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OF QUANTUM VARIABLES AND GENERALIZED DERIVATIVES.

arXiv:hep-th/9303132v2 25 Mar 1993DTP/93/09March, 1993STUDY OF QUOMMUTATORSOF QUANTUM VARIABLES AND GENERALIZED DERIVATIVES.David FAIRLIEDepartment of Mathematical SciencesUniversity of Durham, Durham, DH1 3LE, EnglandJean NUYTSUniversity of Mons, 7000, Mons, BelgiumAbstract :A general deformation of the Heisenberg algebra is introduced with two deformedoperators instead of just one.This is generalised to many variables, and permits thesimultaneous existence of coherent states, and the transposition of creation operators.1

I. IntroductionLet Xi (i = 1, . .

., N) be a set of quantum variables which obey the quommutationrelationsXi ∗Xj = xijXj ∗Xi(1.1a)where the c-numbers xij obviously satisfyxijxji = 1(1.1b)and ∗represents an associative non commutative product.This is not by any meansthe most general starting point[1], but is the most convenient choice as associativity isautomatically satisfied for the Xi.Let Di (i = 1, . .

., N) the corresponding quantumderivativesDi ∗Dj = dijDj ∗Di(1.2a)where againdijdji = 1. (1.2b)It is then useful to introduce the operators Gi (i = 1, .

. ., N) defined for each i byGi = Xi ∗Di(1.3)which as a consequence of (1.1) and (1.2) satisfy similar quommutation relationsGi ∗Gj = gijGj ∗Gi(1.4a)andgijgji = 1.

(1.4b)The operators Gi are to be interpreted as quantum dilatation operators analogous to theclassical dilatation operator usually defined as Xi∂/∂Xi.Since we expect the quommutator of the operator Di with the corresponding Xi toinvolve diagonal neutral operators[2] which we call Ai (i = 1, . .

., N), we writeDi ∗Xj = vijXj ∗Di + δijAi(1.5)where the arbitrary normalisation of the diagonal δij term has been supposed to be nonzero for all i and has been included in the definition of the Ai.The motivation for this study, is connected with the potentiality of representing A bysomething other than the identity operator. This will permit us to overcome a deficiencywhich has been ignored by much of the recent literature on q-deformations,[3,4] and willenable us to demonstrate the existence of coherent states and at the same time permittransposition of the variables Xi.

As far as we are aware, there are only certain particularlysimple parameter choices in some recent work on differential calculus[4,5] which currently2

allow this. A secondary motivation is to write an operator realisation of the symmetricform of the q-derivative, and generalise this to many variables.We can write after multiplication of (1.5) on the left by XiGi ∗Xj = pijXj ∗Gi + δijXi ∗Ai(1.6a)wherepij = vijxij(1.6b)Let us define naive scale invariance as follows : Xi scales as a length [L1], Di as aninverse length [L−1] and hence Gi and Ai as [L0].

This naive scale invariance would allowthe addition of a term ρiXi to the right hand side of (1.6a). We have however shown thatthe associativity restrictions on the operators lead to ρi = 0 in a natural way (except forpathological configurations which we have excluded).In order to construct a complete set of quommutator relations the equations (1.1),(1.4) and (1.6) have to be supplemented by relations between the Ai themselves as well asbetween the Ai and both the Xj and the Gj.For these quommutation relations we proposeAi ∗Aj = aijAj ∗Ai(1.7a)aijaji = 1(1.7b)i.e.

of a form analogous to (1.1), (1.4) andAi ∗Xj = qijXj ∗Ai + δijαiXi ∗Gi(1.8)Gi ∗Aj = rijAj ∗Gi + δij(βiA2i + γiG2i )(1.9)Let us state again that in (1.8) we could have added a term σiXi and in (1.9) a termλiAi + µiGi + νi in agreement with the naive scale invariance and the idea that all termswith degree less or equal to two should be included in these relations. But all these termsturn out to be zero σi = λi = µi = νi = 0, in a natural way, upon the application of theassociativity requirements.

Here again it would be possible to generalise, by intermixingterms with different indices as is done[5−8] , but we take the simplest choice. As we shallsee this will permit us to change the basis to simplify the algebra.

What we have is ageneralisation of previous work[9], with the incorporation of the Ai operators. Once theform (1.8–9) is obtained both Ai and Gi can still be renormalized (rescaled) by the samefactor.The rules given above enables one to rewrite any product of operators in what weshall call a normal order i.e.- the Gi at the right of all the other operators (Xi and Ai)- the Ai at the right of the Xi- within the operators of the same name, the operators are ordered in, say, decreasingorder of their indices.3

There is a however a subtle point connected to (1.9). If one takes that equation fori = j, i.e.Gi ∗Ai = riiAi ∗Gi + βiA2i + γiG2i(1.10)and multiplies it by Ai on the right, one obtains, dropping the index i as in (2.3) below,using (1.10) repeatedly and after rearrangement of the termsG∗A2 = βγG∗A2 +β(1 +r)A3 +γ2(1 +r)G3 +rγ(1 +r)A ∗G2 +r(r +βγ)A2 ∗G (1.11)All the terms in the right hand side, except the first one are normal ordered.

There is thusa condition to be able to write G ∗A2 in normal order : namely thatβγ ̸= 1(1.12)which we shall call the “normal ordering condition”. In this caseG ∗A2 =1(1 −βγ)β(1 + r)A3 + γ2(1 + r)G3+ rγ(1 + r)A ∗G2 + r(r + βγ)A2 ∗G(1.13)The condition (1.12), essential to the consistency of the definition of the normal prod-uct, will be systematically imposed in the next section.In section 2 we solve, in general, the associativity requirements which, as we know,are not automatic for quommutation relations of the form given above.

In section 3 weoutline definitions of symmetric quantum derivatives which give representations of one ofthe solutions obtained in section 2.II. Solutions of the associativity requirements.We present here the general restrictions on the parameters which follow from thebraiding relations which ensure that any product of operators can be rewritten in an un-ambiguous unique way as a normal product.

Let us note that we have excluded pathologicalsolutions such that xij = 0, pij = 0, . .

..After some lengthy computations one finds that associativity for the productsA ∗X ∗X,G ∗X ∗X,G ∗A ∗A,A ∗A ∗X,G ∗G ∗X,G ∗G ∗A(2.1)for all the indices i, j, k require very simple equalities between the non-diagonal coefficientsrij = aij = gij(2.2a)i ̸= jqij = pij(2.2b)4

while the coefficients of the other possible terms in the right hand sides, i.e. the linear(λ, σ, µ, ρ) terms as well as the constant term ν, are forced, for non pathological solutions,to be zero.The treatment of the diagonal case is somewhat more complicated.Indeed the relations imposed by associativity which remain to be checked are thosefor (Gi ∗Ai) ∗Xi = Gi ∗(Ai ∗Xi), i.e.

within one family only. Dropping the index i onehas to find the restrictions on the parameters for the set of quommutatorsG ∗X = pX ∗G + X ∗A(2.3a)A ∗X = qX ∗A + αX ∗G(2.3b)G ∗A = rA ∗G + βA2 + γG2(2.3c)with p = pii, α = αi, .

. ..There are four disconnected cases which we now give explicitelyCase AG ∗X = pX ∗G + X ∗AA ∗X = qX ∗A + pqX ∗GG ∗A = rA ∗G + βA2 + (q −qr −q2β)G2(2.4Aa)with the normal ordering condition (1.12)β(q −qr −q2β) ̸= 1(2.4Ab)Case BG ∗X = pX ∗G + X ∗AA ∗X = qX ∗A + αX ∗GG ∗A = (1 + βp −βq)A ∗G + βA2 −αβG2(2.4Ba)with the restrictionα −pq ̸= 0(2.4Bb)and with the normal ordering condition (1.12)αβ2 + 1 ̸= 0(2.4Bc)Case CG ∗X = pX ∗G + X ∗AA ∗X = −pX ∗A + αX ∗GG ∗A = −A ∗G + βA2 + (αβ −2p)G2(2.4Ca)5

with the restrictionsβp + 1 ̸= 0α + p2 ̸= 0(2.4Cb)and with the normal ordering condition (1.12)β(αβ −2p) ̸= 1(2.4Cc)Case DG ∗X = pX ∗G + X ∗AA ∗X = −pX ∗A −(2p2 + pγ)X ∗GG ∗A = −A ∗G −1pA2 + γG2(2.4Da)with the restrictionp + γ ̸= 0(2.4Db)while the normal ordering condition (1.12) coincides here with (2.8b).Obviously, the situation is somewhat complicated by the fact that for each i, thesolution can be chosen arbitrarily to belong to one of the four cases (A, B, C, D) above.The usual Heisenberg algebra belongs to class B.It may be interesting to ask the question whether, by a suitable linear change of basicoperators, the set of quommutators, which is written in (2.3) with underlying physicalmotivations, cannot be brought in a simpler canonical position. Let us first stress that Xplays a special role and that we are thus restricted to consider linear combinations of Gand A only.

Let G′ (or A′) be given byG′ = vG + wA(2.5)It will fulfill the equationG′X = p′XG′(2.6)provided p′ and λ = wv satisfy the equationsp′2 −(p + q)p′ + (pq −α) =0αλ2 + (p −q)λ −1 =0(2.7)If there are two different roots, say p′ and q′, to (2.7) i.e. if the discriminant ∆whichis the same for both equations is non zero∆≡(p −q)2 + 4α ̸= 0(2.8)6

we see that (2.3) can be brought to the canonical formG′ ∗X = p′X ∗G′(2.9a)A′ ∗X = q′X ∗A′(2.9b)s′G′ ∗A′ = r′A′ ∗G′ + β′A′2 + γ′G′2(2.9c)where the last equation is the most general quadratic combination of na¨ıve zero degree. Ifwe suppose that s′ is non zero, it can be renormalised to 1.If α = pq, i.e.

case A above, one of roots of (2.7), q′ say, is zero.The associativity requirements for (2.9) are (provided that we suppose that s′ and r′are not both zero and remembering that q′ ̸= p′)p′γ′ = 0(2.9d)q′β′ = 0(2.9e)Since at least one of the roots is non-zero, say p′, (2.9d) implies thatγ′ = 0(2.9f)and then (2.9e) leads to two casesCase A′: q′ = 0, γ′ = 0(2.9g)Case B′: β′ = 0, γ′ = 0(2.9h)When the two roots of (2.7) are equal (i.e. if ∆of (2.8) is zero and p′ = (p + q)/2),the system can be brought to the formGX = p′X ∗G + XA′(2.10a)A′ ∗X = p′X ∗A′(2.10b)s′G ∗A′ = r′A′ ∗G + β′A′2 + γ′G2(2.10c)whereA′ = A + p −q2G(2.10d)p′ = p + q2(2.10e)s′ = 1 −β(q −p)2(2.10f)r′ = r + β(q −p)2(2.10g)β′ = β(2.10h)γ′ = γ + β(p −q)24+ (q −p)(r −1)2(2.10i)7

The form of (2.10a), (2.10b) is somewhat similar to that of the well known deformationof the Heisenberg commutation relations of Biedenharn[10] and Macfarlane[11], with theformal identification of a† with X, a†a = N with G and q−N with A. Of course in theirscheme, the operators A and G are not independent, as they are for us.If s′ ̸= 0 it can be renormalized to 1 and the associativity requirements for (2.10) areγ′ = 0(2.10j)p′(s′ −r′) = 0(2.10k)leading to two cases again.Case C′: p′ = 0, γ′ = 0(2.10l)Case D′: r′ = s′, γ′ = 0(2.10m)If s′ = 0, r′ can be renormalized to -1 and the associativity requirements lead top′γ′ = 0(2.10n)p′ −γ′ = 0(2.10o)This leads to the rather uninteresting caseCase E′: p′ = 0, γ′ = 0(2.10p)It is obvious that the associativity conditions were easier to write in the new basisbut we first presented the results in the old one as it is more physical.

The change of basisfacilitates, in certain cases, the discussion of representations.Let us also note that the conditions (2.2) are exactly those which allow the aboveredefinitions in terms of G′ and A′ in a coherent way between different indices.III. Symmetric q-derivatives.We now present a particular case of associative operators which are suited to define asymmetric q-derivative.Let, as above, the Xi be a set of quantum variables.

For a function f(xi) of onevariable alone, let us define(Xif)(xi) = xif(xi)(3.1a)(Dif)(xi) = f(pixi) −f(xi/pi)xi(pi −1/pi)(3.1b)as the symmetrical q-derivative.As a consequence the operator Gi defined in (1.3) has the following action(Gif)(xi) = f(pixi) −f(xi/pi)(pi −1/pi)(3.2)8

The operator Ai defined byDi ∗Xi = piiXi ∗Di + Ai(3.3)has still some arbitrariness but can also be chosen, in a unique way, to have a symmetricaction(Aif)(xi) = f(pixi) + f(xi/pi)2(3.4)From these definitions the quommutation relations within the i family can be deducedGi ∗Xi = piiXi ∗Gi + Xi ∗AiAi ∗Xi = piiXi ∗Ai + (p2ii −1)Xi ∗GiGi ∗Ai = Ai ∗Gi(3.5)wherepii = pi + 1/pi2(3.6)They correspond to Case B above with q = p, α = p2 −1 and β = 0, a particularlysimple case.The connection between the operators coresponding to two different indices i and jstill depend on four a priori independent parameters xij, gij, pij and pji. However theconsistency of the quommutator (1.7) when applied to the function f ≡1 implies aij = 1i.e.

gij = 1 since through (3.4) the action of Ai on 1 is 1 for any i.Let us note also that a exponential can be easily defined as a solution, say for onevariable x only, ofDE(x) = E(x)(3.7)For the symmetrical q-derivative it readsE(x) =∞Xn=0anxn(3.8)wherea0 = 1,a1 = 1an =1Qn−1k=1[k]pn > 1. (3.9)Here[j]qi = qji −q−jiqi −q−1i.

(3.10)9

In order to generalise the q-exponential to many variables it is necessary to choose aparticular set of values for the parameters pij etc. This choice ispij = qij =(qi + q−1i)2,∀j,αi =(qi −q−1i)24,rij =1, γ = 0.

(3.11)It allows xixj to be interchanged; it turns out that(qj + q−1j )xixj = (qi + q−1i)xjxi∀i, j(3.12)may be imposed without affecting the q-exponential. This equation allows all monomialsin xi, xj, .

. .

to be ordered alphabetically.The general q-exponential is given by the simultaneous solution of the equationsDiE(x1, x2, . .

.xN) = E(x1, x2, . .

.xN). (3.13)The general term in the expression E(x1, x2, x3) for example, is given byaYj1[j]q1bYk1[k]q2cYl1[l]q3xa1xb2xc3 2[2]q1a(b+c)2[2]q1bc.

(3.14)The generalisation to an arbitrary number of variables is obvious. Thus this choice ofparameters allows a simultaneous construction of a generalised q-exponential, or coherentstate, and a set of permutable creation operators.

This last feature is admittedly absent inthe work of Greenberg[3] and in most of the literature on coherent states by implication.IV. General q-derivatives.A slight generalisation of the preceding section can be obtained as follows in the caseof one variable only.Suppose we define the X as before on a function f(x)(Xf)(x) = xf(x)(4.1)and try to construct the action of G as(Gf)(x) =MXk=1χkf(λkx)(4.2)10

Equation (2.3a) is then a simple definition of A(Af)(x) =MXk=1χk(λk −p)f(λkx)(4.3)Equation (2.3b) is then a consistency equation which readsMXk=1χkλ2k −(p + q)λk + qp −αf(λkx) = 0. (4.4)Since the f(λkx) are independent for sufficiently general choices of f(x) (4.4) implies thatλ2k −(p + q)λk + qp −α(4.5)for every k (equation (2.7) again).

But since p, q and α don’t depend on k and since also asecond degree equation has only two solutions, there are at most two allowed values of λk.Hence M = 2 and the two λ’s are given in terms of the three free parameters p, q and α.Conversely, if the two values of λ are given λ1 = λ, λ2 = µ, the condition (4.5) givesthe restrictionsp + q = λ + µ(4.6)pq −α = λµ(4.7)So that there are altogether again three free parameters, say λ, µ and p. The remainingones q and α being fixed by (4.6) and (4.7).Finally A and G commuteG ∗A = A ∗G. (4.8)All these equation finally generate a representation of case B with β = 0 but the otherparameters are free.

This representation can be extended in a natural way when there ismore than one variable xi. Obviously these variables are then quantum variables and haveto fulfill quommutation relations in agreement with (1.1a).The representations discussed so far are those acting on an infinite dimensional func-tion space.V.

Representations with G and A diagonal.Let us look now for representations such that both G and A are diagonal and hencecommute.Suppose we start with a vector | 0 >, eigenvector of G and A with eigenvalue g0 anda0 respectivelyG | 0 > = g0 | 0 >A | 0 > = a0 | 0 >(5.1)11

Let us define| n >= Xn | 0 >(5.2)ThenG | n > = gn | n >A | n > = an | n >(5.3)With the vector vn defined byvn =gnan(5.4)and the matrix M defined byM =p1αq(5.4)it is easy to prove thatvn = Mvn−1 = M nv0(5.5)This obviously depends on the precise form of the matrix M and of its eigenvalues,obviously equation (2.7) again.Let us now make the following important remark. Since both G and A are diagonal,they commute for this representation.

But this commutation is not a qualgebra relation.What has to hold is (2.5c) which has still to be satisfied at every stage of the procedure.It reads in the general caseCn ≡βa2n + γg2n + (r −1)angn = 0(5.6)More precisely if, say an and gn are chosen in such a way that Cn = 0 there is acondition on the free parameters to garantee that Cn+1 be zero. Eitherp2βγ −pqr2 + 2pqr + 2pqβγ −pq −prγ + prαβ + pγ −pαβ + q2βγ+ qrγ −qrαβ −qγ + qαβ + γ2 −2αβγ + α2β2 = 0(5.7a)orp2γβ −2pqβγ −prγ + prαβ + pγ −pβα + q2βγ+ qrγ −qrαβ −qγ + qαβ −r2α + 2rα + γ2 + 2αβγ + α2β2 −α = 0(5.7b)It is amusing to note that the product of the two expressions is always identicalllyzero in the four allowed cases A −D.

This fact shows that this representation alwaysexists. Let us stress again that the fact that G and A commute is a simple artefact ofthe representation.

It is analogous for example to the fact that, if the Pauli matrices are2-dimensional representations of SU(2), the fact that σ1σ2 = iσ3 is a simple artefact whichhas nothing to do with the basic commutation relations of the SU(2) algebra.VI. Representations where X has an eigenvector.12

In the preceding paragraph we have chosen to present the case where both G andA are diagonal, as they are simple. However we believe that the physically interestingrepresentations rather correspond to cases where X has an eigenvector | 0 > with eigenvaluex0X | 0 >= x0 | 0 >(6.1)It is now more convenient to re-introduce the linear combinations in terms of which thequommutation relations simplify to (2.9).An infinite set of states | k, l >, k = 1, .

. ., l; l = 1, .

. .

can then be constructedthrough| k, l >= A′kG′l | 0 >,0 ≤k ≤l(6.2)The action of the operators in this infinite dimensional space in the case can then be foundeasily through the equationsX | k, l > = x0q′−kp′−l | k, l >A′ | k, l > =| k + 1, k >G′ | k, l > =| k, l + 1 >(6.3)When G′ and A′ do not commute then the last of these equations requires modification.Using (2.9c), applied to the states | l, 0 >, | l + 1, 1 >,G′ | 1, k >=r | 1, k + 1 > +β | 2, k > +γ | 0, k + 2 >G′ | 2, k >=1(1 −βγ)β(1 + r) | 3, k > +γ2(1 + r) | 0, k + 3 >+ rγ(1 + r) | 1, k + 2 > +r(r + βγ) | 2, k + 1 >(6.4)The last of these equations also follows from (1.13) applied to | l, 0 >. The general resultwill follow upon iteration.VII.

Finite dimensional representations.Finite dimensional representations of operators are always useful to consider.Werestrict ourselves to situations where p ̸= 0, q ̸= 0 and r ̸= 0.One dimensional representations.First, the one dimensional representations are obviously trivial but we present themin order to be able to make the remark following (7.2) below.When X is represented by 1 by na¨ıve rescaling and G = 1 also.ThenX = 1G = 1A = (1 −p)(7.1a)13

We can apparently solve directly for the parameters of (2.3), without going through thefour cases (A-D) one by one.The parameters are restricted by the two relationsα = (1 −p)(1 −q)(7.1b)γ = (1 −p)(1 −r −β + pβ)(7.1c)When X = 0 thenX = 0G = gA = a(7.2a)and the numbers g and a must satisfy(1 −r)ga −βa2 −γg2 = 0(7.2b)It may appear strange at first sight that the representation (7.1) (or (7.2)) looks moregeneral than any of the cases (A–D) above. The justification for this fact is as follows.

Toderive the conditions of associativity we have supposed, rightly, that, once put in normalorder, any product of the starting operators X, G, A, for up to three operators in theproduct, is linearly independent of any other. This is clearly not true for X = G = 1.Hence there are apparently more solutions to the associativity requirements.

These extrasolutions should obviously be rejected as they are not bona fide representations of theabstract quommutators.Two dimensional representations.Henceforth we will restrict ourselves to representations where not all the operatorsare represented by diagonal matrices, i.e. irreducible representations.By performing a general change of basis in the two-dimensional space upon which theoperators act and by using a suitable rescaling of the naive length [L] unit, it is always tobring the operator X in one of the following well-known canonical positions :a) X = 12 is the 2-dimensional unit matrixb) X = diag(1, x) where x ̸= 0 and x ̸= 1c) X = diag(1, 0)d) X = σ+ where σ+ is the Pauli matrix with only non-zero element σ+(1, 2) = 1e) X = 12 + xσ+ where x ̸= 0.The full discussion is rather rich.

Indeed, the allowed representations depend often onparticular and more detailed relations between the parameters than those which define thefour cases (A–D) above. We have thus chosen not to present them though some of themare quite interesting.14

Finite representations with G and A diagonal.Using the results of section V, finite dimensional representations of the qualgebras canbe constructed. Indeed, if we suppose that there exists a positive integer P such that (see(5.2))| P >=| 0 >(7.3)the space on which the representation acts becomes P−dimensional.

In order to reproducethe same eigenvalues for G and A, one must have vP = v0. Consequently one needs (see(5.5))M P = 1(7.4)For this to be the case, the two eigenvalues of M i.e.

p′ and q′ have to be P−roots ofunity. We find once again the occurrence of the important “roots of unity” which play acentral role in qualgebras.In the transformed basis where M is diagonalized, X is essentially a matrix of cyclicpermutation.Other representations can be constructed from these by using suitable direct sums ordirect products of representations.

An example using direct products, usually equivalentto one of the general type constructed above, or reducible to direct sum of them is asfollows : If a and b in are prime numbers, then a representation by ab × ab dimensionalmatrices in the changed basis is given byG =diag{1, p′, p′2, . .

.p′a−1} ⊗diag{1, 1, 1, . .

.1}A =diag{1, 1, . .

.1} ⊗diag{1, q′, q′2 . .

. q′b−1}X =Pa×a ⊗Pb×b(7.5)Here p′a = 1, q′b = 1 and Pa×a and Pb×b are matrix representations of cyclic permutationsof a and b objects respectively.

Here again G and A commute but, as explained above thisis a simple artefact of the representation of a more general qualgebra.VIII. Conclusion.Using a natural set of a priori quommutation relations between quantum variables Xi,quantum derivatives Di or better the related quantum dilatation operators Gi = Xi ∗Diwe have been led to introduce the corresponding neutral operators Ai, in order to givean algebraic realisation of symmetric q-differentiation.

We have outlined all the possiblechoices allowed by the associativity requirements (or braiding relations) within our basicrestriction (1.1a), that the transposition of two quantum variables does not intoduce anyother operators. We have shown that, within a given i (the i family) and taking into accountthe normal ordering condition, there are four disconnected cases, (2.4A-D).

The relationsbeween two families i and j depend, due to (2.2), on four arbitrary parameters xij =1/xji, gij = 1/gji, pij and pji. We have presented some representations of our abstractqualgebras with one X, one G and one A only both in finite and infinite dimensional15

spaces. Having constructed all 2-dimensional representations, we have realized that thespace of representation is apparently very rich.A particular example of a representation for these operators and their action hasbeen shown explicitly for what we have called symmetric or general quantum derivatives.It allows xixj to be interchanged and at the same time permits the construction of amultivariable q-exponential; explicitly the result is obtained that(qj + q−1j )xixj = (qi + q−1i)xjxi∀i, j(8.2)may be imposed without affecting the q-exponential.

This may have some bearing uponattempts to apply quantum groups to quantum optics. We hope to elaborate on thosepoints in the near future.16

References[1] Woronowicz SL, Comm. Math.

Phys. 122 (1989) 125.

[2] Fairlie D.B. and Nuyts J., Neutral and charged quommutators with and withoutsymmetries, Zeitschrift f¨ur Physik 56(1992)237.

[3] Greenberg O.W., Phys. Rev.

Lett. 64 (1990), 705.

[4] Bogoliubov N.M. and Bullough, R. K. J. Phys A A25 (1992) 4057. [5] Pusz, W. and Woronowicz, S.L., Rep.Math.Phys., 27 (1989) 231.

[6] Vokos, S.P., Zumino, B. and Wess, J., Z.Phys. C, 48 (1990) 317.

[7] Wess J. and Zumino B. Nuclear Physics B Supplements 18B (1990) 302.

[8] Zumino B., Mod. Phys.

Lett. A13 (1991) 1225.

[9] Fairlie D.B. and Zachos C.K., Phys.Lett.

256B (1991) 43. [10] Biedenharn, L.C.

J.Phys. A A22 (1989) L873.

[11] Macfarlane, A.J J.Phys. A A22 (1989) 4581.17

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