Fixed order matrix elements of the parafermi operators

Vol. 14 (1978)

REPORTS

ON MATHEMATICAL

FIXED ORDER MATRIX ELEMENTS

No. 3

PHYSICS

OF THE PARAFERMI

OPERATORS

T. D. PALEV Joint Institute for Nuclear Research, Laboratory

of Theoretical

Physics, Dubna, U.S.S.R.

(Received December 8, 1976) Explicit representation order p.

formulae

are found for the parafermi

operators

of arbitrary

In 1953 Green [1] pointed out that the quantization rules can be considerably generalized if one does not impose the postulate for the commutator or the anticommutator of two fields to be a c-number. As a result the Bose and Fermi commutation relations were replaced by more general three linear structure relations. For the parafermi operators a:, i= 1,2 ,..., the double commutation relations can be written in a compact form as follows (E,q, E = f or 11): (1) [[4, a?], 41 = ~(~-&)2~jk~-~(E-E)2~ik~~. All single vacuum irreducible representations of the paraoperators were classified by Greenberg and Messiah [2]. They have shown that to every positive integer p, called the order of the parastatistics, there corresponds an irreducible representation. In principle, it is known how one can fmd the matrix elements of the creation and annihilation operators. For every fixed order of the statistics this could be done through the Green ansatz [l]. In a more general case one can use the interrelations between the parafermi operators and the algebra of the orthogonal group [3]. Nevertheless, general expressions for the representations of the paraoperators were not written. In the present note we write down explicit formulae for the matrix elements of the parafermi operators valid for arbitrary order of the parastatistics. Since the derivation is somewhat complicated whereas the results are comparatively simple, we give only the final expressions. Consider first a finite number a: , . . . , a,’ of parafermi creation and annihilation operators. With any fixed positive integer p we associate a set K = (k) of symmetric matrices k=

k 11 **- k 1n ..... . . ..

I k “1 ... with integer nonnegative matrix elements k,,

(2)

k nn1

satisfying the inequalities

”

0 < Trk 4 p,

O< Ckij,
i =

1, . . . . n.

T. D. PALEV

312 We set up a correspondence

between such matrices k

Ikll,kzz ,... ,k,,,klz,k13,...,kl.,k23,k24,

. . ..kz.,ks‘+,...,k3n

,..., k,-I,,).

(4)

Denote by W, the linear space which is a real linear envelope of all vectors (4) for which (3) holds. Let eij be a square n-dimensional antisymmetric matrix, efj for all i < j = 1, . . . , n. The representation of the parafermi operators of order p is realized in W, through the following relations, a:lkll,

. . . . km-1.n) = I*.., kit+19 +.a)+ Tei,k,ji*..,k,j-l,

.-*,ki,+l,

. ..).

(5a)

I=1

a;lk,, , .-. , kn-l,n> = kii[p-Trk-

~k,~+kii+I]1...,

kit-l,

. ..)+

i=l

+

2 Eljkijl *.-,k,+l,

...,k,-l,

. ..>+

j=l

On the left-hand side of the above relations Ik,, , . . . , k,_ l,n ) is an abbreviation for the vector (4). In the vectors on the right-hand side we indicate only the labels which differ from the corresponding indices on the left. It is to be understood that the vector Ik, 1, . . .) vanishes if some of its labels k,j are negative or do not fulfil inequalities (3). In our notation the vacuum corresponds to the vector jk, 1, . . . , k,_ 1,,) with all labels ki_i= 0, IO) = /o,o, . . . . 0). From (5) we have a; Ui*iO)= 6ijpjO)

(6)

as it should be for the representation with order of parastatistics p. As one may easily see from the relations (5), the parafermi operators can be represented by differential operators in the following way. Introduce the variables xii, where i, j = 1, . . . . n, and for i # jxlj = -Xjie Let 8, = a/ax,. Then UT =

aii [p-

TXjjajjj-l

,,

Ut = Xii+

c

Xij ajj.

i-l i#i

2 x,8,+ 11+ fJjja,-c 2 xkjakkajj, j=l

j=l

i#i

id

j,k=l

i#k#i

MATRIX ELEMENTS OF THE PARAFERMI

OPERATORS

313

In these notations the basis vectors (4) are

The representation formulae (5) can immediately be generalized to the case of infinite n. For this purpose all sums in (5) should be extended to inkity. Because of the inequalities (3), at most a finite number of labels in every vector Ik, 1, . . . , k,_ l,n ) differs from zero. Therefore all infinite sums in (5) will be convergent. Because of the interrelations between a given number IZof pairs of parafermi creation and annihilation operators and the algebra B,, of the orthogonal group in 2n+ 1 dimensions [4], the relations (5) generate an irreducible representation of the algebra B,. The generators of the algebra are given by the operators 4,

[a!,

41;

t,q,d=

t_; i,j, k= I,..., it.

The vacuum is the highest weight of the representation. The orthogonal signature of this representation is (p/2, . . . , p/2), [5]. If one does not impose inequalities (3), formulae (5) nevertheless do define a representation of B.; however, it becomes i&rite dimensional. REFERENCES [I] H. [2] 0. (31 A. [4] C. [5] T.

S. Green: Whys. Rev. 90 (1953), 270. W. Greenberg, A. M. L. Messiah: ibid. 138 (1965), 1155. B. Govorkov: Sequences ofparustutistics, Preprint JINR, E2-7485 (1973). Ryan, E. C. Sudarshan: Nucl. Phys: 47 (1963), 207. Palev: Ann. Inst. Henri PoincarP 23 (1975), 49.