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M. Chaichian1, R. Gonzalez Felipe1,∗) and P. Preˇsnajder2,∗∗)

arXiv:funct-an/9305003v1 21 May 1993HU-TFT-93-28q-Supersymmetric Generalization of von Neumann’sTheorembyM. Chaichian1, R. Gonzalez Felipe1,∗) and P. Preˇsnajder2,∗∗)1High Energy Physics Laboratory, Department of Physics, P.O.

Box 9(Siltavuorenpenger 20 C), SF-00014 University of Helsinki, Finland2Research Institute for Theoretical Physics, University of Helsinki, P.O. Box 9(Siltavuorenpenger 20 C), SF-00014 University of Helsinki, FinlandMay 1993AbstractAssuming that there exist operators which form an irreducible representation of theq-superoscillator algebra, it is proved that any two such representations are equivalent,related by a uniquely determined superunitary transformation.

This provides with a q-supersymmetric generalization of the well-known uniqueness theorem of von Neumann forany finite number of degrees of freedom.∗)ICSC-World Laboratory; On leave of absence from Grupo de F´isica Te´orica, Instituto de Cibern´etica,Matem´atica y Fisica, Academia de Ciencias de Cuba, Calle E No. 309, Vedado, La Habana 4, Cuba.∗∗)Permanent address: Department of Theoretical Physics, Comenius University, Milynsk´a dolina F2,CS-84215 Bratislava, Slovakia.

1IntroductionIn the last few years quantum deformations of Lie groups and Lie algebras havefound several applications in mathematics and theoretical physics (see e.g. refs.

[1]-[3]).These deformations have been subsequently extended to supergroups and superalgebras[4]-[6]. In particular, the bosonic and fermionic q-oscillators [7], [8], [6] have been usedfor the realization of different quantum Lie algebras [7]-[9] and quantum superalgebras[6],[10],[11].A natural question then arises, concerning the relation between different irreduciblerepresentations of the q-deformed algebras.It is well known that in the case of theclassical bosonic and fermionic Heisenberg algebras for harmonic oscillators, this problemis solved by the von Neumann’s theorem (see, e.g.

[12],[13]), which states that irreduciblerepresentations of the bosonic (fermionic) algebra are unitarily equivalent to each other.Similar results also hold for irreducible operator representations of Lie superalgebras [14].Recently, it was proved that the analogue of von Neumann’s theorem is also valid in thecase of q-oscillator algebras [15].In this letter we shall extend the results of [14] and formulate a quantum supersymmet-ric generalization of von Neumann’s theorem for irreducible representations of q-deformedsuperalgebras. We start from a supercovariant system of q-oscillators [11], which are co-variant under the coaction of a supergroup, SUq(n|m).

The latters present the extensionof the covariant system of q-oscillators proposed in [16],[17]. Assuming suitable domainproperties as in [14], we prove that any two representations of the q-deformed superalgebraare connected by a unique superunitary transformation.

A similar result is also proved tobe valid for any finite number, n, of bosonic and fermionic independent q-oscillators. Wepresent the explicit form of the superunitary transformation operator for the cases n = 1and 2.2von Neumann’s theorem and its extension to q-deformed algebrasWe start by recalling the classical von Neumann’s theorem [12],[13].

Let b, b+ andb′, b′+ be two irreducible representations of the Heisenberg algebra,bb+ −b+b = 1 ,(1)in the Hilbert spaces H and H′, respectively, and assume that there exist vectors |0 > inH and |0′ > in H′, such that b|0 >= 0, b′|0′ >= 0. Then there exists a unitary operatorU such thatb′ = UbU+ , b′+ = Ub′U+ ,UU+ = U+U = 1 .

(2)A similar theorem also holds in the case of the fermionic algebra [12],[13],cc+ + c+c = 1 . (3)1

The above theorem can be formulated in a slightly different form, which we shall utilizefurther:Let b, b+ and b′, b′+ satisfy all the conditions of von Neumann’s theorem and let usdefine the operators a, a+ and a′, a′+ asa = ϕ(N)b,a+ = b+ϕ+(N) ,a′ = ϕ(N′)b′,a′+ = b′+ϕ+(N′) ,(4)whereN = b+b , N′ = b′+b′ ,(5)are the number operators and ϕ is a well-behaved function.Then for the representations a, a+ and a′, a′+ in some Hilbert (sub)spaces Ha andH′a (Ha ⊆H, H′a ⊆H′), respectively, von Neumann’s theorem also holds, i.e., a, a+ anda′, a′+ are irreducible representations, there exist vectors |0 > in Ha and |0′ > in H′a suchthat a|0 >= 0, a′|0′ >= 0 and there exists a unitary operator U such thata′ = UaU+ , a′+ = Ua+U+ , N′ = UNU+ . (6)To prove (6) we notice that since b, b+ and b′, b′+ satisfy the conditions of von Neu-mann’s theorem, it follows that there exists a unitary operator U such that relations (2)hold.

From (2) and (5) it follows thatN′ = b′+b′ = Ub+U+UbU+ = UNU+ ,and the same relation is also valid for the function ϕ(N),ϕ(N′) = Uϕ(N)U+ . (7)Now, from the definition of the operators a, a+ (eqs.

(4)) we haveUaU+ = Uϕ(N)bU+ = Uϕ(N)U+UbU+ = ϕ(N′)b′ = a′ ,and, similarly,Ua+U+ = a′+ .Relations (6) are thus proved.It is also clear that in the spaces Ha and H′a there exist vectors |0 > and |0′ > suchthat a|0 >= 0, a′|0′ >= 0. Finally, a, a+ and a′, a′+ are two irreducible representationsin the Hilbert spaces Ha, H′a, respectively.

This follows from the definitions (4) and thefact that b, b+ and b′, b′+ are irreducible representations in H, H′.The same conclusion also holds obviously for the case of fermionic oscillators.From the above theorem, it follows, in particular, that any two irreducible represen-tations of the q-deformed bosonic oscillator algebra [7],[8],[6]aa+ −qa+a = q−N ,[N, a] = −a , [N, a+] = a+ ,(8)are connected by a unique unitary transformation. Indeed, by taking ϕ(N) =q[N+1]N+1in (4), where [n] = qn−q−nq−q−1 , we obtain the q-oscillator algebra (8), provided that a+a =2

[N], aa+ = [N + 1], i.e. in the Fock space of (8) [18].

For the exceptional values of qbeing the m-th root of unity, q = e±iπ/m, [m] = 0 and thus ϕ(m −1) = 0. In such a casethe Hilbert space Ha becomes finite, m-dimensional.

A similar statement is also valid inthe case of the q-deformed fermionic algebra [6]ff + + qf +f = qM ,[M, f] = −f , [M, f +] = f + ,(9)which can be obtained from the algebra (3) by means of the change of operatorsf = qM2 c , f + = c+qM2 .Let us remark that the above statements were proved in ref. [15] by following a proof inthe same line as the original von Neumann’s theorem for usual (nondeformed) oscillatorsand valid also for generic values of q including the exceptional values of m-th root of unity.The same theorem is also valid in the case of the other q-fermionic algebra [19] given bythe commutation relationsff + + qf +f = q−M ,[M, f] = −f , [M, f +] = f + .In fact, we can obtain the latters from the bosonic algebra (1) by means of the transfor-mation,f =s[M + 1]fM + 1 b , f + = b+s[M + 1]fM + 1,where [n]f = q−n−(−1)nqnq+q−1, since in the Fock space we have the relations f +f = [M]f , ff + =[M + 1]f .3q-Supersymmetric von Neumann’s TheoremIn this section we formulate our main result, namely, we prove that an irreduciblerepresentation of the q-deformed superalgebras is, up to a superunitary transformation,unique, in the following sense:Theorem (i)Let Z = {B, B+, F, F +} be an irreducible operator family, which satisfies the q-deformed superalgebra [11]:BF=qFB , B+F + = q−1F +B+ ,(10a)BF +=q−1F +B , B+F = qFB+ ,(10b)F 2=(F +)2 = 0 ,(10c)BB+ −q−2B+B=1 + (q−2 −1)F +F ,(10d)FF + + F +F = 1;(10e)3

where q is a real number, F, F + are bounded operators on the separable Hilbert spaceH and B, B+ are densely defined closed operators in H. Let θ be a Grassmann variable,such that{θ, F} = {θ, F +} = θ2 = 0 ,(11a)[θ, B] = [θ, B+] = 0 . (11b)Let D and G be densely defined closed linear operators and define on a suitable domain[14]B′ = B + θD ,B′+ = B+ −θD+ ,(12)F ′ = F + θG ,F ′+ = F + + θG+ ,G = G00(B, B+) + G11(B, B+)F +F ,D = D10(B, B+)F + + D01(B, B+)F ,(13)where G and D are assumed to be even and odd Grassmann elements, respectively.

As-sume that the operator family Z′ = {B′, B′+, F ′, F ′+} also fulfills the algebra (10) on asuitable domain of definition.Then, under the above conditions, there exists a uniquely determined self-adjoint oddoperator A, such thatG = {A, F} , D = [A, B] ,(14)A = F +G00(B, B+) + G+00(B, B+)F = A+ ,(15)and the transformation (12) is implemented by the superunitary operator eθA such thateθABe−θA = B + θ[A, B] = B′ ,eθAFe−θA = F + θ{A, F} = F ′ . (16)Under the conditions of the above theorem, we haveG11(B, B+) = G00(qB, qB+) −G00(B, B+) ,(17a)D10(B, B+)=qG00(qB, qB+)B −BG00(qB, qB+) ,D01(B, B+)=q−1G+00(B, B+)B −BG+00(B, B+) .

(17b)The transformation eθA = 1 + θA is called superunitary if A is odd and self-adjoint. Inparticular, from this definition it follows that {θ, A} = 0 and the latter is fulfilled onlywhen θ satisfies the commutation relations (11).4

Remark:The Grassmann variable θ, being on the q-plane of the oscillators given in (10), canin general have the q-commutation relations with the elements of the algebra as:θqF + pFθq = 0 ,θqF + + p−1F +θq = 0 ,θqB −rBθq = 0 ,θqB+ −r−1B+θq = 0 ,with p and r real numbers ∗. If however, we use the above q-commutation relations with pand r, instead of commutation relations (11), from the requirement of oddness of operatorA and the superunitarity condition (eθqA)+ = e−θqA, i.e.

A+θq = −θqA, one arrives at thefollowing ”q-self adjointness” condition,A+(B, B+, F, F +) = A(rB, r−1B+, pF, p−1F +) ,instead of the usual one. The latter restriction seems rather unnatural, since a usualself-adjoint operator A on the original algebra of oscillators (10) now acquires restrictionsby the inclusion of an additional auxiliary Grassmann element θq.

Thus for our purposewe can restrict ourselves to the case p = r = 1, i.e., to the commutation relations (11).Proof of the theorem:The proof is actually quite similar to the one given in [14] for the classical Lie super-algebras.From the relations F 2 = F ′2 = 0 and using (11a) we have, according to (12), F ′2 =θ[G, F] = 0, from which we obtain[G, F] = 0 . (18)Relations (10a), (10b) imply that for any function g(B, B+) it holdsg(B, B+)F=Fg(qB, qB+) ,g(B, B+)F +=F +g(q−1B, q−1B+) .

(19)With the ansatz (13) and using (10e) and (19) we find[G, F] = {G00(B, B+) −G00(q−1B, q−1B+) −G11(q−1B, q−1B+)}F . (20)Comparing (18) and (20) we getG11(B, B+) = G00(qB, qB+) −G00(B, B+) ,(21)∗These q-commutation relations, because of associativity, are of course compatible with the supersym-metric q-Jacobi identities[[A, B}(q3,q−13), C}( q1q2 , q2q1 ) + (−1)ηA(ηB+ηC)[[B, C}(q1,q−11), A}( q2q3 , q3q2 )+(−1)ηC(ηA+ηB)[[C, A}(q2,q−12), B}( q3q1 , q1q3 ) = 0 ,where ηZ = 1 if Z is odd, ηZ = 0 if Z is even and [A, B}(p,q) ≡pAB ∓qBA, where we take the plussign when both A and B are odd, otherwise we take the minus sign.

The above expression representsthe most general form of the supersymmetric q-Jacobi identities, which includes three arbitrary complexparameters q1, q2 and q3.5

and substituting it into (13) we obtainG = G00(B, B+)FF + + G00(qB, qB+)F +F . (22)Let us now take G, in the required form (14), asG = {A, F} ,(23)where A is odd.

Assuming that A is self-adjoint, i.e. A+ = A, we can writeA = F +α(B, B+) + α+(B, B+)F(24)and we have from (23),G = α(qB, qB+)F +F + α(B, B+)FF +(25)Comparing (22) and (25), we findα(B, B+) = G00(B, B+) ,and by substituting it back into (24), the relation (15) is obtained.Now to find D, we use relations (10a) and (11).

We haveB′F ′ −qF ′B′ = θ{DF + BG −qGB + qFD} = 0 . (26)Substituting (23) into (26) and using (10a), we will obtain then that{D −[A, B]}F + qF{D −[A, B]} = 0 .

(27a)Similarly, from (10b), (11) and G+ = {A, F +} we obtain{D −[A, B]}F + + q−1F +{D −[A, B]} = 0 . (27b)Thus, from eqs.

(27) we conclude thatD = [A, B] . (28)Finally, the operator eθA is superunitary since A is odd and self-adjoint (see eq.

(24)).Moreover, we haveeθABe−θA = (1 + θA)B(1 −θA) = B + θ[A, B] = B′ ,eθAFe−θA = (1 + θA)F(1 −θA) = F + θ{A, F} = F ′ ,i.e. relations (16) hold.

The proof of the relations (17) is straighforward: eq. (17a) wasalready proved (see eq.

(21)) and relations (17b) follow from (13), (14), (15) and (19).This completes the proof. When q = 1, we reproduce the results of [14].Let us remark that a pair of one bosonic, b, b+, and one fermionic, f, f +, independentq-oscillators (which satisfy the relations (30)), can also be introduced by means of thetransformation [11]b = qN2 +MB,f = qM2 F ,[N, B] = −B,[N, B+] = B+ , N+ = N ,[M, F] = −F,[M, F +] = F + , M+ = M ,(29)6

where B and F are the elements of supercovariant algebra (10).In this case, theuniqueness (up to a superunitary transformation) of any irreducible representation ofb, b+, N, f, f +, M is given by the following theorem.Theorem (ii)Let Z = {b, b+, N, f, f +, M} be an irreducible operator family, which satisfies theq-oscillator algebra[b, f] = 0 , [b, f +]=0 ,(30a)f 2 = (f +)2=0 ,(30b)bb+ −q−1b+b=qN ,(30c)ff + + qf +f=qM ,(30d)[N, b] = −b, [N, b+]=b+ ,(30e)[M, f] = −f, [M, f +]=f + ,(30f)on a suitable domain of definition; q is real. Let θ be a Grassmann variable,{θ, f} = {θ, f +} = θ2 = 0 ,[θ, b] = [θ, b+] = 0 .

(31)Defineb′ = b + θD,f ′ = f + θG,D = D10(b, b+, M)f + + D01(b, b+, M)f ,(32)G = G00(b, b+, M) + G11(b, b+, M)f +f ,where D and G are the even and odd Grassmann elements, respectively, and have the mostgeneral form as in (32). Assume that the operator family Z′ = {b′, b′+, N′, f ′, f ′+, M′}also fulfills the algebra (30) on a suitable domain of definition.Then there exists a uniquely determined self-adjoint odd operator A, such that onecan writeG = {A, f} ,D = [A, b] ,(33)A = f +α(b, b+, M) + α+(b, b+, M)f = A+(34)and the transformation (32) is implemented by the superunitary operator eθA such thateθAbe−θA = b′ ,eθAfe−θA = f ′ .

(35)Under the conditions of the above Theorem (ii), we haveG00(b, b+, M) = α(b, b+, M)qM, G11(b, b+, M) = α(b, b+, M −1) −qα(b, b+, M)(36a)andD10(b, b+, M) = [α(b, b+, M −1), b], D01 = [α+(b, b+, M), b]. (36b)The proof is similar to the one of Theorem (i) and will not be given.7

Let us notice that from the Theorem (ii), it also follows thateθANe−θA = N + θ[A, N] = N′ ,eθAMe−θA = M + θ[A, M] = M′ . (37)Indeed, relations (35) imply that for any functions ϕ(bb+, b+b) and ψ(ff +, f +f), one haseθAϕ(bb+, b+b)e−θA = ϕ(b′b′+, b′+b′) ,eθAψ(ff +, f +f)e−θA = ψ(f ′f ′+, f ′+f ′) ;ϕ = N and ψ = M are just particular cases of these functions (see eqs.

(30c) and (30d)respectively).Here we would like to mention that there exists a relation between the transformationsof Theorem (i) and of Theorem (ii). Indeed, if one takes in (34)α(b, b+, M) = q−M2 G00(B, B+) ,(38)where B = q−N2 −Mb, F = q−M2 f according to (29), it is straighforward to show that thesuperunitary transformation generated by (34), (35) with α given in (38), corresponds tothe superunitary transformation (15), (16) of Theorem (i), and therefore eqs.

(17) aresatisfied.Also it will be interesting to find a direct relation between the superunitary transfor-mations of the usual (undeformed) and of the q-deformed cases obtained here, in the sameway as such a relation exists for the nonsupersymmetric q-oscillators treated in section 2.To summarize, we have shown that the irreducible representations of the q-superoscillatoralgebra are equivalent and are related by a unique superunitary transformation. The q-supersymmetric generalization of von Neumann’s theorem, presented above, is (N = 1supersymmetry) for only one bosonic and one fermionic degrees of freedom.

Our resultscan be extended to the supersymmetric case with any number of bosonic and fermionicdegrees of freedom. The theorem now can be formulated as follows:Theorem (finite degrees of freedom):Let Z = {bk, b+k , Nk, fk, f +k , Mk; k = 1, ..., n} be an irreducible operator set, whichsatisfies the q-oscillator algebrabkb+k −q−1b+k bk = qNk ,fkf +k + q f +k fk = qMk ,[Nk, bk] = −bk , [Nk, b+k ] = b+k ,(39)[Mk, fk] = −fk , [Mk, f +k ] = f +k ,with all the other (anti)commutation relations vanishing (i.e., independent system of q-oscillators); q is real.Assume now that another set of operators Z′ = {b′k, b′+k , N′k, f ′k, f ′+k , M′k; k = 1, ...n}also satisfies the same algebra (39).

Then the two sets Z and Z′ are equivalent, relatedby a unique superunitary transformation such thatb′k = UbkU+ ,f ′k = UfkU+ ,U = eθA ,(40)8

with the same relation between the remaining elements of the two sets. In (40) θ is a Grass-mann variable satisfying the (anti)commutation relations (31) for all the bk, b+k , fk, f +k andA is a self-adjoint odd operator.The proof of the theorem can be most easily performed by the method of induction inthe number of degrees of freedom, n. In addition to the explicit form of the superunitaryoperator for n = 1 given in Theorem (ii), we present below explicitly the formulae for thecase n = 2.Defineb′k = bk + θDk , f ′k = fk + θGk ,k = 1, 2 ,(41)where Gk have the most general formGk = G0k + G1kf +1 f1 + G2kf +2 f2 + G12k f +1 f1f +2 f2+Hkf1f2 + H′kf +1 f +2 + Kkf1f +2 + K′kf +1 f2 ;(42)all the coefficients in (42) can depend on bℓ, b+ℓ, Nℓ, Mℓ(ℓ= 1, 2).From the anticommutation relations {f ′k, f ′ℓ} = {f ′+k , f ′+ℓ} = {f ′1, f ′+2 } = {f ′+1 , f ′2} = 0,we obtain that H′k = K′k = 0 and that the coefficients Gkk, G12k , Hk, Kk (k = 1, 2) can beexpressed in terms of G01, G02, G12, G21 and their hermitian conjugates.

Now if we write Gkin the form Gk = {A, fk}, then it is straightforward to show that there exists a uniqueself-adjoint odd operator A given byA =Xkf +k G0kq−Mk +Xk̸=ℓf +k f +ℓfℓGℓk q−Mk + h.c. ,and from the commutativity between the b′k and f ′ℓ, we obtain the required form Dk =[A, bk].AcknowledgementsIt is our pleasure to thank A. Demichev for useful discussions and several clarifyingremarks. R.G.F.

would like to thank ICSC-World Laboratory for financial support. P.P.is grateful to the Research Institute for Theoretical Physics, University of Helsinki, forthe hospitality.References[1] L.D.

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