i < j

657 Progress of Theoretical Physics, Vol. 100, No.3, September 1998

Coherent States of suq(n)-Covariant Oscillators W. -So

CHUNG

Theory Group, Department of Physics and Research Institute of Natural Science Gyeongsang National University, Jinju, 660-701, Korea (Received February 16, 1998) Two types of coherent states of sU q (n )-covariant oscillators are investigated.

Introduction

, Quantum groups and q-deformed Lie algebra imply some specific deformations of classical Lie algebras. From a mathematical point of view, quantum group is a non-commutative associative Hopf algebra. The structure and representation theory of quantum groups have been developed extensively by Jimbo 1) and Drinfeld. 2) The q-deformation of Heisenberg algebra was carried out by Arik and Coon, 3) Macfarlane,4) Biedenharn 5) and Chaichian et al. 6), 7) Recently there has been some interest in more general deformations involving arbitrary real functions of weight generators and including q-deformed algebras as a special case. 8) - 12) Recently, Spiridonov 13) found the new coherent states of the q- Weyl algebra aa t - qat a = 1, 0 < q < 1 which are defined as eigenstates of the operator at. The purpose of this paper is to find such a coherent state for the suq(n)-covariant oscillator algebra. The suq(n)-covariant differential calculus has been discussed by some physicists. 14) - 16)

§2. Standard realization of 8uq (n)-covariant oscillator algebra The sU q ( n )-covariant oscillator algebra 17) is defined as

(i < j)

(i < j)

! = 1 + qa!ai + (q -

ai a

m

1)

L

a!ak'

(1)

k=i+l

Throughout, ()t denotes the hermitian conjugate of O. Here, at is the hermitian conjugate of ai and q is assumed to be real. When the deformation parameter q' is not real, at is independent of ai, which implies that at cannot be interpreted as a hermitian conjugate of ai. The name suq(n)-covariant oscillator algebra is used because this algebra has the suq(n) symmetry among its step operators.

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§1.

w.

658

-So Chung

The Fock space representation of the algebra (1) can be easily constructed by introducing the hermitian number operators Ni obeying ( i J. = 1 2 ... n)

,

'"

(2)

Let 10,···,0) be the unique ground state of this system satisfying NIO t " ... 0) = 0 ,

ai 10, ... ,0) = 0

( i = 1 " 2 ... , n)

(3)

and let In) be the orthogonal number eigenstate, where we adopt the following notation:

(4) From the algebra (1), one can obtain the relation between the number operators and mode opeartors as

(5) where [xl is called a q-number and is defined as

[xl =

qX - 1.

q-l

Using the relation (5), we have the representation

ailn) = Jl:k=i+1nk[nilln - ei), a!ln) = J qL-k=i+ 1nk [ni where

In + ei)

and

In - ei)

+ llin + ei),

(6)

are defined as

In+ei) = In},···,ni+ 1,···,nn), In - ei) = In},···, ni - 1,···, nn). Then, the number eigenstate In) is obtained by applying the creation operators to the ground state 10,0,·· . ,0) successively: (7)

The coherent state for the suq(n)-covariant algebra is usually defined as

(8)


The number eigenstate In) satisfies the relation

Coherent States of suq(n)-Covariant Oscillators

659

From the sUq ( n )-covariant oscillator algebra, we obtain the following commutation relation between the Zi and z; (where z; is the complex conjugate of Zi): ZiZj

= qZjZi,

(i < j)

* * ZiZj

1 * Zi, = -2j

(i < j)

Zi* Zj

= qZjZi* ,

(i =f j)

z; Zi

=

q

(9)

ZiZ;,

Using these relations, the coherent state becomes

IZl,"', zn)- =

C(ZI,"',

zn) eXPq(zna~) ... eXPq(zlat)IO, ... , 0),

(11)

where 00

eXPq(x) = is a q-deformed exponential function. the relation

18)

n

~ [:]!

A q-deformed exponential function satisfies

eXPq(qx) = [1 - (1 - q)x] expix).

(12)

In order to obtain the normalized coherent state, we should impose the condition -(ZI,"', ZnIZl,"" zn)- = 1. Then, the normalized coherent state is given by

Iz"""

zn)

~

JeXPq(lzI12) 1... eXPq(lznl

2)

exp,(z"al) .. · eXPq(z, a1)10, ... , 0). (13)

§3.

Realization based on the q-hypergeometric series

In this section, we discuss another realization based on the hypergeometric series and obtain its corresponding coherent states. From the algebra (1), we know that there are n mutually commuting operators in this algbera. They read

Hi = a!ai, Hn = a~an -

(i

= 1 " 2 ... , n - 1) (14)

1/,

where 1/ = l~q and 0 < q < 1. Now, we regard Hi as sub-hamiltonians, and the full Hamiltonian is given by n

H =

L k=1

n

Hk =

L k=1

alak -

1/.

(15)


Using Eq. (7), we can rewrite Eq. (10) as

W. -So Chung

660

This hamiltonian has no classical analogue because 1/ is not defined when q is 1. From the algebra (1), we can easily check that [Hi, H j ] = O. Then the commutation relations between sub-hamiltonians and mode operators are given by

Hia} = qa}Hi'

(i < j)

Hia} = qa}Hi'

(i > j) n

HiaI

L

= qaIHi + (q - l)al

.

Hk ;

(16)

k=i+l

Acting with the sub-hamiltonian operators on the number eigenstate gives

where (18)

Then, the total energy of this system reads

Hln) = E(n)ln),

(19)

where

It is worth noting that the energy spectrum is degenerate and negative for o < q < 1. The degenerated states are split by the eigenvalues of the operator

Hi. As noted in Ref. 13), for a positive energy state, it is not ai but rather aI that plays the role of the lowering operator. If we introduce the positive energy eigenstate I(A, n)) as

the representation becomes

Hil(A, n)) = cAiqn 1 - Ai+1qni+1 )1(>" n)), Hnl(A, n)) = Anqnnl(A, n)),

(i

i- n)

i

aII(A, n))

= VAiqni+l - Ai+lqni+ 1 1(A, n + L ek)), k=l

ail (A, n)) =

VAiqni - Ai+lqn

i +1

i

1(A, n -

L

ek)),

(20)

k=l

where the Ai are arbitrary. This representation is a non-highest weight representation of the suq(n)-covariant oscillator algebra. Moreover, this representation has no


(17)

Coherent States of SU q (n) -Covariant Oscillators

661

classical analogue because it is not defined for q ~ 1. Due to this fact, it is natural to define a coherent state corresponding to the representation (20) as the eigenstates of

at :

(21 )

al,

Due to the noncommutativity of the the Zi should also be noncommuting variables. Instead, they satisfy the following commutation relations: 1 (i < j)

(i

=1=

j) (22)

00

L

(23)

nl,"',nn=-OO

where C(n) implies

C(n)

= C(nl,"', nn).

Inserting Eq. (23) into Eq. (21), we find the recurrence relation for coefficients

C(n) (

1

,;q

)ni-1-ni

JAiq-n

i +1

-

Ai+lqni+ 1

= C(n - ei),

(24)

where we have set

An+l =

-1/.

Solving the recurrence relation (24), we have

where the q-shifted factorial (x; q)n is defined as (x; q)n

= (1 - x)(l - qx)··· (1 - qn-l x ).

If we demand that +(zlz)+ = 1 ,we have 00

L

nl,n2,"',nn=-OO

(26)


Because the representation (20) depends on n free parameters Ai, the coherent state IZl, Z2,' " , zn)+ can take different forms. If we assume that the positive energy states are normalizable, i.e. ((A,n)I(A,n)) = bn,m = IIi=l bnimi , and form exactly one series for some fixed set of Ai, then we can write

w.

662

-So Chung

where we have used the identity

(aq-m j q)n = (_a)m q-m(mH)/2(q/aj q)m(a; q)n-m.

(27)

If we make the replacement ni - niH -+ li in Eq. (26) and use the identity (27) we can express the normalization constant C(O) in terms of the bilateral qhypergeometric series: }3)

(28)

where general bilateral q-hypergeometric series is defined by 19)

f=

n=-oo

(a}:q)n···(a r.;q)n((_)n qn(n-1)/2)S-r z n. (b}, q)n· .. (bs , q)n (29)

We can introduce the nonunitary displacement operator D(z}, Z2)

IZ1, Z2,···, zn)+ = D(z},···, zn)IA},···, An).

(30)

Then the operator D(Zl' ... , zn) is given by (31)

For some realizations of the sU q (n )-covariant oscillator algebra, the state (23) belongs to a continuous spectrum. Thus, it is appropriate to consider integrals over the Ai'S for the expansion of IZ}, Z2, ... , z,,)+ in the basis IA1' ... , An) instead of sums: (32) where C(A, Z) = C(A1,···, Anj Z},···, zn). Inserting this expression into Eq. (21) gives

(33)

where the di satisfy

t

d

k

= -l + InJ-(A}··· AI-d A/+ 1 .

k=l

Solving the above equation, we have

In q


ar );q,z) = as

Coherent States of suq(n)-Covariant Oscillators

§4.

663

Conclusion

To conclude, we have obtained two types of realizations of the suq(n)-covariant oscillator algebra. One is a standard realization based on the q-exponential function. This realization has a classical analogue when q goes to 1. It has a Fock representation, and the spectrum is bounded from below, which implies that there exists a ground state killed by the lowering operators (the ai). The second realization discussed in §3 is based on the q-hypergeometric series. This realization has no classical analogue, which means that it does not recover the classical oscillator algebra when q goes to 1. In addition it has a non-Fock representation, and the spectrum is unbounded, so there does not exist a ground state. In this representation, plays the role of a lowering operator. Thus, the two realizations have no internal relations, because when q goes to 1, the first realization, based on the q-exponential function, recovers the ordinary su( n)covariant oscillator algebra but the second realization, based on the q-hypergeometric series, does not for any value of q. In the second realization, we have found that the coherent state for the suq(n)covariant oscillator algebra is given by products of coherent states for the single mode q-oscillator algebra.

a!

The author wants to give many thanks to V. Spiridonov for helpful comments and advice. He also thanks the referees for their useful comments. This paper was supported by the KOSEF (981-0201-003-2) and the present studies were supported by the Basic Science Research Program, Ministry of Education, 1998 (BSRI-98-2413). References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19)

M. Jimbo, Lett. Math. Phys. 10 (1985),63; 11 (1986),247. V. Drinfeld, Pmc. Intern. Congress of Mathematicians (Berkeley, 1986), p. 78. M. Arik and D. Coon, J. Math. Phys. 17 (1976), 524. A. Macfarlane, J. of Phys. A22 (1989), 4581. L. Biedenharn, J. of Phys. A22 (1989), L873. M. Chaichian and P. Kulish, Phys. Lett. B234 (1990), 72. M. Chaichian, P. Kulish and J. Lukierski, Phys. Lett. B262 (1991), 43. A. Polychronakos, Mod. Phys. Lett. A5 (1990), 2325. M. Rocek, Phys. Lett. B225 (1991), 554. C. Daskaloyannis, J. of Phys. A24 (1991), L789. W. S. Chung, K. S. Chung, S. T. Narn and C. 1. Urn, Phys. Lett. A183 (1993), 363. W. S. Chung, J. Math. Phys. 35 (1994),3631. V. Spiridonov, Lett. Math. Phys. 35 (1995), 179. S. Woronowicz, Publ. RIMS Kyoto 2:3, No.1 (1987), 117. J. Wess and B. Zumino, Nucl. Phys. Proc. Suppl. 18B (1991), 302. Yu. 1. Manin, Commun. Math. Phys. 123 (1989), 163. W. Pusz and S. Woronowicz, Rep. Math. Phys. 27 (1989), 231. D. Fairlie and C. Zachos, Phys. Lett. B256 (1991), 43. G. Gasper and M. Rhaman, Basic Hypergeometric Series (Cambridge University Press, 1990).


Acknowledgements