Download Indecomposable Representations of the Square

Commun. Theor. Phys. (Beijing, China) 35 (2001) pp. 513–518
c International Academic Publishers
Vol. 35, No. 5, May 15, 2001
Indecomposable Representations of the Square-Root Lie Algebras of Vector Type and
Their Boson Realizations∗
RUAN Dong,1,2,3 YUAN Jing,1 JIA Yu-Feng1 and SUN Hong-Zhou1,3
1
Department of Physics, Tsinghua University, Beijing 100084, China
2
Key Laboratory for Quantum Information and Measurements of MOE, Tsinghua University, Beijing 100084, China
3
Center of Theoretical Nuclear Physics, National Laboratory of Heavy Ion Accelerator, Lanzhou 730000, China
(Received July 20, 2000)
Abstract The explicit expressions for indecomposable representations of nine square-root Lie algebras of vector type,
Rνλ (ν, λ = 0, ±1), are obtained on the space of universal enveloping algebra of two-state Heisenberg–Weyl algebra,
the invariant subspaces and the quotient spaces. From Fock representations corresponding to these indecomposable
representations, the inhomogeneous boson realizations of Rνλ are given. The expectation values of Rνλ in the angular
momentum coherent states are calculated as well as the corresponding classical limits.
PACS numbers: 02.20.Sv, 11.30.Na
Key words: nonlinear Lie algebra, indecomposable representation, boson realization, angular momentum coherent state
1 Introduction
Recently, much attention has been paid to various
nonlinear Lie algebras and their applications in quantum
systems.[1,2] In a previous paper,[3] we studied boson realizations of square-root Lie algebras of vector and quadratic
types by means of Schwinger’s coupled boson representation of angular momentum theory. In this paper we
shall study indecomposable representations (i.e. reducible
but not completely reducible representations[4] ), inhomogeneous boson realizations and expectation values in the
angular momentum coherent states of the square-root Lie
algebras of vector type, Rνλ (ν, λ = 0, ±1), one of which
is generated by two angular momentum operators J0 , J 2
and any component of an irreducible tensor operator of
rank 1, T (1νλ) (ν, λ = 0, ±1), of an SO(3) group.
Generally, there are two approaches to the indecomposable representations of some algebra g: one is based
upon the universal enveloping algebra of g,[5] another
upon the universal enveloping algebra of the Heisenberg–
Weyl algebra.[6] Due to the appearance of square-root operator in the commutation relations obeyed by the generators of Rνλ , it is convenient for Rνλ to be dealt with
by the second approach. Furthermore, from these representations, we may define the corresponding Fock representations of Rνλ on the Fock spaces, and then obtain
inhomogeneous boson realizations of Rνλ , which, as we
shall see below, include the results of Ref. [3] as a special
case.
This paper is arranged as follows. In Sec. 2, the square∗ The
root Lie algebras of vector type, Rνλ (ν, λ = 0, ±1), are
reviewed. In Sec. 3, the indecomposable representations of
Rνλ on the space of the universal enveloping algebra of the
two-state Heisenberg–Weyl algebra, the quotient spaces
and the invariant subspaces are studied respectively. The
inhomogeneous boson realizations of Rνλ are derived by
means of these indecomposable representations. In Sec. 4,
the expectation values of Rνλ in the angular momentum
coherent states are calculated.
2 Nonlinear Lie Algebras Rνλ
Consider the two-state Heisenberg–Weyl algebra with
five generators: the creation operators a+
i (i = 1, 2), the
†
annihilation operators ai = (a+
)
and
the
unit operator
i
e, that satisfy the commutation relations
[ai , a+
j ] = δij ,
+
[ai , aj ] = [a+
i , aj ] = 0 ,
[ai , e] = [a+
i , e] = 0 .
(1)
Then, according to the Poincaré–Birkhoff–Witt theorem,[4,5] two-state Heisenberg–Weyl basis can be chosen
as the following set of ordered elements
{f (k1 , k2 , k3 , k4 )
k 1 + k2 k3 k4
= (a+
1 ) (a2 ) a1 a2 |k1 , k2 , k3 , k4 ∈ ℵ}
(2)
and f (0, 0, 0, 0) = 1 denotes the identity operator, where
(and afterwards) ℵ stands for a set of non-negative integers. In fact, {f (k1 , k2 , k3 , k4 )} is the basis of the quotient
space Ω = Ω0 /I1 , where Ω0 is the universal enveloping algebra of the two-state Heisenberg–Weyl algebra and I1
project supported by National Natural Science Foundation of China (19905005), Major State Basic Research Development Program
(G2000077400 and G2000077604) and Tsinghua Natural Science Foundation (985 Program, 98JC079)
514
RUAN Dong, YUAN Jing, JIA Yu-Feng and SUN Hong-Zhou
generated by an element e − 1 is the left ideal associated
with Ω0 .
Three generators of any Rνλ (ν, λ fixed) satisfy the following closed nonlinear commutation relations
According to Ref. [3], one of nine square-root Lie algebras of vector type, denoted by R νλ (ν, λ = 0, ±1), is
generated by
Rνλ : {Jz , J 2 , T (νλ)} ,
(3)
respectively, where Jz is the z-component of an ordinary
angular momentum operator J [7] and T (νλ) (abbreviated notation of T (1νλ), ν, λ = 0, ±1) are the ladder
operators.[4,8] They, expressed by four elements {a+
i , ai }
(i = 1, 2) of the two-state Heisenberg–Weyl algebra, are[3]
1
a1 − a+
Jz = (a+
2 a2 ),
2 1
1
1
2
a1 + a+
J 2 = (a+
2 a2 + 1) − ;
4 1
4
+ +
+ +
T (11) = a1 a1 ,
T (10) = a1 a2 ,
+
T (1 − 1) = a+
2 a2 ,
T (−10) = a1 a2 ,
T (01) = a+
1 a2 ,
T (−11) = a2 a2 ,
T (−1 − 1) = a1 a1 ,
T (00) =
T (0 − 1) = a+
2 a1 .
1 +
(a a1 − a+
2 a2 ),
2 1
(4)
Vol. 35
[Jz , J 2 ] = 0,
[Jz , T (νλ)] = T (νλ)λ ,
p
[J 2 , T (νλ)] = T (νλ)(ν 2 + ν 1 + 4J 2 ) .
(5)
We note that (i) R00 in fact is an Abelian Lie algebra,
which has a one-dimensional representation; (ii) the nine
square-root Lie algebras Rνλ (ν, λ = 0, ±1) contain two
common generators Jz and J 2 so that the different properties of their representations result from their respective
remaining generators T (νλ) (ν, λ = 0, ±1).
3 Indecomposable Representations of Rνλ
and Their Boson Realizations
Acting three generators of Rνλ (ν, λ fixed) upon the
two-state Heisenberg–Weyl basis elements f (k1 , k2 , k3 , k4 )
defined by Eq. (2), and re-expressing the resultant basis
elements in terms of Eq. (1), we may obtain a representation of Rνλ . The representations of all generators of Rνλ
(ν, λ = 0, ±1) are listed in Table 1. The representations
ρ are indecomposable in k1 , k2 , k3 and k4 for R1λ , in k3 ,
k4 as well as in the sum k1 + k2 for R0λ , and in k3 and
k4 for R−1λ , respectively since the values for these indices
do not decrease.
Table 1 The representations ρ of Rνλ on the basis f (k1 , k2 , k3 , k4 ).
Xα
Jz
J2
T (11)
T (10)
T (1 − 1)
T (01)
T (00)
T (0 − 1)
T (−11)
T (−10)
T (−1 − 1)
ρ[Xα ]f (k1 , k2 , k3 , k4 )
1
[f (k1
2
1
[f (k1
4
+ 1, k2 , k3 + 1, k4 ) − f (k1 , k2 + 1, k3 , k4 + 1) + (k1 − k2 )f (k1 , k2 , k3 , k4 )]
+ 2, k2 , k3 + 2, k4 ) + 2f (k1 + 1, k2 + 1, k3 + 1, k4 + 1)
+ f (k1 , k2 + 2, k3 , k4 + 2) + (2k1 + 2k2 + 3)f (k1 + 1, k2 , k3 + 1, k4 )
+ (2k1 + 2k2 + 3)f (k1 , k2 + 1, k3 , k4 + 1) + (k1 + k2 + 2)(k1 + k2 )f (k1 , k2 , k3 , k4 )]
f (k1 + 2, k2 , k3 , k4 )
f (k1 + 1, k2 + 1, k3 , k4 )
f (k1 , k2 + 2, k3 , k4 )
f (k1 + 1, k2 , k3 , k4 + 1) + k2 f (k1 + 1, k2 − 1, k3 , k4 )
1
[f (k1 + 1, k2 , k3 + 1, k4 ) − f (k1 , k2 + 1, k3 , k4 + 1) + (k1 − k2 )f (k1 , k2 , k3 , k4 )]
2
f (k1 , k2 + 1, k3 + 1, k4 ) + k1 f (k1 − 1, k2 + 1, k3 , k4 )
f (k1 , k2 , k3 , k4 + 2) + 2k2 f (k1 , k2 − 1, k3 , k4 + 1) + k2 (k2 − 1)f (k1 , k2 − 2, k3 , k4 )
f (k1 , k2 , k3 + 1, k4 + 1) + k1 f (k1 − 1, k2 , k3 , k4 + 1) + k2 f (k1 , k2 − 1, k3 + 1, k4 )
+ k1 k2 f (k1 − 1, k2 − 1, k3 , k4 )
f (k1 , k2 , k3 + 2, k4 ) + 2k1 f (k1 − 1, k2 , k3 + 1, k4 ) + k1 (k1 − 1)f (k1 − 2, k2 , k3 , k4 )
Now let us construct the inhomogeneous boson realization of Rνλ from the indecomposable representation ρ.[6]
k0 ,k0 ,k0 ,k0
Due to the fact that the matrix elements ρ(Xα )k11 ,k22 ,k33 ,k44
(See Table 1) determined by the commutation relations of
Rνλ are related to four independent parameters k1 , k2 , k3
and k4 , we need four sets of independent boson operators
{a+
i , ai } (i = 1, 2, 3, 4) to define a Fock space F with the
basis
k1 + k 2
{|k1 , k2 , k3 , k4 i = (a+
1 ) (a2 )
k3 + k 4
× (a+
3 ) (a4 ) |0i | k1 , k2 , k3 , k4 ∈ ℵ} ,
(6)
where |0i stands for a “vacuum state” and ai |0i = 0. It follows that the corresponding Fock representations of Rνλ
may be obtained from the indecomposable representations
Indecomposable Representations of the Square-Root Lie Algebras of · · ·
No. 5
ρ by directly replacing f (k1 , k2 , k3 , k4 ) in Table 1 with
|k1 , k2 , k3 , k4 i since they have the same matrix elements
k0 ,k0 ,k0 ,k0
ρ(Xα )k11 ,k22 ,k33 ,k44 . With the help of Eqs (1) and (6), the
four-boson realizations of Rνλ may be derived from the
Fock representations as
1
+
a+ − a+
B[Jz ] = [a+
2 a4 + n1 − n2 ],
2 1 3
1
+ 2 + 2
2
B[J 2 ] = [(a+
)2 (a+
3 ) + (a2 ) (a4 )
4 1
+ + +
+
+
+ 2a+
1 a2 a3 a4 + (2n1 a1 + 2n2 + 1)a3
+
+ (2n2 a+
2 + 2n1 + 1)a4
+ (n1 + n2 )(n1 + n2 + 2)];
+
+
B[T (01)] = a+
1 a4 + a1 a2 ,
1 + +
+
[a a − a+
2 a4 + n1 − n2 ],
2 1 3
+
+
B[T (0 − 1)] = a+
2 a3 + a2 a1 ,
+
+
B[T (−11)] = a+
4 a4 + 2a4 a2 + a2 a2 ,
+
+
+
B[T (−10)] = a+
3 a4 + a4 a1 + a3 a2 + a1 a2 ,
+
+
B[T (−1 − 1)] = a+
3 a3 + 2a3 a1 + a1 a1 ,
B[Xα ]† 6= B[Xα ] for all Xα ; (ii) if the representation of
some generator is independent of the index ki (i.e., the
value of ki is invariant under the action of ρ), then the corresponding a±
i do not appear in the boson realization of
this generator. For example, the representations ρ[T (1λ)]
of T (1λ) (λ = 0, ±1) are independent of the indices k3 and
k4 (See Table 1) so that B[T (1λ)] have only two-boson realizations, which are the same as those in Eq. (4) and
hence not given in Eq. (7).
In order to realize Rνλ by using the less bosons, we
consider a quotient space Ω/I2 of Ω, where the left ideal
I2 associated with Ω is generated by two elements a1 −Λ1 ,
a2 −Λ2 (Λ1 , Λ2 are complex numbers). Thus on Ω/I2 with
the basis
{F (k1 , k2 ) ≡ f (k1 , k2 , 0, 0) mod I2 },
B[T (00)] =
(7)
where ni = a+
i ai (i = 1, 2) are the particle number
operators. There are two obvious properties: (i) the
boson realizations (7) of Rνλ are non-Hermitian since
Jz
J2
T (11)
T (10)
T (1 − 1)
T (01)
T (00)
T (0 − 1)
T (−11)
T (−10)
T (−1 − 1)
ρ̃[Xα ]F (k1 , k2 )
1
[Λ1 F (k1
2
1
[Λ21 F (k1
4
+ 1, k2 ) − Λ2 F (k1 , k2 + 1) + (k1 − k2 )F (k1 , k2 )]
+ 2, k2 ) + Λ22 F (k1 , k2 + 2) + Λ1 (2k1 + 2k2 + 3)F (k1 + 1, k2 )
+ Λ2 (2k1 + 2k2 + 3)F (k1 , k2 + 1) + (k1 + k2 )(k1 + k2 + 2)F (k1 , k2 )
+ 2Λ1 Λ2 F (k1 + 1, k2 + 1)]
F (k1 + 2, k2 )
F (k1 + 1, k2 + 1)
F (k1 , k2 + 2)
Λ2 F (k1 + 1, k2 ) + k2 F (k1 + 1, k2 − 1)
1
[Λ1 F (k1 + 1, k2 ) − Λ2 F (k1 , k2 + 1) + (k1 − k2 )F (k1 , k2 )]
2
Λ1 F (k1 , k2 + 1) + k1 F (k1 − 1, k2 + 1)
Λ22 F (k1 , k2 ) + 2k2 Λ2 F (k1 , k2 − 1) + k2 (k2 − 1)F (k1 , k2 − 2)
Λ1 Λ2 F (k1 , k2 ) + k1 Λ2 F (k1 − 1, k2 ) + k2 Λ1 F (k1 , k2 − 1) + k1 k2 F (k1 − 1, k2 − 1)
Λ21 F (k1 , k2 ) + 2k1 Λ1 F (k1 − 1, k2 ) + k1 (k1 − 1)F (k1 − 2, k2 )
In terms of Eqs (1) and (6), we may obtain the two-
+ Λ1 (2n1 a+
1 + 2n2 + 1)
boson realizations of Rµλ from the Fock representations
+ Λ2 (2n2 a+
2 + 2n1 + 1)
which correspond to the induced representations ρ̃,
+ (n1 + n2 )(n1 + n2 + 2)];
1
+
B̃[Jz ] = [Λ1 a+
1 − Λ2 a2 + n1 − n2 ],
2
1
+ +
2
2 + 2
B̃[J 2 ] = [Λ21 (a+
1 ) + Λ2 (a2 ) + 2Λ1 Λ2 a1 a2
4
(8)
we obtain the induced representations ρ̃ for Rµλ (µ,
λ = 0, ±1) from representations ρ, which are given in
Table 2. Moreover, for clarity, the action of the operators ρ̃[Xα ] onto the basis elements F (k1 , k2 ) is shown in
Fig. 1. If replacing the basisp
{F (k1 , k2 )} with a new basis |jmi = F (j + m, j − m)/ (j + m)!(j − m)! and taking Λ1 = Λ2 = 0, then the induced representations ρ̃
in Table 2 will become the standard angular momentum
representations.[3]
Table 2 The induced representations ρ̃ of Rνλ on the basis F (k1 , k2 ).
Xα
515
+
B̃[T (01)] = Λ2 a+
1 + a1 a2 ,
B̃[T (00)] =
1
+
[Λ1 a+
1 − Λ2 a2 + n1 − n2 ],
2
516
RUAN Dong, YUAN Jing, JIA Yu-Feng and SUN Hong-Zhou
+
B̃[T (0 − 1)] = Λ1 a+
2 + a2 a1 ,
B̃[T (−11)] = Λ22 + 2Λ2 a2 + a2 a2 ,
B̃[T (−10)] = Λ1 Λ2 + Λ2 a1 + Λ1 a2 + a1 a2 ,
B̃[T (−1 − 1)] = Λ21 + 2Λ1 a1 + a1 a1 .
(9)
k2
Vol. 35
We observe that the boson realizations given by Eq. (4)
are the special cases of Eq. (9) with Λ1 = Λ2 = 0. In
fact, equation (9) can also be obtained from Eq. (7) by
+
replacing a+
3 and a4 with Λ1 and Λ2 respectively. This is
+
not at all surprising since in Eq. (7) a+
3 and a4 are alone,
without their respective adjoint operators a3 and a4 .
k2
6
6
r d
6
r d
dr
ρ̃[Jz ]
ρ̃[J ]
6
ρ̃[Jz ]
6
2
ρ̃[J ]
r
dr
2
r
ρ̃[T (111)]
r
6
@r
@
I
@
ρ̃[T (10 − 1)]
ρ̃[T (101)]
r
d
r
d
?
?
r
ρ̃[T (11 − 1)]
ρ̃[T (110)]
r @
@
R
@
r
ρ̃[T (111)]
r
@
I
@
@r
@
R
@
dr
dr
dr
ρ̃[T (100)]
dr
ρ̃[T (1 − 1 − 1)]
ρ̃[T (1 − 1 − 1)]
ρ̃[T (1 − 10)]
ρ̃[T (1 − 10)]
?
?
ρ̃[T (1 − 11)]
ρ̃[T (1 − 11)]
0
dr
ρ̃[T (10 − 1)]
ρ̃[T (101)]
ρ̃[T (100)]
ρ̃[T (11 − 1)]
ρ̃[T (110)]
r
@
6
r d
-
k1
0
k1
(a)
(b)
Fig. 1 The basis {F (k1 , k2 )} of Ω/L2 is plotted in the form of the lattice, where each point (k1 , k2 ) of the lattice
denotes a basis vector F (k1 , k2 ). The action of all generators of nine square-root Lie algebras Rνλ (ν, λ = 0, ±1) upon
this basis is indicated by arrows and (or) circles. (a) The case for Λ1 , Λ2 6= 0. (b) The case for Λ1 = Λ2 = 0.
For R1±1 and R−1±1 , we even may realize them by
single boson provided that their respective representation
spaces are further reduced so that only one parameter remains in the basis. It can be seen from Fig. 1b that for
the special case of Λ1 = Λ2 = 0 this requirement may
be satisfied. Now, let us first consider R1±1 . Under the
action of representations ρ̃ of R1±1 , any point (k10 , k20 ) in
the lattice pattern of {F (k1 , k2 )} moves only along a ray
with its started point (k10 , k20 ) and arrives at all the points
(k1 , k2 ) of the lattice for k1 ≥ k10 or/and k2 ≥ k20 , and
these points form the invariant subspaces of Ω/I2 for the
given k10 , k20 , i.e.,
V11 (k10 , k20 ) : {η11 (k1 ) ≡ F (k1 , k20 )|k1 = k10 + 2γ ,
γ = 0, 1, 2, . . .} ,
V1−1 (k10 , k20 ) : {η1−1 (k2 ) ≡ F (k10 , k2 )|k2 = k20 + 2γ,
γ = 0, 1, 2, . . .}
(10)
for R11 , R1−1 respectively. Thus, the representations ρ̃
of R1±1 subduce the infinite-dimensional representations,
i.e.,
1
(k1 − k20 )η11 (k1 ) ,
2
1
ρ̄[J 2 ]η11 (k1 ) = (k1 + k20 )(k1 + k20 + 2)η11 (k1 ) ,
4
ρ̄[T (11)]η11 (k1 ) = η11 (k1 + 2)
(11)
ρ̄[Jz ]η11 (k1 ) =
Indecomposable Representations of the Square-Root Lie Algebras of · · ·
No. 5
1
(n + C2 )(n + C2 + 2) ,
4
B̄[T (−1 ± 1)] = aa
on V11 (k10 , k20 ) for R11 and
B̄[J 2 ] =
1 0
(k − k2 )η1−1 (k2 ) ,
2 1
1
ρ̄[J 2 ]η1−1 (k2 ) = (k10 + k2 )(k10 + k2 + 2)η1−1 (k2 ) ,
4
ρ̄[T (1 − 1)]η1−1 (k2 ) = η1−1 (k2 + 2)
(12)
ρ̄[Jz ]η1−1 (k2 ) =
on V1−1 (k10 , k20 ) for R1−1 . It follows from the Fock representations corresponding to Eqs (11) and (12) that we
may obtain the single-boson realizations of R1±1 as
1
B̄[Jz ] = ± (n − C1 ) ,
2
1
B̄[J 2 ] = (n + C1 )(n + C1 + 2) ,
4
B̄[T (1 ± 1)] = a+ a+ ,
(13)
where C1 is an arbitrary constant, and the subscripts of
ni and a±
i (i = 1 or 2) have been discarded.
Similar approach can be used to treat R−1±1 . Under the action of representations ρ̃ of R−1±1 , any point
(k10 , k20 ) in the lattice pattern of {F (k1 , k2 )} moves only
along a ray with its started point (k10 , k20 ) and arrives at
all the points (k1 , k2 ) of the lattice for k1 ≤ k10 or/and
k2 ≤ k20 , and these points form the corresponding finitedimensional invariant subspaces of Ω/I2 for the given k10
and k20 , i.e.,
V−11 (k20 )
:
{ξ−11 (k10 , k2 )
≡
F (k10 , k2 )|0
≤ k2 =
k20
− 2γ,
γ = 0, 1, 2, . . .},
V−1−1 (k10 )
:
{ξ−1−1 (k1 , k20 )
517
≡
F (k1 , k20 )|0
γ = 0, 1, 2, . . .} ,
≤ k1 =
k10
− 2γ,
(14)
with dimV−11 (k20 ) = [k20 /2]+1, dimV−1−1 (k10 ) = [k10 /2]+1
for R−11 , R−1−1 respectively. The representations ρ̃
of R−1±1 subduce the finite-dimensional representations,
i.e.,
1
ρ̄[Jz ]ξ−11 (k10 , k2 ) = (k10 − k2 )ξ−11 (k10 , k2 ) ,
2
1
2
0
ρ̄[J ]ξ−11 (k1 , k2 ) = (k10 + k2 )(k10 + k2 + 2)ξ−11 (k10 , k2 ),
4
ρ̄[T (−11)]ξ−11 (k10 , k2 ) = k2 (k2 − 1)ξ−11 (k10 , k2 − 2) (15)
on V−11 (k20 ) for R−11 and
1
(k1 − k20 )ξ−1−1 (k1 , k20 ) ,
2
1
ρ̄[J 2 ]ξ−1−1 (k1 , k20 ) = (k1 +k20 )(k1 +k20 +2)ξ−1−1 (k1 , k20 ) ,
4
ρ̄[T (−1−1)]ξ−1−1 (k1 , k20 ) = k1 (k1 −1)ξ−1−1 (k1 −2, k20 )(16)
ρ̄[Jz ]ξ−1−1 (k1 , k20 ) =
on V−1−1 (k10 ) for R−1−1 . From the Fock representations
corresponding to Eqs (15) and (16), we obtain the singleboson realizations of R−1±1 as
1
B̄[Jz ] = ∓ (n − C2 ) ,
2
(17)
with an arbitrary constant C2 .
4 Expectation Values of Rνλ in the Angular
Momentum Coherent States
It is well known that for the simplest but nontrivial
one-dimensional harmonic oscillator,[9] its coherent states
|zi defined as eigenstates of the annihilation operator
a, i.e., a|zi = z|zi (z is a complex number), may be
constructed out of the eigenstates |ni of the harmonicoscillator Hamiltonian H = h̄ω(a+ a+1/2) with the eigenvalues En = h̄ω(n + 1/2).
According to Glauber’s idea,[9] angular momentum
coherent states may be constructed from the angular
momentum eigenstates |jmi by means of two mutually
commuting lowering operators, say here T (−1 − 1) and
T (0 − 1), which, analogous to the annihilation operator a
for the harmonic oscillator, can decrease the values of j
and m respectively.[3] Making use of the matrix representations hj 0 m0 |Xα |jmi of Rνλ ,[3] the angular momentum
coherent states read
|z1 z2 i = √
j
∞ X
X
z1j−m z2j
1
p
|jmi, (18)
cosh χ j=0 m=−j (j + m)!(j − m)!
where z1 , z2 are complex numbers and χ = |z2 |(1 + |z1 |2 ).
Different from other angular momentum coherent states
(or called Bloch coherent states, atomic coherent states),
for example, |z; ji being the eigenstates of only J− pertaining to a given j,[10] the states |z1 z2 i given by Eq. (18) are
in fact the common eigenstates of T (−1 − 1) and T (0 − 1),
i.e.,
T (−1 − 1)|z1 z2 i = z2 |z1 z2 i ,
T (−10)|z1 z2 i = z1 z2 |z1 z2 i .
(19)
It is easy to show that over the entire complex planes of
z1 and z2 these states, |z1 z2 i, form an overcomplete set
similar to that for the harmonic oscillator,[9] i.e.,
ZZ 2
d z1 d2 z2 −χ
e |z1 z2 ihz1 z2 | = I ,
(20)
π
π
where I is the identity operator in the space of the representation j.
It follows by using the explicit expressions for
0 0
hj m |Xα |jmi that the expectation values hz1 z2 |Xα |z1 z2 i
of Rνλ in the angular momentum coherent states |z1 z2 i
may be obtained, which are given in Table 3. Introduce
z1 = e
iφ
tanh θ,
z2 = e
iψ
|z2 | ,
(21)
518
RUAN Dong, YUAN Jing, JIA Yu-Feng and SUN Hong-Zhou
where the parameters θ, φ are the co-latitude, longitude
of the expectation values of angular momentum J respectively, and ψ is related to the nodal angle between the
space-fixed and body-fixed coordinate systems. Thus, taking
1
h̄ → 0, χ → ∞ and h̄χ → Λ ,
(22)
2
where Λ is a real number, gives rise to the classical limits
of the expectation values of Rνλ , which may be found in
Table 3. These results may be used to study the classi-
Vol. 35
cal behavior of the quantum systems with the nonlinear
symmetries described by the square-root Lie algebras discussed in this paper and the quantum correlation effects.
Of course, for each of Rνλ , we can also construct the corresponding coherent states by making use of Gilmore’s
approach,[10] that is, of only one lowering or rising operator, T (νλ) (ν, λ = 0, ±1 but excluding ν = λ = 0). This
work, together with the application of the above results
in quantum computation,[11] is now under way.
Table 3 The expectation values of Rνλ in |z1 z2 i and the corresponding classical limits.
Xα
hz1 z2 |Xα |z1 z2 i
Jz
J2
T (11)
T (10)
T (1 − 1)
T (01)
T (00)
T (0 − 1)
T (−11)
T (−10)
T (−1 − 1)
1
χh̄(tanh χ)(1 − |z1 |2 )/(1
2
1 2 2
h̄ χ + 43 h̄2 χ2 tanh χ
4
h̄z2∗
h̄z1∗ z2∗
h̄(z1∗ )2 z2∗
Classical limits
+ |z1 |2 )
h̄(tanh χ)|z2 |z1
1
χh̄(tanh χ)(1 − |z1 |2 )/(1 + |z1 |2 )
2
h̄(tanh χ)|z2 |z1∗
h̄z12 z2
h̄z1 z2
h̄z2
Λ cos θ
Λ2
Λe−iψ (1 + cos θ)
Λe−i (φ+ψ) sin θ
Λe−i (2φ+ψ) (1 − cos θ)
Λh̄e iφ sin θ
Λ cos θ
Λe−iφ sin θ
Λe i (2φ+ψ) (1 − cos θ)
Λe i (φ+ψ) sin θ
Λe iψ (1 + cos θ)
Acknowledgments
The authors would like to thank Profs Q.Z. HAN and G.L. LONG for their helpful discussions.
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