)ψˆ a,
a, Pk (j) = ψˆ a+ Pk (j)ψˆ
µ < ν, k = 1, 2, . . . , n, 3
(11)
where the transformed sets ψˆ a+ , ψˆ a of the creation and annihilation operators are given by Eqs. (I. (4.1)) and (I. (4.2)). Indeed, it is verified by direct calculations that the operators Yˆ (j) = ψˆ a+ Y (j)ψˆ a satisfy the commutation relations (7) and we may conclude that the operators (11) satisfy the commutation relations of the group SUn+1 (j). The finite group transformation is represented by the operator ˆ s, w, j)), Uˆg (r, s, w, j) = exp(−Z(r,
(12)
where the operator Zˆ is given by Eq. (10) with the generators Qλ (j), Lλ (j), Pk (j) replaced by ˆ λ (j), L ˆ λ (j) and Pˆk (j), respectively. The kernel of the operator Uˆg (r, s, w, j) in a the operators Q coherent state basis is obtained quite analogous to the case of orthogonal groups [11]. When the Cayley-Klein groups SUn+1 (j) are obtained from SUn+1 by only contractions (jk = 1, ιk , k = 1, 2, . . . , n) this kernel is given by U (α∗ , β, r, s, w, j) = exp(α∗ Ξ(−r, −s, −w, j)β),
(13)
[compare with Eq. (I. (4.8)]. When the groups SUn+1 (j) are obtained from SUn+1 by both contractions and analytical continuations the kernel is given by the following equation: ∗
−/2
U (α , β, r, s, w, j) = (det ξ)
1 1 exp α∗ ξ −1 ηα∗ + α∗ ξ −1 β + βη1 ξ −1 β , 2 2
(14)
where the matrices ξ, η, η1 are expressed through the matrix (9) by Eqs. (I. (4.11)).
3
Examples
We shall discuss in detail two sets of unitary Cayley-Klein groups SU2 (j1 ) and SU3 (j1 , j2 ) for which it is possible to obtain the explicit form of the finite group transformation matrix Ξ(r, s, w, j).
3.1
SU2 (j1 ) groups
The map (1), namely ψz0 = z0 , ψz1 = j1 z1 , , j1 = 1, ι1 , i, gives the complex space C2 (j1 ), with the quadratic form (z, z) = |z0 |2 + j12 |z1 |2 . The transformations belonging to the group SU2 (j1 ) keep this quadratic form invariant. The matrices (5) are as follows: Y00 =
1 0 0 0
!
,
0 0 0 1
Y11 =
!
,
0 j12 0 0
Y01 =
!
,
Y10 =
0 0 1 0
!
,
(15)
and the commutation relations [Y00 , Y11 ] = 0,
[Y00 , Y10 ] = −Y10 ,
[Y00 , Y01 ] = Y01 ,
[Y11 , Y01 ] = −Y01 ,
[Y11 , Y10 ] = Y10 ,
[Y01 , Y10 ] = j12 Y00
(16)
are satisfied. The matrix generators of SU2 (j1 ) are given by Eq. (8) in the form i P1 = 2
1 0 0 −1
!
,
Q01
i = 2
0 j12 1 0 4
!
,
L01
1 = 2
0 −j12 1 0
!
,
(17)
and satisfy the following commutation relations [P1 , Q01 ] = L01 ,
[L01 , P1 ] = Q01 ,
[Q01 , L01 ] = j12 P1 .
(18)
The generators (17) for j1 = 1 are equal up to a coefficient to the Pauli matrices. The general element of algebra su2 (j1 ) in accordance with Eq. (10) is given by 1 Z(r1 , s1 , w1 , j1 ) = r1 Q01 + s1 L01 + w1 P1 = 2
iw1 −j12 (s1 − ir1 s1 + ir1 −iw1
!
,
(19)
and for the finite group transformation matrix we have v w1 v 1 v cos + i sin −j12 (s1 − ir1 ) sin v 2 v 2 v 2 v 2 Ξ(r1 , s1 , w1 , j1 ) = E2 cos + Z sin = v w1 v , (20) 2 v 2 (s + ir ) 1 sin v cos − i sin 1 1 v 2 2 v 2
where v 2 (j1 ) = w12 + j12 (r12 + s21 ).
(21)
When j1 = 1, ι1 the following operators: i + ˆ 01 (j1 ) = a ˆ+ Q01 (j1 )ˆ a a ˆ0 + j12 a ˆ+ ˆ1 ), Q a = (ˆ 0a 2 1 1 + ˆ 01 (j1 ) = a ˆ+ L01 (j1 )ˆ L a = (ˆ a a ˆ0 − j12 a ˆ+ ˆ1 ), 0a 2 1 i + ˆ+ P1 (j1 )ˆ Pˆ1 (j1 ) = a a = (ˆ a a ˆ0 − a ˆ+ ˆ1 ), (22) 1a 2 0 satisfy the commutation relations (18) and therefore provide the Jordan-Schwinger representation of SU2 (j1 ). The kernel of the finite group transformation operator is obtained by Eq. (13) with help of the matrix (20) and is given in the form ∗
α0∗ β0
U (α , β, j1 ) = exp
w1 v v w1 v v + α1∗ β1 cos + i sin − cos − i sin 2 v 2 2 v 2
1 v 1 v −α1∗ β0 (s1 + ir1 ) sin + j12 α0∗ β1 (s1 − ir1 ) sin . v 2 v 2 For the contracted group SU2 (ι1 ) we have from Eq. (21) v = w1 and from Eq. (23)
U (α∗ , β, ι1 ) = exp α0∗ β0 e−(i/2)w1 + α1∗ β1 e(i/2)w1 − α1∗ β0 (s1 + ir1 )
1 w1 sin . w1 2
(23)
(24)
When j1 = i we introduce the new parameter ˜j1 as j1 = i˜j1 , ˜j = i, ι1 . The case j1 = ι1 corresponds to the contraction of the pseudounitary group SU2 (i) = SU (1, 1). Equations (I. (4.1)) give ψˆ a = (ˆ a0 , a ˆ+ a = (ˆ a+ a1 ). Then from Eqs. (11) we obtain the operators 1 ), ψˆ 0 , −ˆ i ˆ 01 (ij ˜ 1 ) = ψˆ ˜ 1 )ψˆ Q a+ Q01 (ij a = − (ˆ a1 a ˆ0 + j12 a ˆ+ ˆ+ 0a 1 ), 2 1 ˆ 01 (ij ˜ 1 ) = ψˆ ˜ 1 )ψˆ L a+ L01 (ij a = − (ˆ a1 a ˆ0 − j12 a ˆ+ ˆ+ 0a 1 ), 2 5
i + ˜ 1 ) = ψˆ ˜ 1 )ψˆ a0 a ˆ0 + ˆ a1 a ˆ+ (25) Pˆ1 (ij a+ P1 (ij a = (ˆ 1 ), 2 that provide the Jordan-Schwinger representation of SU2 (i˜j1 ). The diagonal matrices ψ1 (i), ψ2 (i) in Eq.(I. (4.2)) are in the form ψ1 (i) =
!
1 0 0 0
,
0 0 0 −1
ψ2 (i) =
!
,
(26)
and using Eqs. (I. (4.11)) and (20) we obtain the matrices ξ, η, η1 ξ= η= η1 =
1 0 0 1
!
0 1 1 0
!
0 1 1 0
v˜ w1 v˜ cos + i sin , 2 v˜ 2
˜j12 (s1 − ir1 ) 1 sin v˜ , v˜ 2
!
v˜ 1 (s1 + ir1 ) sin , v˜ 2
(27)
v˜2 = w12 − j12 (r12 + s21 ). After some calculations we find the matrices ξ −1 , ξ −1 η, η1 ξ −1 and using Eq.(14) obtain the kernel of the finite group transformation operator of SU2 (i˜j1 ) in the form v˜ w1 v˜ U (α , β, i˜j1 ) = cos + i sin 2 v˜ 2
∗
−
(
exp
cos
v˜ w1 v˜ + i sin 2 v˜ 2
−1
×
1 v˜ 1 × α0∗ β0 + α1∗ β1 − (α0∗ α1∗ + α1∗ α0∗ )˜j12 (s1 − ir1 ) sin + 2 v˜ 2 1 1 v˜ + (β0 β1 + β1 β0 )(s1 + ir1 ) sin . 2 v˜ 2 For the contracted group SU2 (iι1 ) we have from Eq. (27) v = w1 and from Eq. (28)
w1 1 1 U (α∗ , β, iι1 ) = e−(i/2)w1 exp e−(i/2)w1 α0∗ β0 + α1∗ β1 + (β0 β1 + β1 β0 )(s1 + ir1 ) sin 2 w1 2
(28)
.
(29) Comparing Eq. (29) with Eq.(24) they provide different Jordan-Schwinger representations of the same group SU2 (ι1 ) = SU2 (iι1 ). Notice that some recent works [13], [14] were devoted to the connections SU (1, 1) with SU (2) and to the problem of the evolution SU (2) coherent state regarding from a unified point of view.
4
SU3(j1, j2) groups
The SU3 (j1 , j2 ) group consists of all transformations of C2+1 (j1 , j2 ) keeping invariant the quadratic form (z, z) = |z0 |2 + j12 |z1 |2 + j12 j22 |z2 |2 . The matrix generators of the general linear group are given by Eq(5) in the form
Y00
1 0 0 = 0 0 0 , 0 0 0
Y11
0 0 0 = 0 1 0 , 0 0 0 6
Y22
0 0 0 = 0 0 0 , 0 0 1
Y10
0 0 j12 j22 = 0 0 0 , 0 0 0
Y02
0 j12 0 = 0 0 0 , 0 0 0
0 0 0 = 1 0 0 , 0 0 0
Y01
Y21
Y20
0 0 0 = 0 0 0 , 0 1 0
0 0 0 = 0 0 0 , 1 0 0
Y12
(30)
0 0 0 = 0 0 j22 . 0 0 0
Then the matrix generators of SU3 (j1 , j2 ) are given by Eq. (8) as follows:
1 0 0 i P1 = 0 −1 0 , 2 0 0 0 0 j12 0 i = 1 0 0 , 2 0 0 0
Q1 = Q01
Q2 = Q02
L2 = L02
0 0 −j12 j22 1 0 = 0 0 , 2 1 0 0
L3 = L12
0 0 0 1 = 0 0 −j22 , 2 0 1 0
0 0 j12 j22 i = 0 0 0 , 2 1 0 0
Q3 = Q12
L1 = L01
0 −j12 0 1 = 1 0 0 , 2 0 0 0
0 0 0 i P2 = 0 1 0 , 2 0 0 −1
0 0 0 i = 0 0 j22 , 2 0 1 0
(31)
They satisfy the commutation relations [P1 , P2 ] = 0,
[P1 , Q1 ] = L1 ,
[P1 , L1 ] = −Q1 ,
1 1 1 [P1 , Q2 ] = L2 , [P1 , L2 ] = − Q2 , [P1 , Q3 ] = − L3 , 2 2 2 1 1 1 [P1 , L3 ] = Q3 , [P2 , Q1 ] = − L1 , [P2 , L1 ] = Q1 , 2 2 2 1 1 [P2 , Q2 ] = L2 , [P2 , L2 ] = − Q2 , [P2 , Q3 ] = L3 , 2 2 2 [P2 , L3 ] = −Q3 , [Q1 , L1 ] = j1 P1 , [Q3 , L3 ] = j22 P2 , 1 [Q2 , L2 ] = j12 j22 (P1 + P2 ), [Q1 , L2 ] = − j12 Q3 , 2 1 2 1 2 1 [Q2 , L3 ] = j2 Q1 , [L1 , Q2 ] = j1 Q3 , [Q3 , L1 ] = Q2 , 2 2 2 1 1 1 [L2 , Q3 ] = − j22 Q1 , [Q1 , L3 ] = − Q2 , [Q1 , Q3 ] = L2 , 2 2 2 1 1 1 [Q1 , Q2 ] = j12 L3 , [Q2 , Q3 ] = j22 L1 , [L1 , L2 ] = j12 L3 , 2 2 2 1 1 2 [L1 , L3 ] = − L2 , [L2 , L3 ] = j2 L1 . (32) 2 2 It is well known [15] that the structure of group (algebra) is changed under contraction. Let j1 = ι1 , then the simple classical algebra su3 contracts into the algebra su3 (ι1 , j2 ) = 7
T (+u2 (j2 ), where T = {Q1 , L1 , Q2 , L2 } is the commutative ideal and the subalgebra u2 (j2 ) = {P1 , P2 , Q3 , L3 } is the Lie algebra of the unitary group in the complex Cayley-Klein space C2 (j2 ). From Eq. (32) for j1 = ι1 we conclude that [T, u2 (j2 )] ⊂ T as it must be for a semidirect sum. The structure of SU3 (ι1 , j2 ) is the semidirect product SU3 (ι1 , j2 ) = eT (×U2 (j2 ). Such groups are called inhomogeneous unitary groups [16]. From Eq. (10) we have the general element of the algebra su3 (j) in the form −j12 j22 ζ2∗ iw −j12 ζ1∗ 1 1 Z(r, s, w, j) = (rk Qk + sk Lk ) + w1 P1 + w2 P2 = ζ1 i(w2 − w1 ) −j22 ζ3∗ , 2 k=1 ζ2 ζ3 −iw2
3 X
(33)
where ζk = sk +irk , k = 1, 2, 3 and ζk∗ is the complex conjugate. The finite group transformation matrix Ξ(ζ, w, j) is obtained from Eq. (33) by the exponential map (9). We shall find the matrix Ξ by the Cayley-Hamilton theorem [17]. The characteristic equation det(Z − λE3 ) = 0 of the matrix Z is the following cubic equation: λ3 + pλ + q = 0, p = w12 − w1 w2 + w22 + |ζ|2 (j), q = −iw1 w2 (w2 − w1 ) + iw2 j12 |ζ1 |2 − i(w2 − w1 )j12 j22 |ζ2 |2 − −iw1 j22 |ζ3 |2 + 2ij12 j22 Imζ1 ζ2∗ ζ3 ,
(34)
|ζ|2 (j) = j12 |ζ1 |2 + j12 j22 |ζ2 |2 + j22 |ζ3 |2 .
(35)
where The roots of Eq. (34) are as follows [15]:
q λ k = − + 2
q + − − 2
s 2
s 2
q 2
q 2
p + 3
3
p + 3
3
1/3
+
1/3 00
= λ0k + λk ,
(36)
00
where λ0k + λk = −p/3 and λ0k , k = 1, 2, 3 are three distinct cube roots. Then by the CayleyHamilton theorem we obtain Ξ(ζ, w, j) = A · E3 − B · Z + C · Z2 ,
(37)
where −w12 − j12 |ζ1 |2 − j12 j22 |ζ2 |2 −j12 (iw2 ζ1∗ + j22 ζ2∗ ζ3 ) j12 j22 (i(w2 − w1 )ζ2∗ + ζ1∗ ζ3∗ ) 2 ∗ 2 2 2 2 2 iw2 ζ1 − j2 ζ2 ζ3 −(w2 − w2 ) − j1 |ζ1 | − j2 |ζ3 | j22 (iw1 ζ3∗ − j12 ζ1 ζ2∗ ) , −i(w2 − w1 )ζ2 + ζ1 ζ3 −iw1 ζ3 − j12 ζ1∗ ζ2 −w22 − j12 j22 |ζ2 |2 − j22 |ζ3 |2 (38) and the functions A, B, C are expressed in the following way:
1 Z2 = 4
A = D−1 [eλ1 λ2 λ3 (λ2 − λ3 ) − eλ2 λ1 λ3 (λ1 − λ3 ) + eλ3 λ1 λ2 (λ1 − λ2 )], 8
B = D−1 [eλ1 (λ22 − λ23 ) − eλ2 (λ21 − λ23 ) + eλ3 (λ21 − λ22 )], C = D−1 [eλ1 (λ2 − λ3 ) − eλ2 (λ1 − λ3 ) + eλ3 (λ1 − λ2 )], D = (λ1 − λ2 )(λ1 − λ3 )(λ2 − λ3 ).
(39)
From Tr Z = 0 we conclude that det Ξ = 1 and Ξ−1 (ζ, w, j) = Ξ(−ζ, −w, j). We shall only discuss contractions of the special unitary group SU3 , i.e.,j1 = 1, ι1 , j2 = 1, ι2 . ˆ+ , ψˆ ˆ in Eq. (11) and the Jordan-Schwinger representation of the generators Then ψˆ a∗ = a a=a (31) is given by the operators i + Pˆ1 (j) = (ˆ a a ˆ0 − a ˆ+ ˆ1 ), 1a 2 0
i + Pˆ2 (j) = (ˆ a a ˆ1 − a ˆ+ ˆ2 ), 2a 2 1
ˆ 1 (j) = ˆ 1 (j) = i (ˆ a+ a ˆ0 + j12 a ˆ+ ˆ1 ), L Q 0a 2 1 ˆ 2 (j) = i (ˆ ˆ 2 (j) = Q a+ a ˆ0 + j12 j22 a ˆ+ ˆ2 ), L 0a 2 2 ˆ 3 (j) = i (ˆ ˆ 3 (j) = a+ a ˆ1 + j22 a Q ˆ+ ˆ2 ), L 0a 2 2 These operators satisfy the commutation relations (32). resented by the following operator: ˆ w, j) = Z(ζ,
3 X
1 + (ˆ a a ˆ0 − j12 a ˆ+ ˆ1 ), 0a 2 1 1 + (ˆ a a ˆ0 − j12 j22 a ˆ+ ˆ2 ), 0a 2 2 1 + (ˆ a a ˆ1 − j22 a ˆ+ ˆ2 ). (40) 1a 2 2 The general element of su3 (j) is rep-
ˆ k (j) + sk L ˆ k (j)) + w1 Pˆ1 (j) + w2 Pˆ2 (j) = (rk Q
k=1
= +
3 X
1n iw1 a ˆ+ ˆ0 + i(w2 − w1 )ˆ a+ ˆ1 − iw2 a ˆ+ ˆ2 + 0a 1a 2a 2 )
ζk a ˆ+ ˆ0 ka
−
j12 ζ1∗ a ˆ+ ˆ1 0a
−
j12 j22 ζ2∗ a ˆ+ ˆ2 0a
−
j22 ζ3∗ a ˆ+ ˆ2 1a
.
(41)
k=1
The kernel of the finite group transformation operator (12) of SU3 (j) in a coherent state basis is given by Eq. (13), using Eqs. (37)–(39). We shall not write this kernel. We write out only the kernel of operator of the group SU3 (ι1 , ι2 ) that is obtained from SU3 by two-dimensional contraction. This kernel is as follows: (
U (α∗ , β, w, ι1 , ι2 ) = exp A0
3 X
i αk∗ βk + B 0 w1 (α1∗ β1 − α0∗ β0 )+ 2 k=1
i 1 + B 0 w2 (α2∗ β2 − α1∗ β1 ) − C 0 w12 (α0∗ β0 + α1∗ β1 )− 2 4 1 1 1 i − C 0 w22 (α1∗ β1 + α2∗ β2 ) + C 0 w1 w2 α1∗ β1 + w2 C 0 − B 0 ζ1 α1∗ β0 − 4 2 2 2 1 i 1 i 1 − (w2 − w1 )C 0 + B 0 ζ2 α2∗ β0 − w1 C 0 + B 0 ζ3 α2∗ β1 + C 0 ζ1 ζ3 α2∗ β0 , (42) 2 2 2 2 4 where the functions A0 , B 0 , C 0 are given by Eqs. (39 and λ1 , λ2 , λ3 are the roots of Eq. (34) with the following coefficients: p = w12 − w1 w2 + w22 , q = −iw1 w2 (w2 − w1 ).
9
5
Concluding remarks
On the basis of ideas of the previous paper [11] we have regarded the unitary Cayley-Klein groups SUn+1 (j) as the groups of motion (except for translations) of the complex Cayley-Klein spaces Cn+1 (j). The groups SUn+1 (j) have been obtained from the classical group SUn+1 by contractions and analytical continuations of the group parameters. It has been shown that all these groups are described in the unified way by introducing n parameters j = (j1 , j2 , . . . , jn ) each of which were equal to the real, dual, or imaginary units. We have built the Jordan-Schwinger representation of the group under consideration. The only contractions of the Jordan-Schwinger representations permit of the unified description. In the case of analytical continuations each of the Jordan-Schwinger representation is built in a particular way.
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[15] A. . Barut and R. Raczka, Theory of Group Representations and Applications (PWN Polish Scientific Publishers Warsaw, 1977). [16] M. Perroud, J. Math. Phys. 24, 1381 (1983). [17] O. A. Korn and T. M. Korn, Mathematical Handbook (McGraw Hill, New York 1961).
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