Coherent states for SU„3…
Manu Mathura)
S. N. Bose National Centre for Basic Sciences, JD Block, Sector III, Salt Lake City,
Calcutta 700091, India
Diptiman Senb)
Centre for Theoretical Studies, Indian Institute of Science, Bangalore 560012, India
We define coherent states for SU共3兲 using six bosonic creation and annihilation
operators. These coherent states are explicitly characterized by six complex numbers with constraints. For the completely symmetric representations (n,0) and
(0,m), only three of the bosonic operators are required. For mixed representations
(n,m), all six operators are required. The coherent states provide a resolution of
identity, satisfy the continuity property, and possess a variety of group theoretic
properties. We introduce an explicit parametrization of the group SU共3兲 and the
corresponding integration measure. Finally, we discuss the path integral formalism
for a problem in which the Hamiltonian is a function of SU共3兲 operators at each
site.
I. INTRODUCTION
Coherent states have been used for a long time in different areas of physics.1,2 In the area of
quantum optics, coherent states based on the Heisenberg–Weyl group 共which are described later in
this work兲 have been extensively used to study the interaction of a single mode of electromagnetic
radiation with a two-level atomic system 共for instance, the Jaynes–Cummings model兲.3 Coherent
states based on the noncompact Lie group SU共1,1兲 have also been used to study certain problems
in quantum optics.4 In condensed matter physics, coherent states for the Lie group SU共2兲 have
been very useful for studying Heisenberg spin systems using the path integral formalism.5– 8 These
studies have been generalized to systems with SU(N) symmetry, although such studies have
usually been restricted to the completely symmetric representations.6,9 However, there is a recent
discussion of coherent states for arbitrary irreducible representations of SU共3兲 in Ref. 10. The
purpose of our work is to discuss a coherent state formalism which is valid for all representations
of SU共3兲, and to give an explicit characterization of them in terms of complex numbers and the
states of some harmonic oscillators. 共Our work differs in this respect from Ref. 10 which does not
use harmonic oscillator operators to define the basis states.兲 As we will see, this way of characterization is very similar to those used for the Heisenberg–Weyl and SU共2兲 coherent states. But,
there are certain additional features 共such as tracelessness兲 which are redundant in the simpler case
of SU共2兲.
One can imagine various possible applications of coherent states for SU共3兲. In quantum
optics, coherent states for SU共3兲 may turn out to be useful for studying the interaction of a
three-level atomic system with three modes of electromagnetic radiation 共corresponding to the
three possible energy differences of the atom兲. We should also mention that there have been many
other studies of SU共3兲 in the recent mathematical physics literature, including the geometric phase
for three-level systems11 and the study of Clebsch–Gordon coefficients and the outer multiplicity
problem.12 These studies do not use coherent states; however, our work is likely to shed new light
on some of these studies. For instance, we will use two triplets of complex numbers z and w which
a兲
Electronic mail: manu@boson.bose.res.in
Electronic mail: diptiman@cts.iisc.ernet.in
b兲
are similar to the ones used in Ref. 12, except that we will normalize the triplets to unity.
Similarly, it is well known that the geometric phases in the different representations of SU共2兲 may
be obtained by integrating around a closed loop the overlap of two coherent states which differ
infinitesimally from each other.7,8 In the same way, it should be possible to derive the geometric
phases for SU共3兲 representations from the coherent states discussed below.
The organization of the article is as follows. Section II will motivate our ideas and techniques
using two examples which are simpler than the SU共3兲 group. We start with the standard group
theoretical definitions of the coherent states of the Heisenberg–Weyl and SU共2兲 groups. We then
discuss another way of defining SU共2兲 coherent states using the Schwinger or Holstein–Primakoff
representation of the Lie algebra of SU共2兲13 in terms of harmonic oscillator creation and annihilation operators. This definition is discussed in some detail as it can be extended to the SU共3兲
group. We then establish its equivalence with the standard group theoretical coherent state
definition.2 In Sec. III, we generalize the SU共2兲 Lie algebra in terms of harmonic oscillators to the
SU共3兲 group, and construct the irreducible representations of SU共3兲. We describe the structure of
SU共3兲 matrices in an explicit way, and provide an integration measure for this eight-dimensional
manifold. In Sec. IV, we use this group structure to construct a set of SU共3兲 coherent states which
are explicitly characterized by a set of complex numbers which are equivalent to eight real
variables. We prove various identities expected for coherent states such as the resolution of
identity and a transformation from a particular coherent state to the general coherent state. In Sec.
V, we provide an alternative set of coherent states for SU共3兲 which require only five real variables;
although these share some of the features of the coherent states defined in Sec. IV, they have a few
limitations arising from the smaller number of variables used. In Sec. VI, we discuss how coherent
states can be used to develop a path integral formalism for problems involving SU共3兲 variables.
II. HEISENBERG–WEYL AND SU„2… COHERENT STATES
There are many definitions of coherent states used in the literature. However, the most essential ingredients common in all these definitions are the continuity and completeness properties.1
共1兲 These are states in a Hilbert space H associated which are characterized by a set of continuous
variables 兵 zជ 其 , and the coherent states 兩 zជ 典 are strongly continuous functions of the labels 兵 zជ 其 .
共2兲 There exists a positive measure d ␮ (zជ ) such that the unit operator I admits the resolution of
identity
I⫽
冕
d ␮ 共 zជ 兲 兩 zជ 典具 zជ 兩 .
共1兲
Given a group G, the coherent states in a given representation R are functions of q parameters
denoted by 兵 z 1 ,z 2 ,...,z q 其 , and are defined as
兩 zជ 典 ⬅T R 共 g 共 zជ 兲兲 兩 0 典 R .
共2兲
Here T R (g(zជ )) is a group element in the representation R, and 兩 0 典 R is a fixed vector belonging to
R. In the simplest example of the Heisenberg–Weyl group, the Lie algebra contains three generators. It is defined in terms of creation annihilation operators (a,a † ) satisfying
关 a,a † 兴 ⫽I,
关 a,I兴 ⫽0,
关 a † ,I兴 ⫽0.
共3兲
This algebra has only one infinite dimensional irreducible representation which can be characterized by occupation number states 兩 n 典 ⬅(a † ) n / 冑n! 兩 0 典 with n⫽0,1,2, . . . . A generic group element in 共2兲 can be characterized by T(g)⫽exp(i␣I⫹za † ⫺z̄a) with an angle ␣ and a complex
parameter z. Therefore,
⬁
兩 ␣ ,z 典 ⬁ ⫽exp共 i ␣ 兲 兩 z 典 ,
兩 z 典 ⫽exp共 za † ⫺z̄a 兲 兩 0 典 ⫽
兺
n⫽0
F n共 z 兲兩 n 典 ,
共4兲
where the sum runs over all the basis vectors of the infinite dimensional representation, and
F n共 z 兲 ⫽
zn
冑n!
exp共 ⫺ 兩 z 兩 2 /2兲
共5兲
are the coherent state expansion coefficients. This feature, i.e., an expansion of the coherent states
in terms of basis vectors of a given representation with analytic functions of complex variables
(F n (z)) as coefficients, will also be present in the case of SU共2兲 and SU共3兲 groups. It is easy to
see that Eq. 共4兲 provides a resolution of identity as in 共1兲 with the measure d ␮ (z)⫽dzdz̄.
We now briefly review the next simplest example, i.e., the coherent states associated with the
SU共2兲 group. The SU共2兲 Lie algebra is given by a set of three angular momentum operators 兵 Jជ 其
⬅ 兵 J 1 ,J 2 ,J 3 其 or equivalently by 兵 J ⫹ ,J ⫺ ,J 3 其 , (J ⫾ ⬅J 1 ⫾iJ 2 ) satisfying
关 J 3 ,J ⫾ 兴 ⫽⫾J ⫾ ,
关 J ⫹ ,J ⫺ 兴 ⫽2J 3 .
共6兲
The SU共2兲 group has a Casimir operator given by Jជ •Jជ , and the different irreducible representations are characterized by its eigenvalues j( j⫹1), where j is an integer or half-odd-integer. A
given basis vector in representation j is labeled by the eigenvalue m of J 3 as 兩 j,m 典 . We characterize the SU共2兲 group elements U by the Euler angles, i.e., U( ␪ , ␾ , ␺ )
⬅exp (i␾J3) exp(i␪J2) exp(i␺J3). The standard group theoretical definition 共2兲 takes 兩 0 典 j in 共2兲 to
be the highest weight state 兩 j, j 典 and is of the form
兩 n̂ 共 ␪ , ␾ 兲 典 j ⫽U 共 ␪ , ␾ , ␺ 兲 兩 j, j 典 ,
⫹j
⫽
兺
m⫽⫺ j
C m 共 ␪ , ␾ 兲 兩 j,m 典 ,
共7兲
In 共7兲, the coefficients C m ( ␪ , ␾ ) are given by
C m 共 ␪ , ␾ 兲 ⫽e im ␾
冑
冋 册 冋 册
␪
共 2 j 兲!
sin
2
共 j⫹m 兲 ! 共 j⫺m 兲 !
j⫺m
␪
cos
2
j⫹m
,
共8兲
where we have ignored possible phase factors.
The algebra in Eq. 共6兲 can be realized in terms of a doublet of harmonic oscillator creation and
annihilation operators aជ ⬅(a 1 ,a 2 ) and aជ † ⬅(a †1 ,a †2 ), respectively.13 They satisfy the simpler
bosonic commutation relation 关 a i ,a †j 兴 ⫽ ␦ i j with i, j⫽1,2. The vacuum state is 兩 0,0典 . In terms of
these operators,
J a ⬅ 12 a †i 共 ␴ a 兲 i j a j ,
共9兲
where ␴ a denote the Pauli matrices. 共We will generally use the convention that repeated indices
are summed over兲. It is easy to check that the operators in 共9兲 satisfy the SU共2兲 Lie algebra with
the Casimir Jជ •Jជ ⬅ 41 aជ † •aជ (aជ † •aជ ⫹2). Thus the representations of SU共2兲 can be characterized by the
eigenvalues of the occupation number operator; the spin value j is equal to (N 1 ⫹N 2 )/2 where N 1
and N 2 are the eigenvalues of a †1 a 1 and a †2 a 2 , respectively.
With these harmonic oscillator creation and annihilation operators, another definition of SU共2兲
coherent states is obtained by directly generalizing 共4兲. We define a doublet of complex numbers
(z 1 ,z 2 ) with the constraint 兩 z 1 兩 2 ⫹ 兩 z 2 兩 2 ⫽1; this gives three independent real parameters which
define the sphere S 3 . Let us parameterize
z 1 ⫽cos ␹ e i ␤ 1
and z 2 ⫽sin ␹ e i ␤ 2 ,
共10兲
where 0⭐ ␹ ⭐ ␲ /2 and 0⭐ ␤ 1 , ␤ 2 ⬍2 ␲ . The integration measure on this space takes the form
d⍀ S 3 ⫽
1
␲
2
dz 1 dz̄ 1 dz 2 dz̄ 2 ␦ 共 兩 z 1 兩 2 ⫹ 兩 z 2 兩 2 ⫺1 兲 ⫽
1
2␲2
cos ␹ sin ␹ d ␹ d ␤ 1 d ␤ 2 ,
共11兲
where we have introduced a normalization factor so that 兰 d⍀ S 3 ⫽1. The SU共2兲 coherent state in
the representation N is now defined as
兩 z 1 ,z 2 典 N⫽2 j ⫽ ␦ aជ † •aជ ,N 冑N!exp共 zជ •aជ † 兲 兩 0,0典 ⫽
兺
N 1 ,N 2
⬘ F N 1 ,N 2 兩 N 1 ,N 2 典 j .
共12兲
In the second equation in 共12兲, the 兺 ⬘ implies that only the terms satisfying the constraint a † •a
⫽N⬅2 j are included or equivalently that
N 1 ⫹N 2 ⫽N.
共13兲
With 共13兲, the states 兩 N 1 ,N 2 典 j form a (2 j⫹1)-dimensional representation of SU共2兲. The expansion coefficients F N 1 ,N 2 (z 1 ,z 2 ) are analytic functions of (z 1 ,z 2 ) and are given by
F N 1 ,N 2 ⬅
冉
N!
N 1 !N 2 !
冊
1/2
N
N
共14兲
z 1 1z 2 2 .
Equations 共12兲 and 共14兲 are similar to 共4兲 and 共5兲, respectively. This will be generalized to the
SU共3兲 case in Sec. III. It is easy to check that 共12兲 provides the resolution of identity with the
measure given in 共11兲, namely,
冕
1
d⍀ SU(2) 兩 z 1 ,z 2 典 j j 具 z 1 ,z 2 兩 ⫽
2 j⫹1
j
兺
m⫽⫺ j
兩 j,m 典具 j,m 兩 .
共15兲
Now we change variables from N 1 and N 2 ⫽2 j⫺N 1 to m⫽ 21 (N 1 ⫺N 2 ), and define
␻⬅
␪
z1
⫽e i ␾ cot .
z2
2
共16兲
These parameters are related to the ones given in 共10兲 as ␪ ⫽2 ␹ and ␾ ⫽ ␤ 1 ⫺ ␤ 2 . We now
consider an unit sphere S 2 with its south pole touching the point ␻ ⫽0. The sphere is characterized
by ( ␪ , ␾ ) where ␪ and ␾ are the polar and azimuthal angles, respectively. Using the stereographic
projection, it is easy to verify that
兺 冑共 j⫹m 兲 ! 共 j⫺m 兲 ! 共 ␻ 兲 (m⫺ j)兩 j,m 典 ⫽ 兩 n̂ 共 ␪ , ␾ 兲 典 j ,
m⫽⫺ j
j
兩 z 1 ,z 2 典 j ⫽ 共 z 1 兲
2j
共 2 j 兲!
共17兲
where we have again ignored possible phase factors. Equation 共17兲 can also be written as
兩 z 1 ,z 2 典 j ⫽ 共 z 1 兲 2 j exp
冉 冊
z2
J 兩 z ⫽1, z 2 ⫽0 典 j ,
z1 ⫺ 1
共18兲
where 兩 z 1 ⫽1, z 2 ⫽0 典 N⫽2 j ⫽ 兩 j, j 典 and we have used the fact that J ⫺ ⫽a †2 a 1 . Equations 共17兲 and
共18兲 establish the equivalence between the group theoretical theoretical definition 共7兲 and the one
using Schwinger bosons 共12兲.
The stationary subgroup of a particular coherent state is defined as the subgroup H of the full
group G which leaves that coherent state invariant up to a phase; the coherent states are functions
of the coset space G/H. 2 It is clear from the previous discussion that the stationary subgroup of
the SU共2兲 coherent states is U共1兲; therefore the coherent states correspond to the coset space
SU(2)/U(1)⫽S 2 which is parametrized by the angles ( ␪ , ␾ ).
III. SU„3… AND ITS REPRESENTATIONS
Let us first discuss a parametrization of SU共3兲 matrices, i.e., 3⫻3 unitary matrices with unit
determinant. To motivate this, let us first consider a parametrization of SO共3兲 matrices. Consider
a real vector of unit length of the form
冉 冊
sin ␪ cos␾
pជ ⫽ sin ␪ sin ␾ .
cos ␪
共19兲
The most general real vector q of unit length which is orthogonal to p is given by
冉
冊
cos ␹ cos ␪ cos ␾ ⫹sin ␹ sin ␾
ជq ⫽ cos ␹ cos ␪ sin ␾ ⫺sin ␹ cos ␾ .
⫺cos ␹ sin ␪
共20兲
Finally, we define a third unit vector rជ ⫽ pជ ⫻qជ , i.e., r 1 ⫽ p 2 q 3 ⫺ p 3 q 2 , etc. Then a 3⫻3 matrix
whose columns are given by the vectors p,q and r is an SO共3兲 matrix.
We will now generalize the previous construction to obtain an SU共3兲 matrix. A complex vector
of unit norm is given by
zជ ⫽
冉
sin ␪ cos ␾ e i ␣ 1
sin ␪ sin ␾ e i ␣ 2
cos ␪ e
i␣3
冊
共21兲
,
where 0⭐ ␪ , ␾ ⭐ ␲ /2 and 0⭐ ␣ 1 , ␣ 2 , ␣ 3 ⬍2 ␲ . Then the integration measure for zជ , which is
equivalent to the sphere S 5 , is given by
d⍀ S 5 ⫽
⫽
2
␲3
1
␲3
dz 1 dz̄ 1 dz 2 dz̄ 2 dz 3 dz̄ 3 ␦ 共 兩 z 1 兩 2 ⫹ 兩 z 2 兩 2 ⫹ 兩 z 3 兩 2 ⫺1 兲
sin3 ␪ cos ␪ cos ␾ sin ␾ d ␪ d ␾ d ␣ 1 d ␣ 2 d ␣ 3 ,
共22兲
ជ of unit norm
which has been normalized to make 兰 d⍀ S 5 ⫽1. The most general complex vector w
ជ ⫽0 is given by
satisfying zជ •w
ជ⫽
w
冉
e i( ␤ 1 ⫺ ␣ 1 ) cos ␹ cos ␪ cos ␾ ⫹e i( ␤ 2 ⫺ ␣ 1 ) sin ␹ sin ␾
e i( ␤ 1 ⫺ ␣ 2 ) cos ␹ cos ␪ sin ␾ ⫺e i( ␤ 2 ⫺ ␣ 2 ) sin ␹ cos␾
⫺e
i( ␤ 1 ⫺ ␣ 3 )
cos ␹ sin ␪
冊
,
共23兲
where 0⭐ ␹ ⭐ ␲ /2 and 0⭐ ␤ 1 , ␤ 2 ⬍2 ␲ just as in the integration measure for S 3 in 共11兲. We may
ជ , where z̄ជ ⬅zជ쐓 . Then we can check that
now define a third complex vector of unit norm as vជ ⫽z̄ជ ⫻w
a 3⫻3 matrix whose columns are given by z, w̄ and v , i.e.,
S⫽
冉
z1
w̄ 1
z̄ 2 w 3 ⫺z̄ 3 w 2
z2
w̄ 2
z̄ 3 w 1 ⫺z̄ 1 w 3
z3
w̄ 3
z̄ 1 w 2 ⫺z̄ 2 w 1
冊
共24兲
is an SU共3兲 matrix.
The integration measure for the group SU共3兲 is given by a product of 共22兲 and 共11兲 as14,15
d⍀ SU(3) ⫽
1
2␲5
sin3 ␪ cos ␪ cos ␾ sin ␾ cos ␹ sin ␹ d ␪ d ␾ d ␹ d ␣ 1 d ␣ 2 d ␣ 3 d ␤ 1 d ␤ 2 ,
共25兲
which is normalized so that 兰 d⍀ SU(3) ⫽1. To prove Eq. 共25兲, we note that the matrix in 共24兲 can
be written as a product of two SU共3兲 matrices, i.e., S⫽A 3 A 2 , where
A 3⫽
and
冉
冉
sin ␪ cos ␾ e i ␣ 1
cos ␪ cos ␾ e i ␣ 1
⫺sin ␾ e ⫺i ␣ 2 ⫺i ␣ 3
sin ␪ sin ␾ e i ␣ 2
cos ␪ sin ␾ e i ␣ 2
cos ␾ e ⫺i ␣ 1 ⫺i ␣ 2
cos ␪ e
i␣3
1
A 2⫽ 0
0
⫺sin ␪ e
i␣3
0
0
cos ␹ e
⫺sin ␹ e
冊
0
⫺i ␤ 1
sin ␹ e
⫺i ␤ 2 ⫹i ␣ 1 ⫹i ␣ 2 ⫹1 ␣ 3
i ␤ 2 ⫺i ␣ 1 ⫺i ␣ 2 ⫺i ␣ 3
cos ␹ e
i␤1
共26兲
,
冊
.
共27兲
The structure of the matrix A 3 is determined entirely by the three-dimensional complex vector
which forms its first column; hence the integration measure corresponding to it is given by 共22兲.
The matrix A 2 is determined by the two-dimensional complex vector which forms its second
column; its contribution to the integration measure is therefore given by 共11兲. Note that although
the parameter appearing in A 2 is ␤ 2 ⫺ ␣ 1 ⫺ ␣ 2 ⫺ ␣ 2 instead of only ␤ 2 as in 共10兲, this makes no
difference in the product measure given in 共25兲 since the differentials d ␣ i already appear in the
integration measure coming from A 3 . Incidentally, this procedure generalizes to any SU(N); the
integration measure is given by a product of measures for S 2N⫺1 , S 2N⫺3 , . . . , S 3 . 14
ជ ⫽(w 1 ,w 2 ,w 3 ) in 共21兲
In short, we have defined two complex vectors zជ ⫽(z 1 ,z 2 ,z 3 ) and w
and 共23兲. These satisfy the constraints
z̄ជ •zជ ⫽ 兩 z 1 兩 2 ⫹ 兩 z 2 兩 2 ⫹ 兩 z 3 兩 2 ⫽1,
ជ •w
ជ ⫽ 兩 w 1 兩 2 ⫹ 兩 w 2 兩 2 ⫹ 兩 w 3 兩 2 ⫽1,
w̄
共28兲
and
ជ ⫽z 1 w 1 ⫹z 2 w 2 ⫹z 3 w 3 ⫽0.
zជ •w
共29兲
ជ
These constraints leave eight real degrees of freedom as required for SU共3兲. We will take zជ and w
쐓
to transform respectively as the 3 and 3 representation of SU共3兲. Thus an SU共3兲 transformation
acts on the matrix S in Eq. 共24兲 by multiplication from the left.
Let us now define two triplets of harmonic oscillator creation and annihilation operators
(a i ,b i ), i⫽1,2,3, satisfying
关 a i ,a †j 兴 ⫽ ␦ i j ,
关 a i ,b j 兴 ⫽0,
关 b i ,b †j 兴 ⫽ ␦ i j ,
关 a i ,b †j 兴 ⫽0.
共30兲
We will often denote these two triplets by (aជ ,bជ ) and the two number operators by N a (⬅aជ † •aជ ) and
N b (⬅bជ † •bជ ). Similarly, their vacuum state is denoted by 兩 0ជ a ,0ជ b 典 . Henceforth, we will ignore the
subscripts a,b and will denote the vacuum state by 兩 0ជ ,0ជ 典 , and the eigenvalues of N a , N b by N and
M, respectively.
Now let ␭ a , a⫽1,2, . . . ,8 be the generators of SU共3兲 in the fundamental representation; they
satisfy the SU共3兲 Lie algebra 关 ␭ a ,␭ b 兴 ⫽i f abc ␭ c . Let us define the following operators
Q a ⫽a † ␭ a a⫺b † ␭ * a b,
共31兲
a
where a † ␭ a a⬅a †i ␭ ai j a j , and b † ␭ * a b⬅b †i ␭ *
i j b j . To be explicit,
Q 3 ⫽ 21 共 a †1 a 1 ⫺a †2 a 2 ⫺b †1 b 1 ⫹b †2 b 2 兲 ,
Q 8⫽
1
2 冑3
共 a †1 a 1 ⫹a †2 a 2 ⫺2a †3 a 3 ⫺b †1 b 1 ⫺b †2 b 2 ⫹2b †3 b 3 兲 ,
Q 1 ⫽ 21 共 a †1 a 2 ⫹a †2 a 1 ⫺b †1 b 2 ⫺b †2 b 1 兲 ,
i
Q 2 ⫽⫺ 共 a †1 a 2 ⫺a †2 a 1 ⫹b †1 b 2 ⫺b †2 b 1 兲 ,
2
共32兲
Q 4 ⫽ 21 共 a †1 a 3 ⫹a †3 a 1 ⫺b †1 b 3 ⫺b †3 b 1 兲 ,
i
Q 5 ⫽⫺ 共 a †1 a 3 ⫺a †3 a 1 ⫹b †1 b 3 ⫺b †3 b 1 兲 ,
2
Q 6 ⫽ 21 共 a †2 a 3 ⫹a †3 a 2 ⫺b †2 b 3 ⫺b †3 b 2 兲 ,
i
Q 7 ⫽⫺ 共 a †2 a 3 ⫺a †3 a 2 ⫹b †2 b 3 ⫺b †3 b 2 兲 .
2
It can be checked that these operators satisfy the SU共3兲 algebra among themselves, i.e.,
关 Q a ,Q b 兴 ⫽i f abc Q c . Further,
关 Q a ,a †i 兴 ⫽␭ aji a †j ,
关 Q a ,b †i 兴 ⫽⫺␭ *ji a b †j ,
关 Q a ,a † •a 兴 ⫽0,
关 Q a ,b † •b 兴 ⫽0,
关 Q a ,a † •b † 兴 ⫽0,
关 Q a ,a•b 兴 ⫽0.
共33兲
From Eqs. 共33兲, it is clear that the three states a †i 兩 0ជ ,0ជ 典 with (N⫽1, M ⫽0) and b †i 兩 0ជ ,0ជ 典 with
(N⫽0, M ⫽1) transform respectively as the fundamental representation 共3兲 and its conjugate
representation (3 쐓 ). By taking the direct product of N aជ † ’s and M bជ † ’s we can now form higher
representations. We now define an operator
i i . . . iN
⬅a i† a i† .
1 2
1 2 . . . jM
O j1 j2
. . a i† b †j b †j . . . b †j .
N
1
2
M
共34兲
i i ...i
Under SU共3兲 transformation the states defined as 兩 ˜␺ 典 (N,M ) ⬅O j1 j2 . . . jN 兩 0ជ ,0ជ 典 will all have N a
1 2
M
⫽N and N b ⫽M , and will transform among themselves. Further, 兩 ˜␺ 典 ⫽N 兩 ˜␺ 典 and N b 兩 ˜␺ 典
⫽M 兩 ˜␺ 典 . However, these do not form an irreducible representation because aជ •bជ and aជ † •bជ † are
SU共3兲 invariant operators 关see 共33兲兴. A general basis vector in the irreducible representation
(N,M ) is obtained by subtracting the traces and completely symmetrizing in upper and lower
indices.16 More explicitly, a state in (N,M ) representation is given by
i ,i , . . . i
冋
N
i i . . . iN
⫹L 1
1 2 . . . jM
兩 ␺ 典 j1 , j2 , . . . ,Nj ⬅ O j1 j2
1
2
M
兺 兺
兺
兺
i i ..i
k1
i
i
i
k1
i i ..i
k2
兺
兺
..i l ⫺1 i l ⫹1 ..i N
2
2
i
i
i
i i ..i
k1
k2
k3
i i ..i
M
兺
i
..i l ⫺1 i l ⫹1 ..i l ⫺1 i l ⫹1 ..i N
2
2
3
3
k 1 ⫺1 k 1 ⫹1 .. j k 2 ⫺1 j k 2 ⫹1 .. j k 3 ⫺1 j k 3 ⫹1 . . . j M
1 2
兺
i
l 1 ,l 2 ,l 3 ,..,l Q ⫽1 k 1 ,k 2 ,k 3 ,..,k Q ⫽1
⫻O j1 j2 .. jl 1 ⫺1 lj1 ⫹1
i
␦ jl 1 ␦ jl 2 ␦ jl 3 O j1 j2 .. jl 1 ⫺1 lj1 ⫹1
N
⫹ . . . ⫹L Q
i
k 1 ⫺1 k 1 ⫹1 .. j k 2 ⫺1 j k 2 ⫹1 . . . j M
1 2
M
l 1 ,l 2 ,l 3 ⫽1 k 1 ,k 2 ,k 3 ⫽1
i
i
␦ jl 1 ␦ jl 2 .. ␦ jl Q
k1
..i l ⫺1 i l ⫹1 ..i l ⫺1 i l ⫹1 ..i N
2
2
Q
Q
k 1 ⫺1 k 1 ⫹1 .. j k 2 ⫺1 j k 2 ⫹1 .. j k Q ⫺1 j k Q ⫹1 . . . j M
1 2
..i N
k 1 ⫺1 k 1 ⫹1 . . . j M
1 2
␦ jl 1 ␦ jl 2 O j1 j2 .. jl 1 ⫺1 lj1 ⫹1
l 1 ,l 2 ⫽1 k 1 ,k 2 ⫽1
N
⫹L 3
i
␦ jl 1 O j1 j2 .. jl 1 ⫺1 lj1 ⫹1
l 1 ⫽1 k 1 ⫽1
M
N
⫹L 2
M
册
k2
kQ
兩 0ជ ,0ជ 典 ,
共35兲
where Q⫽Min(N,M ),
L q⬅
共 ⫺1 兲 q 共 a † •b † 兲 q
,
q! 共 N⫹M ⫹1 兲共 N⫹M 兲共 N⫹M ⫺1 兲 ••• 共 N⫹M ⫹2⫺q 兲
共36兲
and all the sums in 共35兲 are over different indices, i.e., l 1 ⫽l 2 •••⫽l q and k 1 ⫽k 2 ⫽•••⫽k q . The
coefficients in Eq. 共36兲 are chosen to satisfy the tracelessness condition
3
i
i ,i , . . . i
␦ j 兩␺典 j , j , . . . , j
兺
i , j ⫽1
l
k
l
1
2
k
1
2
N
M
⫽0,
for all l⫽1,2, . . . N,
and k⫽1,2, . . . M .
共37兲
For future purposes, a more compact notation for describing all the states given above is to write
i i . . . iN
⬅ 共 a †1 兲 N 1 共 a †2 兲 N 2 共 a †3 兲 N 3 共 b †1 兲 M 1 共 b †2 兲 M 2 共 b †3 兲 M 3 ,
1 2 . . . jM
共38兲
O j1 j2
where (N i ,M i ) denote all the possible eigenvalues of the occupation number operators
(a †i a i ,b †i b i ) satisfying
N 1 ⫹N 2 ⫹N 3 ⫽N
and M 1 ⫹M 2 ⫹M 3 ⫽M .
共39兲
The action of 共38兲 on the vacuum is given by
N N N3 ជ ជ
N N N
兩 0 ,0 典 ⫽ 共 N 1 !N 2 !N 3 !M 1 !M 2 !M 3 ! 兲 1/2兩 M1 M2 M3 典 .
1 2M 3
1 2 3
共40兲
O M1 M2
We can now write the basis vectors of the representation (N,M ) as
i ,i , . . . i
N N N3
⫽
1 2M 3
兩 ␺ 典 j1 , j2 , . . . ,Nj ⬅ 兩 ␺ 典 M1 M2
1
2
M
冋
Q
N N N3
⫹
1 2M 3
O M1 M2
兺
q⫽1
Lq
兺
ជ]
[␣
q
册
N1
M
C ␣1N2C ␣2N3C ␣ 1C ␣1M 2C ␣2
3
N ⫺␣ N ⫺␣2N3⫺␣3 ជ ជ
兩 0 ,0 典 .
1
1 2⫺␣2M 3⫺␣3
⫻ M 3 C ␣ 3 ␣ 1 ! ␣ 2 ! ␣ 3 !O M1 ⫺ ␣1 M2
共41兲
ជ 兴 q denotes the sets of three non-negative integers ( ␣ 1 , ␣ 2 , ␣ 3 ) satisfying ␣ 1
In this equation, 关 ␣
⫹ ␣ 2 ⫹ ␣ 3 ⫽q, and N i ⫺ ␣ i ⭓0, M i ⫺ ␣ i ⭓0 for i⫽1,2,3. The 兺 [ ␣ជ ] q denotes a summation over all
sets of three such integers. In the notation of Eq. 共41兲, the tracelessness condition 共37兲 for the
(N⫹1, M ⫹1) representation takes the form
兺
[ ␥ជ ]
N ⫹␥ N ⫹␥2N3⫹␥3
⫽0.
1
1 2⫹␥2M 3⫹␥3
兩 ␺ 典 M1 ⫹ ␥1 M2
共42兲
1
The definition in 共41兲 satisfies the condition given in 共42兲. This can be verified by using the
identity
兺兺
[ ␥ជ ] [ ␣ជ ]
1
␣ 1! ␣ 2! ␣ 3! N1⫹␥1C ␣1N2⫹␥2C ␣2N3⫹␥3C ␣3M 1⫹␥1C ␣1M 2⫹␥2C ␣2M 3⫹␥3C ␣3
q
N ⫹␥ ⫺␣ N ⫹␥2⫺␣2N3⫹␥3⫺␣3
1
1
1 2⫹␥2⫺␣2M 3⫹␥3⫺␣3
⫻O M1 ⫹ ␥1 ⫺ ␣1 M2
冋
⫽ 共 N⫹M ⫹2⫺q 兲
兺
[ ␣ជ ] q⫺1
⫹ 共 aជ † •bជ † 兲
兺
[ ␣ជ ] q
册
N ⫺␣ N ⫺␣2N3⫺␣3
.
1
1 2⫺␣2M 3⫺␣3
⫻ ␣ 1 ! ␣ 2 ! ␣ 3 ! N 1 C ␣ 1 N 2 C ␣ 2 N 3 C ␣ 3 M 1 C ␣ 1 M 2 C ␣ 2 M 3 C ␣ 3 O M1 ⫺ ␣1 M2
共43兲
The dimension D(N,M ) of the representation (N,M ) can be obtained as follows. For the
(N,0) representation, no tracelessness condition needs to be imposed, and the dimension is simply
given by the number of states in Eq. 共40兲 which satisfy 兺 i N i ⫽N and 兺 i M i ⫽0. This gives
D(N,0)⫽(N⫹1)(N⫹2)/2. Similarly, D(0,M )⫽(M ⫹1)(M ⫹2)/2. Now D(N,M ) is given by
the number of states satisfying 兺 i N i ⫽N, 兺 i M i ⫽M , which is equal to the product
D(N,0)D(0,M ), minus the number of states satisfying 兺 i N i ⫽N⫺1, 兺 i M i ⫽M ⫺1, which is equal
to D(N⫺1,0)D(0,M ⫺1); the subtraction is because of the tracelessness condition. This gives
D 共 N,M 兲 ⫽ 21 共 N⫹1 兲共 M ⫹1 兲共 N⫹M ⫹2 兲 .
共44兲
IV. SU„3… COHERENT STATES
We now observe that the states in Eq. 共35兲 can be extracted from the following generating
function,
ជ 典 (N,M ) ⬅ 冑N!M ! exp共 zជ •aជ † ⫹w
ជ •bជ † 兲 兩 0ជ ,0ជ 典 ,
兩 zជ ,w
共45兲
where we have to project onto the subspace of states with aជ † •aជ ⫽N and bជ † •bជ ⫽M to obtain the
representation (N,M ). More explicitly,
ជ 典 (N,M ) ⫽
兩 zជ ,w
ជ •bជ † 兲 M
共 zជ •aជ † 兲 N 共 w
冑N!
冑M !
兩 0ជ ,0ជ 典 ⫽
兺⬘
N 1 ,N 2 ,N 3
兺⬘
N N N3
典.
1 2M 3
F Nជ ,Mជ 共 z 1 ,z 2 ,z 3 ;w 1 ,w 2 ,w 3 兲 兩 M1 M2
M 1 ,M 2 ,M 3
共46兲
ជ ) are
In 共46兲, 兺 ⬘ implies that the occupation numbers (N i ,M i ) satisfy Eq. 共39兲, and F Nជ ,Mជ (zជ ,w
given by
ជ 兲⫽
F Nជ ,Mជ 共 zជ ,w
冉
N!M !
N 1 !N 2 !N 3 !M 1 !M 2 !M 3 !
冊
1/2
N
N
N
M
M
M
z 1 1z 2 2z 3 3w 1 1w 2 2w 3 3 .
共47兲
N
N
N
M
M
M
On expanding the right hand side of 共46兲, the coefficients of z 1 1 z 2 2 z 3 3 w 1 1 w 2 2 w 3 3 give the basis
vectors of SU共3兲 in the representation (N,M ). It is important to note that the tracelessness conditions in Eq. 共35兲 are automatically satisfied by the state in 共46兲. This is because we can always
N N N
N N N
replace 兩 M1 M2 M3 典 by the SU共3兲 basis vectors 兩 ␺ 典 M1 M2 M3 defined in 共41兲.
1
2
3
1
2
3
It is instructive to consider a specific example here. The coherent state of the representation
(1,1), i.e., the adjoint representation of SU共3兲, is given by
3
ជ 典 (1,1) ⫽
兩 zជ ,w
兺
i, j⫽1
z i w j a †i b †j 兩 0ជ ,0ជ 典 .
共48兲
100
010
001
We then see that the sum of the coefficients of the three states 兩 100
典 , 兩 010
典 and 兩 001
典 is zero due to
the constraint in Eq. 共29兲. Hence there are only eight linearly independent states on the right hand
side of Eq. 共48兲 as there should be; these eight states can be taken to be
兩 V 1典 ⫽
1
冑2
100
010
共 兩 100
典 ⫺ 兩 010
典 ),
兩 V 2典 ⫽
1
冑6
100
010
001
共 兩 100
典 ⫹ 兩 010
典 ⫺2 兩 001
典 ),
100
010
兩 V 3 典 ⫽ 兩 010
典 , 兩 V 4 典 ⫽ 兩 100
典,
100
兩 V 5 典 ⫽ 兩 001
典,
共49兲
001
兩 V 6 典 ⫽ 兩 100
典,
010
001
兩 V 7 典 ⫽ 兩 001
典 , 兩 V 8 典 ⫽ 兩 010
典.
The states defined in Eq. 共46兲 will be called the coherent state of the representation (N,M ).
Note that Eqs. 共39兲, 共46兲, and 共47兲 are analogous to the corresponding SU共2兲 equations 共13兲, 共12兲,
and 共14兲 respectively. The SU共3兲 coherent states 共46兲 are normalized to unity, i.e.,
ជ ជ ជ ជ
(N,M ) 具 z ,w 兩 z ,w 典 (N,M ) ⫽1.
共50兲
To prove this, we use the operator identities
e A e B ⫽e B e A e [A,B]
and e A Be ⫺A ⫽B⫹ 关 A,B 兴 ,
共51兲
which hold if 关 A,B 兴 commutes with both A and B. We find that
具 0ជ ,0ជ 兩 exp关 z̄ជ •aជ ⫹w̄ជ •bជ 兴 exp关 zជ •aជ † ⫹wជ •bជ † 兴 兩 0ជ ,0ជ 典 ⫽exp关 z̄ជ •zជ ⫹w̄ជ •wជ 兴 .
共52兲
ជ •w
ជ ) M on both sides of this equation and using the definiOn comparing terms of order (z̄ជ •zជ ) N (w̄
tion in 共46兲, we obtain Eq. 共50兲. In the same way, we can show that
ជ ជ ជ
ជ ជ
ជ
(N,M ) 具 z ,w 兩 z ⫹dz ,w ⫹dw 典 (N,M ) ⫽1⫹N
兺i z̄ i dz i ⫹M 兺i w̄ i dw i ,
共53兲
ជ denote small deviations from zជ and w
ជ . This equation will be used to derive the
where dzជ and dw
path integral formalism6,7 in Sec. V, and it would also be useful for obtaining the geometric phase
for systems with SU共3兲 symmetry.11
We can prove that the states defined in Eq. 共46兲 satisfy the resolution of identity, i.e.,
冕
ជ 典 (N,M )(N,M ) 具 zជ ,w
ជ 兩⫽
d⍀ 兩 zជ ,w
1
D 共 N,M 兲
D(N,M )
兺
i⫽1
兩 V i 典具 V i 兩 ,
共54兲
where V i denotes a set of orthonormal basis vectors of (N,M ). 关See Eq. 共49兲 for the explicit
example of the representation 共1,1兲.兴 To verify the normalization on the right-hand side of Eq.
N00
共54兲, it is convenient to look at a particular basis vector 兩 0M
0 典 . 关This has the maximum eigenvalue
(N⫹M )/2 of the operator Q 3 given in Eq. 共32兲.兴 From Eq. 共46兲, the coefficient of this vector in the
coherent state is given by z N1 w 2M . Integrating the modulus squared of this using Eqs. 共21兲–共25兲, we
obtain the factor of 1/D(N,M ) in Eq. 共54兲. This is as it should be so that taking the trace of both
sides of 共54兲 gives unity.
A second property of coherent states is that they are overcomplete. This is clear for the states
ជ ), while the dimension
in 共46兲 since they are continuous functions of the complex variables (zជ ,w
of the representation (N,M ) is finite.
The coherent states in 共46兲 have a third property which is group theoretical, and is analogous
to Eq. 共18兲 for the SU共2兲 coherent states. Namely, we can go from a particular coherent state, say,
N00
兩 z 1 ⫽1,w 2 ⫽1 典 (N,M ) ⫽ 兩 0M
0 典 to the general coherent state 兩 z,w 典 (N,M ) by acting with an exponential
of certain combinations of the SU共3兲 generators Q a . First of all, we can check that
冋
兩 z,w 典 (N,M ) ⫽z N1 w 2M exp
册
z2 †
z3
w1 †
w3 †
a 2 a 1 ⫹ a †3 a 1 ⫹
b 1b 2⫹
b b 兩 z ⫽1,w 2 ⫽1 典 (N,M ) .
z1
z1
w2
w2 3 1 1
共55兲
Then we can use Eq. 共51兲 and the constraint 共29兲 to rewrite this in the form10
冋
兩 z,w 典 (N,M ) ⫽z N1 w 2M exp
册
z3
w3 6
z2 1
共 Q ⫺iQ 2 兲 ⫹ 共 Q 4 ⫺iQ 5 兲 ⫺
共 Q ⫹iQ 7 兲 兩 z 1 ⫽1,w 2 ⫽1 典 (N,M ) ,
z1
z1
w2
共56兲
which is similar in structure to Eq. 共18兲.
Another property of these coherent states which is important for their path integral applications is that the expectation value of the SU共3兲 operators 共32兲 in a coherent state should be given
ជ ) and their complex conjugates. We find that
by an SU共3兲 covariant function of (zជ ,w
ជ ជ
(N,M ) 具 z ,w 兩 Q
a
a
ជ 典 (N,M ) ⫽Nz̄ i ␭ ai j z j ⫺M w̄ i ␭ *
兩 zជ ,w
ij wj .
共57兲
This can be proved by using the identities in Eq. 共51兲 to show that
具 0ជ ,0ជ 兩 exp关 z̄ជ •aជ ⫹w̄ជ •bជ 兴 a †i a j exp关 zជ •aជ † ⫹wជ •bជ † 兴 兩 0ជ ,0ជ 典 ⫽z̄ i z j exp关 z̄ជ •zជ ⫹w̄ជ •wជ 兴 ,
共58兲
and a similar identity for the expectation value of b †i b j in terms of w̄ i w j . Equation 共57兲 can now
be obtained by comparing terms of order z̄ N z N w̄ M w M on the two sides of Eq. 共58兲.
The stationary subgroup of the coherent states defined in this section is generally U共1兲⫻U共1兲,
ជ by independent phase factors. These coherent
corresponding to multiplying the vectors zជ and w
states are therefore functions of the coset space SU共3兲/U共1兲⫻U共1兲.10 However, for the completely
symmetric representations (N,0) and (0,M ), the coherent states use only three complex numbers
ជ ) which define the space SU(3)/SU(2)⬃S 5 ; the stationary subgroup is then U(1) which
(zជ or w
corresponds to multiplying that complex vector by a phase factor. In those cases, the coherent
states are functions of the coset space SU共3兲/SU共2兲⫻U共1兲.
V. AN ALTERNATIVE DEFINITION OF SU„3… COHERENT STATES
The SU共3兲 coherent states discussed in Sec. IV involve eight real parameters, and satisfy some
simple group theoretic properties similar to the SU共2兲 coherent states of Sec. II. It is possible that
there may be some applications of coherent states which do not require so many parameters. In
this section, we will discuss an alternative kind of coherent state which only requires five real
parameters. We will see later that these coherent states suffer from some problems and they seem
to lack some of the group theoretic properties precisely because they use fewer parameters.
We observe that the states in 共35兲 can be extracted from the following generating function:
冋
兩 zជ ,z̄ជ 典 ⬅exp共 zជ •aជ † 兲 exp共 z̄ជ •bជ † 兲 1⫹
册
Q
兺
q⫽1
L q 兩 0ជ ,0ជ 典 ,
共59兲
and we have to project onto the subspace of states with aជ † •aជ ⫽N and bជ † •bជ ⫽M to obtain the
representation (N,M ). To be explicit,
冋
册
Q
共 zជ •aជ † 兲 N 共 z̄ជ •bជ † 兲 M
共 zជ •aជ † 兲 N⫺q 共 z̄ជ •bជ † 兲 M ⫺q
⫹
兩 zជ ,z̄ជ 典 (N,M ) ⫽
Lq
兩 0ជ ,0ជ 典 .
N!
M!
共 N⫺q 兲 ! 共 M ⫺q 兲 !
q⫽1
兺
共60兲
On expanding the right hand side of 共60兲, the coefficients of the tensors z i 1 z i 2 . . . z i N z̄ j 1 z̄ j 2 . . . z̄ j M
give the basis vectors of SU共3兲 in the representation (N,M ).
The SU共3兲 coherent states in the representation (N,M ) are defined as in Eq. 共60兲,
兩 zជ ,z̄ជ 典 (N,M ) ⬅
⫽
1
N!M !
兺 兺
,i , . . . j , j , . . .
i1 2
1
兺
2
N
i i . . . iN
1 2 . . . jM
z i 1 z i 2 . . . z i N z̄ j 1 z̄ j 2 . . . z̄ j M 兩 ␺ 典 j1 j2
N
N
M
M
M3
z 1 1 z 2 2 z 3 3 z̄ 1 1 z̄ 2 2 z̄ 3
兺
N 1 !N 2 !N 3 !M 1 !M 2 !M 3 !
N 1 ,N 2 ,N 3 M 1 ,M 2 ,M 3
N N N3
.
1 2M 3
兩 ␺ 典 M1 M2
共61兲
To give a specific example, the coherent state of the representation (1,1) is given by
兩 zជ ,z̄ជ 典 (1,1) ⫽
3
兺
i, j⫽1
3
z i z̄ j a †i b †j 兩 0ជ ,0ជ 典 ⫺
1
a † b † 兩 0ជ ,0ជ 典 .
3 i⫽1 i i
兺
共62兲
We will now prove that the states defined in 共61兲 satisfy the resolution of identity,
冕
d⍀ S 5 兩 zជ ,z̄ជ 典 (N,M )(N,M ) 具 zជ ,z̄ជ 兩 ⫽1.
共63兲
To prove this, we use the definition 共41兲 and the integration measure for zជ given in 共22兲. We find
that
冕
冉 冉
3
共 N ⫹M ⫹ ␦ 兲 !
i
i
i
兺
兺
兿
N
⫹
␦
!
M
⫹
兲
共
共
N ,M
␦
i⫽1
i
i
i ␦i兲!
d⍀ S 5 兩 z,z̄ 典具 z,z̄ 兩 ⫽C
i
i
i
冊
N ⫹␦ N ⫹␦2N3⫹␦3
1
1 2⫹␦2M 3⫹␦3
兩 ␺ 典 M1 ⫹ ␦1 M2
冊
N1N2N3
␺兩,
M 1M 2M 3具
共64兲
where the ␦ i are integers satisfying
3
兺 ␦ i ⫽0,
i⫽1
共65兲
and the constant C is determined shortly. We now use the following property,
冉
3
共 N ⫹M ⫹ ␦ 兲 !
兺␦ i⫽1
兿 共 N i ⫹i␦ i 兲 ! 共iM i ⫹i ␦ i 兲 !
i
冊
N ⫹␦ N ⫹␦2N3⫹␦3
N N N
⫽ 兩 ␺ 典 M1 M2 M3 ,
1
1 2⫹␦2M 3⫹␦3
1 2 3
兩 ␺ 典 M1 ⫹ ␦1 M2
共66兲
which is a consequence of Eq. 共37兲 for the basis vectors of a representation of SU共3兲. Thus Eq.
共64兲 can be simplified to
冕
兺
N ,M
d⍀ S 5 兩 z,z̄ 典具 z,z̄ 兩 ⫽C
i
N N N3 N1N2N3
具␺兩.
1 2M 3M 1M 2M 3
兩 ␺ 典 M1 M2
i
共67兲
The normalization constant C in Eq. 共67兲 can be fixed by looking at one particular basis vector of
the representation (N,M ), say,
N00
兩 ␺ 典 0M
0.
共68兲
From Eq. 共61兲, the coefficient of this vector in the coherent state 兩 zជ ,z̄ជ 典 is z N1 z̄ 2M /(N!M !). Integrating this as in 共22兲, we find that
C⫽
2
.
N!M ! 共 N⫹M ⫹2 兲 !
共69兲
Finally, let us consider the analog of the property given in Eq. 共57兲 for the (z,w) coherent
states. We can prove that
ជ ជ
(N,M ) 具 z ,z̄ 兩 Q
a
兩 zជ ,z̄ជ 典 (N,M ) ⫽ 共 N⫺M 兲 z̄ i ␭ ai j z j .
共70兲
To prove this, we use the identities in 共51兲 to show that
具 0ជ ,0ជ 兩 exp关 z̄ជ •aជ ⫹zជ •bជ 兴 a †i a j exp关 zជ •aជ † ⫹z̄ជ •bជ † 兴 兩 0ជ ,0ជ 典 ⫽z̄ i z j exp关 2z̄ជ •zជ 兴 .
共71兲
On expanding this equation and comparing terms which are of order N in both z i and z̄ i , we find
that the expectation value of Q a in the representation (N,0) satisfies Eq. 共70兲. In a similar way, we
can prove Eq. 共70兲 in the representation (0,M ). Finally, we can generalize the proof to the
representation (N,M ) by using Eq. 共33兲; since Q a commutes with aជ •bជ and aជ † •bជ † , it commutes
with the operators L q which are require to enforce tracelessness in Eq. 共35兲.
Note that 共70兲 vanishes for the self-conjugate representations in which N⫽M . There is a
similar problem for the differential change in overlap analogous to Eq. 共53兲. We find that the
coherent states defined in this section satisfy
具 zជ ,z̄ជ 兩 zជ ⫹dzជ ,z̄ជ ⫹dz̄ជ 典
具 zជ ,z̄ជ 兩 zជ ,z̄ជ 典
⫽1⫹N
兺i z̄ i dz i ⫹M 兺i dz̄ i z i
共72兲
in the representation (N,M ). The left hand side of this equation is equal to 1 if N⫽M due to the
constraint 兺 i z̄ i z i ⫽1. These two problems imply that the (z,z̄) coherent states are unlikely to be
useful for path integral applications in the representations with N⫽M .
For the (z,z̄) coherent states, we have not yet found the construction of the group theoretical
property analogous to 共56兲 in the general representation (N,M ). This would be an interesting topic
for future studies.
The stationary subgroup of the coherent states defined in this section is U(1)⫽S 1 , corresponding to multiplying zជ by a phase factor. These coherent states are therefore functions of the
manifold S 5 /S 1 .
VI. PATH INTEGRAL FORMALISM
We will now use the (z,w) coherent states presented in Sec. IV to derive the path integral for
a problem which has SU共3兲 variables in some representation (N,M ). 共For convenience, we will
drop the subscript (N,M ) on the coherent states in this section.兲 We begin by discussing a problem
involving the Hamiltonian of a single site with a SU共3兲 variable. For any Hamiltonian which is a
function of the SU共3兲 operators Q a , we define its coherent state expectation value to be
E 共 z,z̄,w,w̄ 兲 ⬅ 具 z,w 兩 Ĥ 兩 z,w 典 .
共73兲
If the Hamiltonian is linear in the SU共3兲 operators, i.e.,
8
Ĥ⫽
兺
a⫽1
共74兲
c aQ a,
then Eq. 共73兲 can be found using Eq. 共57兲. But if the Hamiltonian is not linear in the SU共3兲
operators, then Eq. 共73兲 has to be evaluated separately.
Let us now consider the propagator in imaginary time
G 共 z (F) ,w (F) ,z (I) ,w (I) ;T 兲 ⫽ 具 z (F) ,w (F) 兩 exp共 ⫺TĤ 兲 兩 z (I) ,w (I) 典 ,
共75兲
where the superscripts I and F denote initial and final states, respectively, and we are suppressing
the subscripts i (⫽1,2,3) on z and w for the moment. We write the exponential in 共75兲 as a product
of N terms, and use the resolution of identity in 共54兲 to insert a complete set of states between
each pair of terms. A typical term looks like
具 z (n⫹1) ,w (n⫹1) 兩 exp共 ⫺ ⑀ Ĥ 兲 兩 z (n) ,w (n) 典 ,
共76兲
where ⑀ ⫽T/N. We are eventually interested in taking the limit N→⬁ holding T fixed. In that
(n⫹1)
case, we may assume that (z (n⫹1) ,w (n⫹1) ) is close to (z (n) ,w (n) ) in 共76兲, so that dz (n)
i ⫽z i
(n)
(n)
(n⫹1)
(n)
⫺z i and dw i ⫽w i
⫺w i are small. Using Eqs. 共53兲 and 共73兲, we can write 共76兲 as
具 z (n⫹1) ,w (n⫹1) 兩 exp共 ⫺ ⑀ Ĥ 兲 兩 z (n) ,w (n) 典
冋
⫽exp N
(n)
(n)
(n)
(n) (n)
,z̄ ,w (n) ,w̄ (n) 兲
兺i z̄ (n)
i dz i ⫹M 兺 w̄ i dw i ⫺ ⑀ E 共 z
i
册
共77兲
to first order in ⑀ , dz (n)
and dw (n)
i
i . In the limit ⑀ ⫽d ␶ →0, we can write the propagator in 共75兲 in
the path integral form
G 共 z (F) ,w (F) ,z (I) ,w (I) ;T 兲 ⫽
冕
D⍀ SU(3) 共 ␶ 兲 exp共 ⫺S 关 z,w 兴 兲 ,
where
S 关 z,w 兴 ⫽
冕
T
0
冋
d ␶ ⫺N
dz
dw
兺i z̄ i d ␶i ⫺M 兺i w̄ i d ␶ i ⫹E 共 z,z̄,w,w̄ 兲
册
,
共78兲
and
D⍀ SU(3) 共 ␶ 兲 ⬅
兿n d⍀ SU(3)共 n 兲 ,
and (z,w) are functions of ␶ which satisfy the boundary conditions (z(0),w(0))⫽(z (I) ,w (I) ) and
(z(T),w(T))⫽(z (F) ,w (F) ). Note that we have written the functional integral measure in 共78兲 in
terms of the measure given in Eq. 共25兲. Alternatively, we can write the functional integral measure
in terms of DzDz̄DwDw̄ if we introduce appropriate Lagrange multiplier fields in the action S to
enforce the constraints in Eqs. 共28兲 and 共29兲 at each time ␶ .
We can now generalize the above construction to a problem involving several sites which are
labeled by a parameter x, provided that the Hamiltonian is linear in the SU共3兲 variables at each
site. We introduce a coherent state at each site, and write the energy functional as
E 关 z,z̄,w,w̄ 兴 ⫽ 具 z,w 兩 Ĥ 兩 z,w 典 ,
where 兩 z,w 典 ⬅
兿x 兩 z 共 x 兲 ,w 共 x 兲 典 .
共79兲
Then we can show that
具 z (F) 共 x 兲 ,w (F) 共 x 兲 兩 exp共 ⫺TĤ 兲 兩 z (I) 共 x 兲 ,w (I) 共 x 兲 典 ⫽
S 关 z,w 兴 ⫽
冕
T
0
冋 兺再
d␶ ⫺
N
x
兺i z̄ i共 x 兲
dz i 共 x 兲
⫺M
d␶
D⍀ SU(3) 共 x, ␶ 兲 ⬅
冕
D⍀ SU(3) 共 x, ␶ 兲 exp共 ⫺S 关 z,w 兴 兲 ,
兺i w̄ i共 x 兲
冎
册
dw i 共 x 兲
⫹E 关 z,z̄,w,w̄ 兴 ,
d␶
共80兲
d⍀ SU(3) 共 x,n 兲 .
兿
x,n
Note that the first two terms in the actions S given in Eqs. 共78兲 and 共80兲 are purely imaginary due
to the constraints in 共28兲. To show this explicitly, we can rewrite those terms as
1
兺i z̄ i dz i ⫽ 2 兺i 共 z̄ i dz i ⫺dz̄ i z i 兲 ,
兺i
1
w̄ i dw i ⫽
2
共81兲
兺i 共 w̄ i dw i ⫺dw̄ i w i 兲 .
As an example of a problem to which this formalism can be applied, we can consider the
SU共3兲 invariant Hamiltonian
Ĥ⫽
J x,y 兺 Q a 共 x 兲 Q a 共 y 兲 .
兺
x,y
a
共82兲
This is called the SU共3兲 Heisenberg model. It has been discussed extensively in the literature for
the completely symmetric representations (N,0); 6 for those representations, we can use the simpler measure d⍀ S 5 given in Eq. 共22兲 instead of d⍀ SU(3) . Our construction of coherent states now
allows a study of the Heisenberg model in any representation (N,M ).
VII. SUMMARY AND DISCUSSION
In this article we have exploited the representation of the SU共3兲 Lie algebra in terms of six
harmonic oscillator creation and annihilation operators to generate all the representations of
SU共3兲. This harmonic oscillator form of the algebra enables us to define the SU共3兲 coherent states
in terms of two triplets of complex numbers. In this sense the SU共2兲 definition 共12兲 and SU共3兲
definition 共45兲 are analogous to that of the Heisenberg–Weyl coherent states 共4兲. The SU共3兲
coherent states are characterized by two triplets of complex numbers with four real constraints.
This explicit construction in terms of complex numbers can be used to derive the geometrical
phase of SU共3兲. Further, the path integral formalism discussed in the previous section can be used
to obtain the field theory for the SU共3兲 Heisenberg model and study its topological aspects as in
the SU共2兲 case.17 Work in this direction is in progress and will be reported elsewhere.
For any group G, we can use a certain number of harmonic oscillator operators to construct
the group operators as in Eqs. 共32兲 and 共33兲. If we can find the appropriate set of complex numbers
which transform according to that group and satisfy the necessary constraints, we can use our
method to provide an explicit complex number parametrization of the corresponding coherent
states.
ACKNOWLEDGMENTS
We 共MM and DS兲 would like to thank N. Mukunda and H. S. Sharatchandra for some
discussions. MM would like to thank Samir Paul, Debashish Gangopadhyay, and Ranjan
Choudhary for discussions on the SU共2兲 coherent states.
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S. Sachdev, in Low Dimensional Quantum Field Theories for Condensed Matter Physicists, edited by Y. Lu, S.
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13
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F. D. M. Haldane, Phys. Rev. Lett. 61, 1029 共1988兲.
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