multiboson expansions for the q-oscillator and su(1,1)q

ELSEVIER

9 May 1994

Physics Letters A 188 (1994) 1-10

PHYSICS'LETTERS A

Multiboson expansions for the q-oscillator and su ( 1,1 ) q

Angel Ballesteros, Javier Negro Departamento de Ftsica Te6rica, Universidad de Valladolid, 47011 Valladolid, Spain

Received 4 October 1993; revised manuscript received 28 February 1994; accepted for publication 1 March 1994 Communicated by J.P. Vigier

Abstract

All the Hermitian representations of the "symmetric" q-oscillator are obtained by means of expansions. The same technique is applied to characterize in a systematic way the k-order boson realizations of the q-osciilator and su (1,1)q. The special role played by the quadratic realizations of su ( 1,1 )¢ in terms of boson and q-boson operators is analysed and clarified.

It is well known that a way - in some sense opposite to contractions - to relate a pair of non-isomorphic algebras are the expansion transformations [ I ]. The underlying idea of this method is that the generators of the first algebra can be, in principle, expressed as functions of the second ones. Only in some special cases these functions are included in the enveloping algebra of the latter (i.e., they are polynomials) but, 'in general, expan- sions often contain roots, rational functions or formal power series of operators for which clear commutation rules as well as conditions on their representation spaces must be given. A way to avoid many problems in this respect is to work just with one representation - where expansions are well defined - and obtain one precise representation of the final algebra. The general validity of the so-derived relations can be studied a posteriori.

This method has been successfully applied in the representation theory of quantum algebras [2,3 ] and is known in this context as the "deforming functional" approach [ 4-10 ]. This paper explores the possibilities of the procedure described in Refs. [9,10] in order to build k- order expansions relating su( l, 1 )q and the q-oscil- lator (with q being always a real positive parameter) . As a by-product, the classical relations between represen- tations of these algebras are obtained in the q--, 1 limit.

Section l begins with an application of the expansion method that illustrates its deep connection with repre- sentations. We ask for nontrivial expansions of the q-oscillator algebra into itself. We will see that the solutions to this problem correspond to the non-equivalent irreducible representations of this quantum algebra. These realizations are explicitly computed, and it is shown that expansions "intertwine" all of them.

In Section 2 we compute the expansions of the q-oscillator Os( 1 )q in terms of k-order powers of classical boson operators {N, a, a ÷ } (the so-called kth order generalized q-boson operators [11 ] ). The coherent and squeezed states of this algebra have been recently calculated [ 1 l - 14 ] and, in their non-deformed version [ 15 ], these operators have been shown to describe non-classical multiphoton states in quantum optics [ 16 ]. The new representations of Os( 1 )q found in Section 1 play a central role and allow us to classify the solutions in their

E-mail: fteorica@cpd.uva.es.

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2 A. Ballesteros, J. Negro ~Physics Letters A 188 (1994) 1-10

most geheral form. In particular, we find that the original representation space of the classical oscillator supports k independent irreducible representations of Os ( 1 )q, which are explicitly given.

Section 3 carries out the same program for realizations ofsu( 1,1 )q in terms of k-order boson operators. There are several remarkable results: Firstly, we get also k independent representations in a similar way to the oscillator case. Secondly, it is possible to obtain in this way any discrete series representation of su ( 1,1 )q (both positive l>0 and negative l<0 ) . Finally, it is shown that only for the non-deformed k = 2 case it is possible to define expansions within the enveloping algebra. In particular, for l= - ~ we find the realization given by

K 3 = - ½ ( N + ½ ) , K+ =½a 2, K_ =½(a+) 2 .

This is a discrete negative series dual of a well known positive relation that holds both in quantum and classical cases [ 17,18 ]. The "quantum" part of this statement is proved in Section 4, where all q-boson realizations of su ( 1,1 )q are derived. As a result, second order expansions are shown again to be the only ones that can be written in the abovementioned form. Some comments in the final section will end the paper.

1. The expansion method and q-oscillator representations

We shall study the representations of the q-oscillator [ 19,20] algebra, given by the commutators

[w, ~ + 1 = ~ + , [w, ~ ¢ ] = - ~ , [~¢, d + ] = { w + ½ } q , (1.1)

where {X}q = ( qX + q -x ) / ( q ~/2 + q - ~/2 ). This definition means that we have a "symmetric" q-oscillator: q and q - ~ give the same commutation relations. Representations for other non-symmetric cases have been studied in Refs. [21,22].

Firstly we characterize the representations where W is a Hermitian operator, while d + is the Hermitian con- jugated of d . It is possible to prove by a lengthy but straightforward computation that Cq= [ W ] q - d + d is a Casimir operator for this algebra (here the standard notation [X]q = ( q X - q - X ) / ( q _ q - l ) has been used). Let 12 ) be an eigenstate of W with eigenvalue 2. Then, making use of the commutations ( 1.1 ) we can see that (~¢ + ) k 12 ) , if non-null, will be an W eigenvector with eigenvalue 2 + k, while ( ~ ) k' 12 ) will have eigenvalue ; t - k'. Let I # ) be a non-null eigenvector, X I/z) =/~ I/z), obtained from 12) in this way. We compute

(].Llfql~)=Cq,

= (~1 [w]~-~¢ +all ~) = [~]~- (~1 ~ + ~ ¢ 1 ~ ) • (1 .2 )

Since [/Z]q= ( q ~ ' - q - J ' ) / ( q - q - ~ ), where # and q (> 0) are real numbers, then [ / z ]q~-oo i f / z ~ - ~ . Taking into account that ( / z l ~ + d l p ) is positive and, in order to be consistent with the first equation of (1.2), it is necessary that some minimum value kl such that (~¢)k112) =0 exists. Let us call 10) the normalized X eigen- state with lowest v eigenvalue, i.e., XI 0) = v l0) . The whole representation space can be generated out of this vector by applying ~¢+ repeatedly. Hereafter, we shall call this the Vq-representation. It is easy to show that the underlying Hilbert space is just the number state space ~e. The explicit action on the { I n) }-basis can be com- puted in a standard way and reads

~¢+~ i n ) = x / [ v + n + l ] q - [ v ] q i n + l ) , ~ + l n ) = x / [ v + n ] q - [ v ] q i n - 1 ) ,

A ~ l n ) = ( n + v ) l n ) . (1.3)

The Casimir eigenvalue for this representation is Cq = [ v ] q. When v = 0 we have the standard "nq-representation" ofOs(1 )q [ 19,20].

Now, we shall see how expansions can be used to get these representations, as well as the q-Casimir starting from the usual nq realization. Let us take

A. Ballesteros, J. Negro ~Physics Letters A 188 (I 994) 1-10 3

~¢+ =~¢+f + (Y) , ~ = f - (Jff)~, ~4r~=Y+ v , (1.4)

where f + (X) , f - ( X ) are functions to be determined and ~ + , ~ , X are the operators of the standard repre- sentation nq, which act in the following way,

~¢ln)= [x/~qln-1), ~ ¢ + l n ) = ~ ] q l n + l ) , ./V'ln)=nln), n = 0 , 1 , 2 , . . . . (1.5)

We want the operators defined by ansatz (1.4) to reproduce the same algebra given by ( 1.1 ). This means that in fact we are looking for expansion transformations inside the same algebra. By imposing ( 1.1 ) on the basis vectors the following recurrence relation is originated,

[n+ 1 l j + ( n ) f - ( n ) - [n]qf + ( n - 1 ) f - ( n - 1 ) = { n + v+ l}q. (1.6)

The general solution of (1.6) depends on the arbitrary parameter v and has the following form,

A(nq ] vq) = f + ( Y ) f - (A/') = [Jff+ v+ 1 ] q - [vla (1.7) [ X + l l q

The operator A (nql Vq) that displays an infinite number of expansion solutions (note that (1.7) defines only the product f + ( X ) f - ( X ) ) will be called the "characteristic functional" [9]. The arguments of A are the original and final representation, respectively. The Hermiticity relation between ~¢ and ~¢+ selects one partic- ular realization,

f + ( ~ 4 r ) = f - ( J f f ) = N/[y+v+l]q-[vlq[X+l]q (1.7')

Now if we let the operators ( 1.4) with the functions f ± defined by ( 1.7' ) act on the basis { [ n ) } by means of ( 1.5 ), we get all the representations (1.3) previously found. In order to deal with the Casimir operator, we start with the usual representation characterized by Cq = [ X ] q - ~ ÷ ~ = 0. Then, if we substitute expressions ( 1.4 ) we will find

[A~.-vlq-A(nql -1 + Vq) ~ . ~ , = 0 , (1.8)

which taking into account (1.7) gives [ X + v ] q - [ v ] - ~¢ + a¢~ = O, or better

[ ~ ] ~ - ~¢+ ~ = [ v ] ~ , ( 1.8' )

as it should be. Observe that in the limit q- , 1 the Vq-representations ( 1.3 ) of the q-oscillator turn into the corresponding ones

for the classical oscillator,

a+ln)=q"-n--+lln+l), a ln )=q / -n ln -1 ) , N l n ) = ( n + v ) l n ) . (1.9)

In the next sections we shall see that the representations analysed here will appear quite naturally when we study the links between the quantum algebras su( 1,1 )q and Os( 1 )o by using arbitrary powers of their raising and lowering operators.

2. k-order q-oscillator realizatlons

We start from the usual representation v = 0 ofOs( 1 ) in order to get the commutation rules ofOs( 1 )q. In this case we formulate the expansion in the following way,

1 ~ + = X = -~ (N+p), (a+)kg~(N), ~ = g F ( N ) a k (2.1)

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We immediately see that, with this definition, the original representation space of Os ( 1 ) gives rise to k different invariant subspaces. The highest weight vector for each expansion is, respectively, [ 0 ) , I l ) , ..., [ k - 1 ). The whole support space for these k representations is generated by applying the operators (2.1) on the correspond- ing fundamental state.

We impose the commutation rules (1.1) to hold on every basis vector In) . Taking into account (2.1) we obtain

n! ( n _ k ) [ g . ~ ( n _ k ) g ~ ( n _ k ) ( n + k ) ! n! g ~ ( n ) g ~ ( n ) = { ( l / k ) ( n + p ) + t } u . (2.2)

This recurrence can be solved (see Ref. [ 9 ] ) within each m-subspace generated by I n) if n = ks + m, with s any positive integer number and m fixed such that 0 ~ m < k. Each m-subspace supports an expansion characterized by an arbitrary value p that will be denoted by Pro, m=0, 1, 2, ..., k - 1 . Thus, we obtain the functional A~m( n l mq ) = g ~ ( n ) g E ( n ) whose matrix elements read

n! A~m(n lmq)= ( n + k ) ! { [ ( 1 / k ) ( n + p m ) + t ] q + [ ( 1 / k ) ( n - k + p m ) + 1 ] q

+ [ ( 1 / k ) ( n - 2 k + p m ) + I ]q +. . .+ [ (1 /k) ( m + p m ) + 1 ]q}

n! - (n+k)------~v " [ ( 1 / 2 k ) ( n - m ) + t ] q [ ( 1 / 2 k ) ( n + m + 2 p , , ) + t ] q ( q l / 2 + q - l / 2 ) . (2.3)

We shall write (2.3) in another form, which will be more appealing for its interpretation,

1 A ~ " ( n l m q ) = ( N + I ) . . . ( N + k ) { [ ( 1 / k ) ( N + p m ) + l ] q - [ ( l / k ) ( m + p , , , ) ] q . (2.4)

If we rename Nm = ( 1 / k ) ( N - m ) and v,, = ( 1 / k ) ( m + p,, ), this expression turns into

1 a~m(nl Vmtq)) = ( N + 1 ) . . . ( N + k ) ( [Nm + Vm + 1 ]q-- [Vm]q) • (2.5)

With this notation, a Hermitian solution for each rn representation space is

~V~ = Nm + V m , ,/ 1 d~=(a+)k (N+I)...(N+k) ([N'~+vm+ll°-[Vml°)'

sl,~= (N+I)...(N+k) ([Nm+Vm+llq--[Vmlq) ak" (2.6)

It is clear from (2.6) and ( 1.3 ) that the m-subspace supports a representation similar to one of those deduced in Section 1 and characterized by the value Vm = ( 1/k) ( m +Pro). This method gives rise to k independent (in- finite) families of realizations ofOs( 1 )q (note that we obtain any vq representation ofOs( 1 )o, and not only the standard "'no").

Remark that, since Pm is an arbitrary real number, we can choose Pm = -- m, for m = 0, 1, ..., k - 1. In this way we get v,,, = 0, V.m. We then have the k-order generalization of the standard representation of the q-oscillator vo= [0 ]q, whose Hermitian expansion has the form

~ = N m ,

~ ¢ + = ( a + ) k ~ 1 ~ . 1 [ N m + l ] q ( a ) k (2.7) ( N + I ) . . . ( N + k ) [Nm + l ]q ' ~ m = ( N + I ) . . . ( N + k )

A. Ballesteros, £ Negro ~Physics Letters A 188 (1994) I-I0 5

If we compare (2.7) with expression ( 11 ) given in Ref. [ 11 ] (where the notation { b, b + } is used, instead of our present {a, a + }), we realize that both are formally identical. However, the generalized number operator Nm has now an explicit form in terms of N that depends on the selected subspace and avoids the use of the integer part function given in Ref. [ 11 ]. In particular for k= 2 we have two Hermitian realizations of the standard "nq" representation,

( m = 0 )

X=½N, ~ + = ( a + ) 2 J - 1 [½N+l]q , .~1= Q/ 1 [½N+l ]qa 2 (N+ 1 ) (N+2) (N+ 1 ) (N+2) '

( m = l )

1 7 ' X = ½ N - ½, ~ + = ( a + ) 2 (N+ 1 ) (N+2) [½N+½]¢, M= (N+ 1) (N+2)

When q~ 1 these two realizations turn into

(i) Xf½N, d+=(a+)2~/1/2(N+2) , d=x /1 /2 (N+2) a 2

(ii) ../V'=½N-½, ,&+=(a+)2~/1/Z(U+l), ~ = ~ / 1 / 2 ( U + l ) a 2.

(2.8a)

[½N+ ½ ]q a 2 .

(2.8b)

(2.9)

3. su0 ,1 )q boson realizations of order k

The quantum deformation of su ( 1,1 ) has the following commutation rules [ 17 ],

[ ~ , ac± ] = + ac~, [~c+, at_ ] = - [ 2 ~ ] q . (3.1)

Its discrete series representations are labelled by/, the eigenvalue of ~ on the highest weight vector. The Casimir operator is given by

Cq = [Ae~3 - ½ ]] - ~ + YC_ = [ 1 - ½ ]~ . ( 3 . 2 )

Discrete series representations of su ( 1,1 )¢ have been studied from an expansion point of view in Refs. [ 9,10]. Now we talk the classical oscillator and its v = 0 representation space, and we define the deformed generators of su( 1,1 )q as follows (see also, in this respect, an equivalent formulation in Ref. [ 11 ] ),

9¢'~3=(1/k)(N+p), .~+ =(a+)kg~(N), aqg "_ =g~(N)a k . (3.3)

(We have to include the factor 1/k factor to preserve the commutation relations IX]3, 4 ] = + ~ (k= 1, 2, ...). ) By imposing the last commutation rule in (3.1) on the operators defined by (3.3), we obtain

n ! (n ~k)i g~ (n-k)g~ ( n - k ) - (n+k)! n! g~(n)g~(n)=-[(2 /k) (n+p)]q . (3.4)

Again we get a recurrence relation where n=ks+m, with k, s, m integers and 1 <~m<k. Hence, there will be k independent expansion constants, Pro. A straightforward computation shows that the matrix elements of the characteristic functional A, in the oscillator basis { I n ) }, are

n! Af"(nll,) =g~ (n)g~ ( n ) - (n+k)~. [ (1/k) (n+ 2p,,, + m) ]q[ ( I / k ) ( n - m + k ) ]q. (3.5)

6 A. Ballesteros, J. Negro /Physics Letters A 188 (1994) I-10

In conclusion, we can say that the original representation of the classical harmonic oscillator gives rise to a reducible representation of the quantum algebra su ( 1,1 )q that can be decomposed into k irreducible represen- tations - each of them labelled by Pm - by means of creation and anihilation operators of degree k. We shall study some cases separately.

3.1. First order realizations

If k= 1, then m = 0. Any representation l=-po ( > 0) is realizable and we obtain

1 A~(nllq) =g+ (N)gi- (N) = (N+ 1 ) [N+2I]q[N+ 1 ]q. (3.6)

The classical limit of (3.6) is limq~l At (nl lq) = (N+2l ) [9] and relates the usual representation of the har- monic oscillator with any/-representation of the Lie algebra of su ( 1,1 ) (for instance, Holstein-Primakoff real- izations [ 23 ] are included within this class).

3.2. Quadratic realizations

Let us take k= 2. We have two options (m = 0, 1 ) that originate the following characteristic functionals for Po,

Pl ,

1 A~°(nllq)= ( N + Z ) ( N + I ) [½(N+2po)]q[½(N+Z)]q ( m = 0 ) , (3.7a)

1 A~'(nllq)= ( N + 2 ) ( N + I ) [½(N+Zpl+l)]q[½(N+l)]q ( r e = l ) . (3.7b)

Both functionals coincide only when Po = 1 and Pl = ½. In that case, the corresponding irreducible representations of su ( 1,1 )q are lo = 41 and l~ = 3, and the unique functional is given by

1 A2(nllq) =g~ (N)g£ ( N ) = ( N + 2 ) ( N + 1) [½ (N+ 1 )]q[½ ( N + 2 ) ] q . (3.8)

A Hermitian realization (g~" (N) =gfi- (N) ) will act on the number state space in the form

~ln>=½(n+½)ln>, ~ff+ln>=x/[½(n+l)]q[½(n+2)] q In+Z>,

Yg_ In>= x/[½n]q[½(n-1)]qln-2) . (3.9)

The classical limit q~ 1 of (3.7a), (3.7b) is

AO2(nll) = N+2po ( m = 0 ) , (3.7'a) 4 ( N + 1 )

A~(nll)= N+2p~ +1 ( r e = l ) . (3.7'b) 4 ( N + 2 )

Moreover, it is clear that i fpo=Pl = ½ both solutions (3.7 'a) , (3.7 'b) give the well known classical quadratic n 1 realizations of su ( 1,1 ) associated to the representatio s lo = 4, Ii = ~ [ 11,17 ],

K3=½(N+½), K+ =½(a+) 2, K_ =½a 2. (3.10)

Then, what is so special about the l = 14, ~ representations? They are the only ones where the quotients in (3.7'a), (3.7 'b) equal 1 and g+ (N) ( =g~- (N) = 1 ) can be chosen within the oscillator enveloping algebra. Any other l-

A. Ballesteros, J. Negro ~Physics Letters A 188 (1994) 1-10 7

representation of su( 1,1 ) can be obtained from the "n" representation of Os( 1 ) provided that more general funct ionsgf are allowed.

3.3. The general case k> 2

We take the following notation: lm = (Pro -I" m ) / k and Nm = ( N - m) / k . Then, within each irreducible subspace o~°,,, the operator N,, takes integer values 0, 1, 2 .... and we have

1 A~(nl lq)= ( N + I ) . . . ( N + k ) [Nm+21m]q[Nm+llq. (3.11)

Note that this expression resembles strongly the case k= 1. The Casimir operator of su ( 1,1 ) for each subspace oaffm reads

[ 4 - ½ ]~ - ~+ :I"_ =[lm -- ½ 12q. (3.12)

3.4. Negative discrete series representations

Upper bounded (1< 0) discrete representations can be constructed in the same way. In order to see this ex- plicitly let us define

o~C+ =akg~(N) , .,~g'_=g~(N)(a+) k, o~3=- (1 / k ) (N+pm) . (3.13)

In this case the matrix elements of the characteristic functional are given by

( n - k ) ! [ ( 1 / k ) ( n + m + 2 p m ) - l ] q [ ( 1 / k ) ( n - m ) ] q , (3.14) A~"(nllq)= n!

where m and Pm have the same meaning as in the previous case. If k=2, only for the value po= t, Pl = ½ (the representations of su ( 1, I ) q are 1o = - ~ and ll = - 4 ~ respectively), both functionals coincide and take the form

1 A~m(nllq)- N ( N - 1 ) [½N]q[½(N-1)]q . (3.15)

In the classical limit q--. 1, (3.15) gives the usual N-independent expression: A2 (NI lq) = ~. However, the reali- zation of the su ( 1,1 ) algebra is slightly different from (3.10); here we have

1(.3= -½(N+½), K+ =½a 2, K_ =½(a+) 2 (3.16)

4. ¢- I~on realizations

In the previous section we discussed the realizations of su ( 1,1 )¢ as k-order powers of classical bosons. Here we will address the same question but using q'-bosons. It must be remarked that, in principle, q and q' are independent deforming parameters.

We can solve this problem by means of the characteristic functional method taking into account results al- ready obtained in the previous sections. As a in'st step, we recall the expression (3.3) where the su( 1,1 )¢ gen- erators are written as k-order bosons,

~ = ( 1 / k ) ( N + p ) , aY'+ =(a+)kg~(N) , oY'_ = g ~ ( N) a k . (4.1)

The associated functional A~"(nllq) is given by (3.5). Afterwards, we can substitute these classical bosons in terms of the q' bosons by inverting (2.1) with k= 1 and p = O,

8 A. Ballesteros, J. Negro ~Physics Letters A 188 (1994) 1-10

N = X , a+ = d + f + ( N ) , a = f _ ( N ) d . (4.2)

The resultant expansion is given by the characteristic functional A (n, In) =f+ (N)f_ (N) = (N+ 1 ) / [ N + 1 ]q, [ 9 ]. It is easy to check that we can generalize this statement to the k-order case by writing

( a + ) k = (M + )k f+(N) f+(N+ 1 ) . . f + ( N + k - 1) ,

(a)k=f_ (N) f_ (N+ 1 ). . f_ ( N + k - 1 ) (~¢)k. (4.3)

This expansion is characterized by

( N + I ) ( N + 2 ) ( N + k ) Ak(n, I n ) = [N+ 1 ]q-----~ IN+2 ]q"--------~ "'" [N+k]q, " (4.4)

Finally, we substitute (4.3) into (4.1) to get the complete expansion. The global functional is the product of the expansion components (3.5) and (4.4),

A~m( nq , l/q) =Ak( nq, I n)A~m(nllq)

= (N+I_______~)(N+2_______~) ( N + k ) [ ( 1 / k ) ( N + 2 p , n + m ) ] q [ ( l / k ) ( N - m + k ) ] q (4.5) I N + 1 ]q, [ N + 2 ] q , "'" [N+k]q, (N+ 1 ) ( N + 2 ) . . . ( N + k )

Since both (3.5) and (4.4) contain two q-numbers, it is easy to see that the case k= 2 is the only one whose functional could be, in principle, simplified. We have already shown that - just for this case - A~ m ( n I lq) can be taken independent of m (3.8). With this prescription

[½ (N+ 1 )]q[ ½ ( N + 2 ) ] q (4.6) A2(nq, llq)= [N+ I )q,[N+ 2]q,

If q z= q,, (4.6) turns into A2 (nllq) = (q + q - ~ ) - 2, which corresponds to the known quadratic q-boson realiza- tion given by Kulish and Damashinski [ 17 ],

~3 = ~ (d~/'.~_ ½) ' d~¢~+ = 1 ( ~ ¢ + ) 2 ~ _ = ½ ( M ) 2 . (4.7)

We remark that with (4.7) we obtain two different su ( 1,1 )q representations ( l= ~ and l= ] ) depending on the number subspace we are acting on (this fact was already indicated in Ref. [ 17 ] ). Moreover, this representation dependence is enhanced by the fact that this relation is not valid if we start with Vm ~ 0 representations of the q'- bosons. On the other hand, the same characterization (4.6) can be reproduced for the quadratic negative series expansion (3.15 ), obtaining another Kulish-Damashinski type realization,

~d3 = - ½ ( Y + ~), 3f"+ =½M2, j~fr_ = ½ ( d + ) 2 (4.8)

The same composition of expansions can be used to obtain all q-generalized boson realizations. If we take into account (2.7), this expansion leads us to replace the functional (4.4) by the inverse of (2.5) (with Vm = 0). A straightforward computation shows that the analogue of (4.5) is

A~"(nkqllq) = [X~ +2lm]q • (4.9)

TO compare this inf'mite family of realizations with Ref. [ 11 ] we have to recall the considerations made in Section 3. Functional (4.9) contains Holstein-Primakoff and Dyson realizations used in Ref. [ 13 ].

To end, we emphasize that the technique we have outlined here is quite general and can be applied to many other algebras. For instance, k-order realizations of su (2)q (either in terms of (q)-boson operators or starting from su ( 1,1 ) discrete series ) can be obtained (as has been done in Ref. [ 11 ] ). Some other interesting problems, which are in progress, arise if one wants to generalize the characteristic functional approach to simple quantum algebras with higher rank.

A. Ballesteros, J. Negro ~Physics Letters A 188 (1994) 1-10 9

5. Conclusions and remarks

We have analyzed the process o f building Os ( 1 ) q and su ( 1,1 ) q representations in terms o f k-products o fboson (or q-boson) operators by means o f the expansion method. The original representation space ~ o f O s ( 1 ) splits into k representation spaces Yfm, m = 0,..., k - 1, just in the same way as at the classical level is described by Luis and S~chez -So to [ 16 ]. We remark that the procedure allows us to construct any irreducible representation, but some of them take a particularly simple expression (that only when k= 2 can be defined inside the enveloping algebra). Things can be arranged so that all the isomorphic split spaces Yfm ~ ~ support representations equiv- alent to the initial one on Yr. Thus, ~ can be written in the form Yf= ~ @ ~ , where ~ i s a k-dimensional space, whose vectors can be characterized, for instance, by a representation o f the cyclic group of order k, Cgk.

Our study was intended to clarify the representations that k-order operators can give rise to. These consider- ations must be taken into account when they are used in coherent states (CS) or squeezed states (SS) construc- tions that have become so usual lately. In principle, there are different choices for the vacuum (corresponding to the spaces Yfm) that can give rise to a rich wealth o f coherent states as has been shown Ref. [ 14 ]. For example, in our language the framework of Ref. [ 14 ] can be summarized as follows. A standard q-CS ~ (z ) can be decom- posed using the projector P,n on each spacein Yt'm by ~m(Z) --Prn¥(z). The point is that if a representation o f Os( 1 )q is given in the space o f functions on z, i.e., Yf, by {a-.z, a+~Dq(z), N-,z/dz}, then {a~D(a)®z, a+--.D(a+)®Dz(z), N--,D(N)®z/dz} will define a representation in ~ ® ~ , where D:{a, a +, AT}--. cgk is a certain asignation on the cyclic group and ~m (Z)eYfm ~ ~ . This point o f view fits well with the discussion of Ref. [ 16 ] in the q-deformation context.

Multiboson CS have been studied in Ref. [24 ] concerning their squeezing properties. When q is complex, the squeezing has been examined in Ref. [ 12 ] in a way that can be readily extended to the multiboson case. Other applications o f the mult iboson formalism about the q-deformed Jaynes -Cumming model is given in Ref. [25 ], or in the mul t imode model o f Schumaker-Caves in quantum optics by Ref. [26] .

Acknowledgement

This work has been partially supported by a D G I C Y T project (PB91-O 196) f rom the Ministerio de Educa- ci6n y Ciencia de Espafia.

References

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