lt7-vanderjeugt

8/13/2019 lt7-vanderjeugt

1/12

Lie Theory and Its Applications in Physics VIIed. V.K. Dobrev et al, Heron Press, Soa, 2008

Unitary representations of the Liesuperalgebra osp (1|2n ) and parabosons

S. Lievens 1 , N.I. Stoilova 1,2 , J. Van der Jeugt 11 Department of Applied Mathematics and Computer Science, Ghent University,

Krijgslaan 281-S9, B-9000 Gent, Belgium.E-mail : [email protected]

2 Institute for Nuclear Research and Nuclear Energy,Boul. Tsarigradsko Chaussee 72, 1784 Soa, Bulgaria

Abstract

It is known that there is a close connection between the Fock space of npairs of boson operators B i (i = 1 , 2, . . . , n ) and the so-called meta- plectic representation V (1) of the Lie superalgebra osp (1|2n) with lowestweight (1/ 2, 1/ 2, . . . , 1/ 2) . On the other hand, the dening relations of osp (1|2n) are equivalent to the dening relations of n pairs of parabosonoperators bi . In particular, with the usual star conditions, this implies thatthe parabosons of order p correspond to a unitary irreducible (innite-dimensional) lowest weight representation V ( p) of osp (1|2n) with lowestweight ( p/ 2, p/ 2, . . . , p / 2) . Apart from the simple cases p = 1 or n = 1 ,these representations had never been constructed due to computational dif-culties, despite their importance.

We have now managed to give an explicit and elegant construction of theserepresentations V ( p) , and can present explicit actions or matrix elementsof the osp (1|2n ) generators. Essentially, V ( p) is constructed as a quotientmodule of an induced module. In all steps of the construction and for thechosen basis vectors, the subalgebra u(n ) of osp (1|2n) plays a crucial role.

1 Introduction

Green [1] generalized the classical notion of Bose operators or bosons to para-bose operators or parabosons. The rst applications were in quantum eld the-ory [24] and quantum statistics (para-statistics) [1,5]. The generalization of theusual boson Fock space is characterized by a parameter p, referred to as the orderof the paraboson. For a single paraboson, n = 1 , the structure of the parabosonFock space is well known [6]. Surprisingly, for a system of n parabosons withn > 1, the structure of the paraboson Fock space is not known, even though itcan in principle be constructed by means of the so-called Green ansatz [1,5].

1


2/12

2 Paraboson representations

In this paper, we analyse the structure of the paraboson Fock space, forarbitrary p and n. An important step in the solution was made many yearsago by Ganchev and Palev [7], who discovered the relation between n pairsof parabosons and the dening relations for the orthosymplectic Lie superalge-bra osp (1|2n) [8]. From their result it follows that the paraboson Fock space of order p is a certain innite-dimensional unitary irreducible representation (unir-rep) V ( p) of osp (1|2n). Here, we construct this representation V ( p) explic-itly. Our solution uses group theoretical techniques, in particular the branchingosp (1|2n) sp(2n) u(n). This allows us to construct a proper Gelfand-Zetlin (GZ) basis for some induced representation [9], from which the basis forthe irreducible representation V ( p) follows. For the representation V ( p) we givean orthogonal GZ-basis, the action (matrix elements) of the paraboson operatorsin this basis, and also the character. More details of the analysis presented herecan be found in the extended paper [10].

2 The paraboson Fock space V ( p)

For a single pair ( n = 1 ) of paraboson operators b+ , b [1], the dening relationis a triple relation (with anticommutator {., .}and commutator [., .]) given by

[{b , b+ }, b ] = 2b . (1)The paraboson Fock space [6] is a Hilbert space with vacuum vector |0 , char-acterized by the following conditions:

0|0 = 1 , b |0 = 0 , (b ) = b ,{b , b+ }|0 = p |0 , (2)

and by irreducibility under the action of b+ , b . Herein, p is a parameter, knownas the order of the paraboson. In order to have a genuine inner product for theHilbert space, p should be positive and real: p > 0. A set of basis vectors forthis space, which will be denoted by V ( p), is given by

|2k = (b+ )2k

2k k!( p/ 2)k |0 , |2k + 1 =

(b+ )2k+1

2k k!2( p/ 2)k+1 |0 . (3)

This basis is orthogonal and normalized; the symbol (a)k = a(a + 1) (a +k 1) is the common Pochhammer symbol.If one considers b+ and b as odd generators of a Lie superalgebra, then theelements {b+ , b+ }, {b+ , b }and {b , b }form a basis for the even part of thissuperalgebra. Using the relations (1) it is easy to see that this superalgebra is theorthosymplectic Lie superalgebra osp (1|2), with even part sp(2) = sp(2, R ).The paraboson Fock space V ( p) is then a unirrep of osp (1|2). It splits as thedirect sum of two positive discrete series representations of sp(2) : one withlowest weight vector |0 (lowest weight p/ 2) and basis vectors |2k , and onewith lowest weight vector |1 (lowest weight 1+ p/ 2) and basis vectors |2k +1 .For p = 1 the paraboson Fock space coincides with the ordinary boson Fock space.


3/12

S. Lievens, N.I. Stoilova and J. Van der Jeugt 3

Let us now consider the case of n pairs of paraboson operators bj ( j =1, . . . , n ). The dening triple relations for such a system are given by [1]

[{bj , b

k}, bl ] = ( ) jl b

k + ( ) kl b

j , (4)

where j ,k, l {1, 2, . . . , n } and ,, {+ , } (to be interpreted as +1and 1 in the algebraic expressions and ). The paraboson Fock space V ( p) is the Hilbert space with vacuum vector |0 , dened by means of ( j, k = 1, 2, . . . , n )0|0 = 1 , bj |0 = 0 , (bj ) = bj ,{bj , b+k }|0 = p jk |0 , (5)

and by irreducibility under the action of the algebra spanned by the elements b+j ,

b

j ( j = 1 , . . . , n ), subject to (4). The parameter p is referred to as the order of the paraboson system. In general p is thought of as a positive integer, and for p = 1 the paraboson Fock space V ( p) coincides with the ordinary n-boson Fock space. We shall see that also certain non-integer p-values are allowed.

Constructing a basis for the Fock space V ( p) turns out to be a difcult prob-lem. Even the simpler question of nding the structure of V ( p) (weight struc-ture) was not solved until now. In [10] we unravel the structure of V ( p), deter-mine for which p-values V ( p) is actually a Hilbert space, construct an orthogo-nal (normalized) basis for V ( p), and give the actions of the generators bj on thebasis vectors. A summary will be presented here.

3 Relation with the Lie superalgebra osp (1|2n)The orthosymplectic superalgebra osp (1|2n) is one of the basic classical Liesuperalgebras [8]. It consists of matrices of the form

0 a a1a t1 b c

a t d bt, (6)

where a and a1 are (1 n)-matrices, b is any (n n)-matrix, and c and d aresymmetric (n n)-matrices. The even elements have a = a1 = 0 and the oddelements are those with b = c = d = 0 . It will be convenient to have the rowand column indices running from 0 to 2n (instead of 1 to 2n + 1 ), and to denoteby eij the matrix with zeros everywhere except a 1 on position (i, j ). Then theCartan subalgebra h of osp (1|2n) is spanned by the diagonal elements

h j = ejj en + j,n + j ( j = 1 , . . . , n ). (7)In terms of the dual basis j of h, the odd root vectors and corresponding rootsof osp (1|2n) are given by:

e0,k en + k, 0 k , k = 1 , . . . , n ,e0,n + k + ek, 0 k , k = 1 , . . . , n .


4/12


The even roots and root vectors are

ej,k en + k,n + j j k , j = k = 1 , . . . , n ,ej,n + k + ek,n + j j + k , j k = 1 , . . . , n ,en + j,k + en + k,j j k , j k = 1 , . . . , n .

If we introduce the following multiples of the odd root vectors

b+k = 2(e0,n + k + ek, 0), bk = 2(e0,k en + k, 0) (k = 1, . . . , n ) (8)then it is easy to verify that these operators satisfy the triple relations (4). Sinceall even root vectors can be obtained by anticommutators {b

j , b

k}, the followingholds [7]

Theorem 1 (Ganchev and Palev) As a Lie superalgebra dened by generatorsand relations, osp (1|2n) is generated by 2n odd elements bk subject to the fol-lowing (paraboson) relations:

[{bj , b

k}, bl ] = ( ) jl b

k + () kl b

j . (9)

The paraboson operators b+j are the positive odd root vectors, and the bj are the

negative odd root vectors.Recall that the paraboson Fock space V ( p) is characterized by (5). Further-

more, it is easy to verify that

{bj , b+j }= 2 h j ( j = 1 , . . . , n ). (10)

Hence we have the following:Corollary 2 The paraboson Fock space V ( p) is the unitary irreducible repre-sentation of osp (1|2n) with lowest weight ( p2 , p2 , . . . , p2 ).

In order to construct the representation V ( p) [11] one can use an inducedmodule construction. The relevant subalgebras of osp (1|2n) are easy to describeby means of the odd generators bj .Proposition 3 A basis for the even subalgebra sp (2n) of osp (1|2n) is given bythe 2n2 + n elements

{bj , bk } (1 j k n), {b+j , bk } (1 j, k n). (11)The n2 elements

{b+j , bk } ( j, k = 1, . . . , n ) (12)are a basis for the sp (2n) subalgebra u(n).

Note that with {b+j , bk } = 2E jk , the triple relations (9) imply the relations[E ij , E kl ] = jk E il li E kj . In other words, the elements {b+j , bk }form, upto a factor 2, the standard u(n) or gl(n) basis elements.


5/12


So the odd generators bj clearly reveal the subalgebra chain osp (1|2n) sp(2n)

u(n). Note that u(n) is, algebraically, the same as the general linearLie algebra gl(n). But the conditions (bj )

= bj imply that we are dealing herewith the compact form u(n).

The subalgebra u(n) can be extended to a parabolic subalgebra P of osp (1|2n) [11]:

P = span{{b+j , bk }, bj ,{bj , bk } | j, k = 1 , . . . , n }. (13)Recall that {bj , b+k }|0 = p jk |0 , with {bj , b+j } = 2h j . This meansthat the space spanned by |0 is a trivial one-dimensional u(n) module C |0of weight ( p2 , . . . , p2 ). Since bj |0 = 0 , the module C |0 can be extended toa one-dimensional P module. Now we are in a position to dene the inducedosp (1|2n) module V ( p):

V ( p) = Ind osp (1 | 2n )P C |0 . (14)This is an osp (1|2n) representation with lowest weight ( p2 , . . . , p2 ). By thePoincar e-Birkhoff-Witt theorem [9, 11], it is easy to give a basis for V ( p):

(b+1 )k 1 (b+n )k n ({b+1 , b+2 })k 12 ({b+1 , b+3 })k 13 ({b+n 1 , b+n })k n 1 ,n |0 , (15)k1 , . . . , k n , k12 , k13 . . . , kn 1,n Z + .

The difculty comes from the fact that in general V ( p) is not a simple module(i.e. not an irreducible representation) of osp (1|2n). Let M ( p) be the maximalnontrivial submodule of V ( p). Then the simple module (irreducible module),corresponding to the paraboson Fock space, is

V ( p) = V ( p)/M ( p). (16)

The purpose is now to determine the vectors belonging to M ( p), and hence tond the structure of V ( p). This is not our only goal: we also want to nd explicitmatrix elements of the osp (1|2n) generators bj in an appropriate basis of V ( p).4 Paraboson Fock representations of osp (1|2n)We shall start our analysis by considering the induced module V ( p). Computingthe action of the generators bj in the basis (15) turns out to be very difcult:we managed to do this for osp (1|4) (i.e. n = 2 ) but not for larger values of n .Furthermore, from this action itself it was still very hard to determine the vectorsbelonging to M ( p). For this reason, we have introduced a new basis for V ( p).In this new basis, matrix elements can be computed, and from these expressionsit will be clear which vectors belong to M ( p).

The old basis for V ( p) has been given in (15). From this basis, it is easyto write down the character of V ( p): this is a formal innite series of termsx j 11 x j nn , with ( j1 , . . . , j n ) a weight of V ( p) and the dimension of thisweight space. So the vacuum vector |0 of V ( p), of weight ( p2 , . . . , p2 ), yields


6/12


a term xp21 x

p2n = ( x1 xn ) p/ 2 in the character char V ( p). The weight of a

general vector follows from (15), and contributes a termxk 11 xk nn (x1x2)k 12 (x1x3)k 13 (xn 1xn )k n 1 ,n (x1 xn ) p/ 2

in the character of V ( p). Summing over all basis vectors (i.e. over allk1 , . . . , k n , k12 , k13 . . . , kn 1,n ) yields

char V ( p) = (x1 xn ) p/ 2n

i=1 (1 x i ) 1 j


7/12


Now we develop a technique to compute the matrix elements of bi in thisbasis. By the triple relations, one obtains

[{b+i , bj }, b+k ] = 2 jk b+i .With the identication {b+i , bj }= 2 E ij in the standard u(n) basis, this is equiv-alent to the adjoint action E ij ek = jk ei . In other words, the triple relationsimply that (b+1 , b+2 , . . . , b+n ) is a standard u(n) tensor of rank (1, 0, . . . , 0). Thismeans that one can attach a unique GZ-pattern with top line 10 0 to every b+j ,corresponding to the weight + j . Explicitly:

b+j

10 00010 00 0 0 0

, (22)

where the pattern consists of j 1 zero rows at the bottom, and the rst n j +1rows are of the form 10 0. The tensor product rule in u(n) reads([m]n )(10 0) = ([ m]n+1 )([m]n+2 ) ([m]n+ n ) (23)

where ([m]n ) = ( m1n , m 2n , . . . , m nn ) and a subscript k indicates an incre-ment of the kth label by 1:([m]n k ) = ( m1n , . . . , m kn 1, . . . , m nn ). (24)

In the right hand side of (23), only those components which are still partitions

(i.e. consisting of nondecreasing integers) survive.A general matrix element of b+j can now be written as follows:

(m |b+j |m) = [m]n+ k

|m )n 1b+j

[m]n

|m)n 1

= [m]n

|m)n 1 ;

10 0010 0 0[m]n+ k

|m )n 1 ([m]n+ k ||b+ ||[m]n ).

(25)

The rst factor in the right hand side is a u(n) Clebsch-Gordan coefcient [16],the second factor is a reduced matrix element. By the tensor product rule, therst line of |m ) has to be of the form (24), i.e. [m ]n = [m]n+ k for some k-value.The special u(n) CGCs appearing here are well known, and have fairly sim-ple expressions. They can be found, e.g. in [16]. They can be expressed bymeans of u(n)-u(n1) isoscalar factors and u(n1) CGCs, which on their turnare written by means of u(n 1)-u(n 2) isoscalar factors and u(n 2) CGCs,etc. The explicit form of the special u(n) CGCs appearing here is given in Ap-pendix A of [10]. The actual problem is now converted into nding expressions


8/12


for the reduced matrix elements, i.e. for the functions F k ([m]n ), for arbitraryn -tuples of non increasing nonnegative integers [m]n = ( m

1n, m

2n, . . . , m

nn):

F k ([m]n ) = F k (m1n , m 2n , . . . , m nn ) = ([ m]n+ k ||b+ ||[m]n ). (26)So one can write:

b+j |m) =k,m

[m]n

|m)n 1 ;

10 0010 0 0[m]n+ k

|m )n 1F k ([m]n )

[m]n+ k

|m )n 1,

(27)

bj |m) =k,m

[m]n k

|m )n 1 ;

10 0010 0 0[m]n

|m)n 1F k ([m]n k )

[m]n k

|m )n 1.

(28)For j = n , the CGCs in (27)-(28) take a simple form [16], and we have

b+n |m) =n

i =1

n 1k =1 (mk,n 1 m in k + i 1)n

k= i =1 (mkn m in k + i)1/ 2

F i (m1n , m 2n , . . . , m nn ) |m)+ in ; (29)

bn |m) =n

i =1 n 1k =1 (mk,n 1 m in k + i)n

k = i =1 (mkn m in k + i + 1)1/ 2

F i (m1n , . . . , m in 1, . . . , m nn ) |m) in . (30)In order to determine the n unknown functions F k , one can start from the fol-lowing action:

{bn , b+n }|m) = 2 hn |m) = ( p + 2(n

j =1

m jn n 1

j =1

m j,n 1))|m). (31)

Expressing the left hand side by means of (29)-(30), one nds a system of cou-pled recurrence relations for the functions F k . Taking the appropriate boundaryconditions into account, we have been able to solve this system of relations. Thecalculations were still very hard, and without the use of Maple we would nothave found the solution. We only present the nal result here:

Proposition 5 The reduced matrix elements F k appearing in the actions of bjon vectors

|m) of V ( p) are given by:

F k (m1n , m 2n , . . . , m nn ) = ( 1)m k +1 ,n + + m nn(mkn + n + 1 k + E m kn ( p n)) 1/ 2

n

j = k =1

mjn mkn j + km jn mkn j + k Om jn m kn

1/ 2

,

(32)


9/12


where E and Oare the even and odd functions dened by

E j = 1 if j is even and 0 otherwise ,Oj = 1 if j is odd and 0 otherwise . (33)

The proof consists of verifying that all triple relations (9) hold when acting onany vector |m). Each such verication leads to an algebraic identity in n vari-ables m1n , . . . , m nn . In these computations, there are some intermediate veri-cations: e.g. the action {b+j , bk }|m) should leave the top row of the GZ-pattern|m) invariant (since {b+j , bk }belongs to u(n)). Furthermore, it must yield (upto a factor 2) the known action of the standard u(n) matrix elements E jk in theclassical GZ-basis.

The next task is to deduce the structure of V ( p) from the general expressionof the matrix elements in V ( p). For this purpose, it is essential to investigate

for which basis elements |m) the reduced matrix element ([m

]n

||b+

||[m]n

)becomes zero. This will be governed by the factor

(mkn + n + 1 k + E m kn ( p n))in the expression of F k ([m]n ), since this is the only factor in the right handside of (32) that may become zero. If this factor is zero or negative, the assignedvector |m ) belongs to M ( p). Recall that the integers m jn satisfy m1n m2n mnn 0. If mkn = 0 (its smallest possible value), then this factor in F ktakes the value ( p k + 1) . So the p-values 1, 2, . . . , n 1 will play a specialrole. Let us depict these factors in a scheme (here shown for n = 3 ):

2,

2,

1

0,

0,

0 1,

0,

0 2,

0,

0 4,

0,

0 5,

0,

0 6,

0,

0

1,

1,

0 2,

1,

0 3,

1,

0 4,

1,

0 5,

1,

0

2,

2,

0 3,

2,

0 4,

2,

0

2,

1,

1 3,

1,

1 4,

1,

1

3,

3,

0

3,

2,

1

2,

2,

2

3,

0,

0

1,

1,

1


10/12


In this diagram, there is an edge between two partitions ([m]n ) and ([m ]n ) =([m]n

+ k) if the reduced matrix element ([m]n

+ k ||b+

||[m]n ) = F

k([m]n ) is in

general nonzero. For the boldface lines, the relevant factor (mkn + n + 1 k +E m kn ( pn)) is equal to p1; for the dotted lines, this factor is equal to p2. Asa consequence, for p = 1 the irreducible module V ( p) corresponds to the rstline of the scheme only, i.e. only partitions ([m]n ) of length 1 appear (the lengthof a partition is the number of nonzero parts). For p = 2 , the irreducible moduleV ( p) is composed of partitions ([m]n ) of length 1 or 2 only. This observationholds in general, due to the factor ( p k + 1) for F k . This nally leads to thefollowing result:Theorem 6 The osp (1|2n) representation V ( p) with lowest weight ( p2 , . . . , p2 )is a unirrep if and only if p {1, 2, . . . , n 1}or p > n 1.For p > n 1 , V ( p) = V ( p) and

char V ( p) = (x1 xn ) p/ 2

i (1 x i ) j


11/12


5 Conclusions

In this contribution, we have presented a solution to a problem that has beenopen for many years, namely giving the explicit structure of paraboson Fock representations. In order to solve this problem, we have used a combinationof known techniques and new computational power. We used in particular:the relation with unirreps of the Lie superalgebra osp (1|2n), the decomposi-tion of the induced module V ( p) with respect of the compact subalgebra u(n),the known GZ-basis for u(n) representations, the method of reduced matrix el-ements for u(n) tensor operators and known expressions for certain u(n) CGCsand isoscalar factors.

The solution given here is also the explicit solution that would be obtainedby means of Greens ansatz [1]. The method of Greens ansatz is easy to de-scribe, but difcult to perform, and has not lead to the explicit solution of the

paraboson Fock representations, as presented here. In representation theoreticterms, Greens ansatz amounts to considering the p-fold tensor product of theordinary boson Fock space, V (1) p , and extracting in this tensor product theirreducible component with lowest weight ( p2 , . . . ,

p2 ).

Since the actions of all osp (1|2n) generators bj are known explicitly, onecan also determine the action of all sp(2n) basis elements {bj , bk }. Underthis sp (2n) action, V ( p) is in general not irreducible. It is not very difcult todetermine its irreducible sp (2n) components. For p > n 1, V ( p) has n + 1irreducible sp (2n) components. For p {1, 2, . . . , n 1}, V ( p) has only p + 1irreducible sp (2n) components. The characters of these sp (2n) unirreps can beobtained from the character of V ( p), see [10].

AcknowledgmentsThe authors would like to thank Professor T.D. Palev for his interest. NIS wassupported by a project from the Fund for Scientic Research Flanders (Bel-gium) and by project P6/02 of the Interuniversity Attraction Poles Programme(Belgian State Belgian Science Policy).

References

[1] H.S. Green, Phys. Rev. 90, 270 (1953).[2] K. Dr uhl, R. Haag, J.E. Roberts, Comm. Math. Phys. 18, 204 (1970).[3] Y. Ohnuki, S. Kamefuchi, Quantum Field Theory and Parastatistics (Springer,

Berlin, 1982).

[4] F. Mansouri and Xi Zeng Wu, J. Math. Phys. 30, 892 (1989).[5] O.W. Greenberg and A.M.L. Messiah, Phys. Rev. B 138 , 1155 (1965).[6] T.D. Palev, J. Math. Phys. 23, 1778 (1982).[7] A.Ch. Ganchev and T.D. Palev, J. Math. Phys. 21, 797 (1980).[8] V.G. Kac, Adv. Math. 26, 8 (1977).[9] V.G. Kac, in: Lecture Notes in Math., vol. 626, pp. 597-626 (Berlin, Springer,

1978).


12/12


[10] S. Lievens, N.I. Stoilova and J. Van der Jeugt, The paraboson Fock space and unitary irreducible representations of the Lie superalgebra osp (1|2n) (preprintarXiv:0706.4196v1 [hep-th]).

[11] T.D. Palev, Rep. Math. Phys. 31, 241 (1992).[12] D.E. Littlewood, The theory of Group Characters and Matrix Representations of

Groups (Oxford University Press, Oxford, 1950).[13] I.G. Macdonald, Symmetric Functions and Hall Polynomials (Oxford University

Press, Oxford, 2nd edition, 1995).[14] I.M. Gelfand and M.L. Zetlin, Dokl. Akad. Nauk SSSR 71, 825 (1950).[15] G.E. Baird and L.C. Biedenharn, J. Math. Phys. 4, 1449 (1963).[16] N.Ja. Vilenkin and A.U. Klimyk, Representation of Lie Groups and Special Func-

tions, Vol. 3 (Kluwer Academic Publishers, Dordrecht, 1992).[17] V.K. Dobrev and R.B. Zhang, Phys. Atom. Nuclei 68, 1660 (2005).

Documents

lt7-vanderjeugt