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“Infinite statistics” and its two-parameter q-deformation
23 May 1994
r, r~__,
PHYSICS LETTERS A
Physics Letters A 188 ( 1994 ) 210-214
1A ~St,VII'.R
" I n f i n i t e s t a t i s t i c s " a n d its t w o - p a r a m e t e r
q-deformation
A.K. Mishra 1, G. Rajasekaran Institute of Mathematical Sciences, C.I. T. Campus, Madras 600 113, India
Received 10 February 1994; accepted for publication 22 March 1994 Communicated by J.P. Vigier
Abstract
We generalize Greenberg's q-mutator algebra for infinite statistics to the most general form invariant under unitary transformations in the indices and study its consequences. Constraints on the parameter space arising from the positivity of the metric in the underlying Fock space are obtained.
Recently, Greenberg [ 1 ] studied the q-mutator algebra
coc~ - q,c~co = ~,,~,
(1t
where c and c t are annihilation and creation operators and ql is a real parameter. Fivel [2] and Zagier [3] proved that the underlying Fock space has a positive metric for - 1 ~< qt ~< 1. In this note, we generalize the above to a two-parameter algebra CaCtB--qlCtBCc~-q2t~c~#ZC~C,,, = ~,#,
c,C'~
where q2 is another real p a r a m e t e r and study the consequences. This algebra is invariant under the unitary transformation U~,,,~co,
utu
the parameters, addition of a third term p5,~ }-~. c:.c~ (where p is a real parameter) does not lead to a more general algebra. A more detailed discussion of this point is given in Ref. [4]. We first study some special points and regions in the (qL, q2) plane represented by Fig. 1. The points B and F respectively correspond to Bose or Fermi statistics. What about C and G? At C Eq. (2) becomes
(2)
7
c"' = Z
is easy to obtain a linear relation between ~ , q,c~ and ~ . c~q, and hence, barring some exceptional values of
= UU* = 1.
d~,fl Z
C,!C:, = O',,/s.
~'4)
Let us define the total number operator ?v = Z c ~ c : , , 7
(3)
which satisfies In fact, one can show that under certain conditions Eq. (2) is the most general bilinear algebra of c¢, and c~ invariant under this unitary transformation. In particular, from any cc t algebra such as Eq. (1) or (2) it E-mail: graj @imsc.ernet.in.
IN, c,,] = - c ,
6)
as can be verified using Eq. (4). In Fock space, 1 + A is a positive-definite diagonal operator. So, one is allowed to define new annihilation and creation operators by
0375-9601/94/$07.00 (~) 1994 Elsevier Science B.V. All rights reserved SSD1 0375-9601 ( 9 4 ) 0 0 2 2 5 - E
A.K. Mishra, G. Rajasekaran /Physics Letters A 188 (1994) 210-214
211
Although N can still be defined by Eq. (5), the transformation corresponding to Eq. (7) will now involve the operator 1 - N which is not positive definite and hence one cannot obtain a and a S from c and c t. So, it is best to leave Eq. (10) as it stands. Further, as will be explained below, there exists a cc relation at G,
C
tk c~cp = 0 ,
,,<
I?,
for a l l a a n d f l .
(11)
As a consequence of Eq. (1 1 ), the eigenvalues of N are restricted to 0 and 1 only. The algebra defined by Eqs. (10) and (11) has been used to construct a generalization of supersymmetric quantum mechanics named orthosupersymmetric quantum mechanics [6]. Let us now consider the algebra on the line GC (ql = 0),
d
cnc~ - q2~n# Z
C~Cr = ~nt~.
(12)
Y
Fig. 1. The (ql,q2) plane. The region of positivity in the two and three particle sectors with the index taking only two values is the triangular region lying above and bounded by the straight lines GF and GB both extended to infinity.
The n-particle inner product for this case is given by . . . c
c
c,
cl
. . . to)
= [1(I + q 2 ) ( 1 + q 2 + q ~ ) . . . co = (1 + N)l/2ao,
c~ = a~(l + N) 1/2
(7a) × (1 + q2 + q~ + ... + q ; - ' ) ]
or
an = (1 +
x ~o~6#u~~. . . . .
N)-I/2cn,
a~ = c~(l + N ) -1/2,
(7b)
so that Eq. (4) can be cast into the form
a~,a~ = & p .
(8)
Eq. (8) is in some sense the simplest algebra of creation and destruction operators and it was first proposed by Greenberg [5] as an example of "infinite statistics". Thus, Eq. (4) provides a different algebraic representation for the infinite statistics. One also notes that N is a simple quadratic expression of c and c t whereas, in terms of a and a t, it is known to be an infinite series [ 5 ]:
N = Z atar + Z a t a ~ a ' a ' + . . . .
The q2-dependent factor on the right-hand side of Eq. (13) is positive for qz > - I . We thus see that Fock space with positive-definite metric exists all along the line ql = 0, q2 > -- I, i.e. the infinite line GC starting with G and extending beyond C to q2 ~ +c~ (Fig. 1 ). For q2 = - 1, the right-hand side of (13) vanishes except for n = 1, which is consistent with Eq. ( 11 ). For the above values o f q2, c,~ct~is a positive operator and so is I + q2 ~ r c~cr because of the identity
C~C~ = 1 + q2 E
ctrcr,
(14)
which follows from (12). Now we can define \ -1/2
(9)
an=
l + q2 Z
c~cr ) Y
At the point G, we have
(13)
cn
(15)
/
and by repeating the earlier argument, we can again convert Eq. (12) to
ana~ = 8ha.
(8)
A.K. Mishra, G. Rajasekaran / Physics Letters A 188 (1994) 210-214
212
Thus, the whole line qt = 0, - 1 < q2 < oc is mapped on to the single point ql = 0, q2 = 0 and this generalizes the earlier mapping of the point q~ = 0, q2 = 1 to the point q~ = 0, q2 = 0. Within the framework of Fock space there exists [4] a systematic method to derive cc relations, if any, from the cc t relation such as Eq. (2). Using this method it is straightforward to prove that cc relations exist only at B, F and G and nowhere else in the (ql,q2) plane. We get Eq. (11) for the point G and the relations c,~cBJzc~c,~ = 0,
c~c~ - (q, 5- q2)c~ca - q 2 c t G = 1, CaCta -- (ql 5- q2)ctOco -- q2ctaCo = 1,
For q2 = 0, qx = ±1, this is a system of two independent bosonic or fermionic oscillators while for q~ = 0, q2 = :t: 1, they represent a new type of oscillators which may be called "orthobosonic" or "orthofermionic" oscillators [7 ]. An arbitrary state-vector of this system can be defined by (18)
For the dual state-vector, we shall specify the order of the indices by ( m , h, p, s . . . . I =- (01 • • - c~dc~,com.
( 19 )
The inner products can be calculated directly from the algebra (17). A general result of interest is (mtm)
!1), il),
]2), Ill), ]11), ]2),
13), 121), II11), 112), i12), II11),
!21), 13).
(21}
Let us now consider the positivity of the inner products. We have, for the two-particle sector, (212) = (212) = l + q] + q2,
{22)
(illll)
/Jl[lI~
=
q~
1
+q2
). {23,
The matrix (23) has eigenvectors
with corresponding eigenvalues 1 + q~ + q2 and l + q: q~. Hence positivity of the two-particle sector allows the infinite triangular region bounded by the straight linesGF(!+ql+q2 = 0)andGB(l+q2-q~ = O) (see Fig. l ). For the three-particle sector, we have the innerproducts (313) = (3{3) = (l+ql+q2)[l+ql+q2+{qj+q2)
2]
(24)
and a 3 × 3 matrix of inner products, ((2112i) (1]1121) (i212i)
(211111) (1]111il) (i2llil)
(2ili2)) (1111]2) (i2Ii2)
,
where
(2il2i) = (l+q2)(l+ql+q2+qd+qlq2)Wq(q2,
(25)
~lilllll) = (1 + q2)(l + q2 + q~_} + q~(ql + q2),
(26)
12q12)
= (ml/n)
----- (1 + ql + q2)(l + q2 + q2 5- qlq2),
= (1 + q ) ( 1 + q 5- q 2 ) . . . ( 1 + q + . . . + q m - J ) ,
(20) where q = q~ + q2.
[0),
(16)
the upper and lower signs being for the points F and B respectively. What about the positivity for the other regions in the (ql, q2 ) plane? To study this problem, we consider the simpler situation when the index a ranges over only two values which we shall call a and 6. So, we have c~,, co and their Hermitian conjugates satisfying the algebra
Im, h , p , g , . . . ) ~ (c~)m(c~t)n(ca~)P(cfo)~...[O).
Up to the three-particle level, we have the following set of states,
(27)
(2111~) = (111121) = qj(l + q~ + 2q2 + q~q2 + 2q 2)
(28)
A.K. Mishra, G. Rajasekaran / Physics Letters A 188 (1994) 210-214
(21112) = (i212i) = q~(1 + ql + q2),
(29)
(111112) = (121111) = q l ( 1 + q~ + q 2 ) ( l
+ q2).
(30)
We have another 3 × 3 matrix of inner products defined by exchanging barred and unbarred indices in the above matrix, but the values o f the inner products are identical to those in ( 2 5 ) - ( 3 0 ) . The determinant D o f this matrix turns out to be of the form D=(l+ql+q2)(l+q2-ql)f(ql,q2),
(31)
where f (ql, q2 ) is a polynomial which does not have any zeroes inside the triangular region bounded by G F and GB (Fig. 1 ). Thus, D vanishes only along the two lines G F and GB of Fig. 1. Also, the factor 1 + ql + q2 + (ql + q 2 ) 2 occurring in (24) does not have any zero in this region. Hence, we conclude that the positivity of the norm in the three-particle sector is preserved over the whole region for which the norm in the two-particle sector was positive. However positivity for the n-particle sector remains an open question except for the lines ql = 0, q2 >/ - 1 and q2 = 0, - 1 ~< ql ~< 1 where positivity has been proved. It is possible to get representations of the creation and destruction operators in terms of the above state vectors and their duals. The results, up to the threeparticle level, are
+ 1i)(1i I + d [ ( 1 + q2)[il) - q ~ l l i ) ] ( l i l l
+ d[(1 + q2)lil)-
+ q2)]li)-qllil)](ill[
ql[li)](El[
+ (1 + ql + q2)-~[2)(121 + . . . .
(33)
where d = (1 + q l + q 2 ) - t ( 1 - q l
+ q 2 ) -1.
for rn >I 2.
(35)
All the matrix elements in ( 2 4 ) - ( 3 0 ) are zero. There are only four nonvanishing states : [0), I1), Ii) and I 1 i) with Ill) = - [ 1 i ) . Hence,
= 40)
(36)
ca = IO)(il + II>(ill.
(37)
co
Alternatively, co and ca can be represented by 4 x 4 matrices. Orthofermi [4,7]: ql = O, q2 = - 1 , f o r m t> 2.
(38)
(32)
ca = 10)(il + 1i)(2l + (1 + qt + q2)-~12)(31 + I1)(il I + d [ ( l
(mlm) = (rhlrh) = 0
Now all the matrix elements in ( 2 2 ) - ( 3 0 ) vanish and there are only three nonvanishing states : 10), [1), [i). Hence,
+ d [ ( l + q2)lli) - q~lil)](2il + (1 + ql + q2)-~12)(121 + . . . .
The representations o f ca gwen in (32) and (33) appear to become singular for 1 + q~ + q2 ----- 0 and 1 - q~ + q2 = 0. As shown earlier these are the straight lines bounding the triangular region of positive definite norm for the two- and three-particle sectors (Fig. 1 ). The singular terms drop out, because the state vectors rA) or their duals (B] which occur in the form IA)(BI in these singular terms become null vectors on the boundaries. In general, Eqs. (32) and (33) contain an infinite number of terms on the right-hand side ; alternatively, one may represent c, and c~ by infinite-dimensional matrices. However, for two special cases, we have only a finite number of terms in (32) and (33) and c, and ca become finite-dimensional matrices. These are the Fermi-Dirac and orthofermi cases [4,7]. Fermi-Dirac: qt = - 1 , q2 = 0,
( m l m ) = (rh]gt) = 0
c,~ = I0)(1[ + 11)(2[ + (1 + ql + q2)-112)(3[
213
(34)
c~ and ca* can be obtained by taking the Hermitian conjugate of (32) and (33).
co = J0)(ll,
(39)
ca = 10)(i[.
(40)
Alternatively, ca and co can be represented by 3 × 3 matrices. Although at the two points F and G (in Fig. 1 ), co and ca are represented by finite-dimensional matrices, finite-dimensional representation of the algebra (17) is not possible along the line F G connecting these points. One may ask whether there exists an algebra which can interpolate between the Fermi-Dirac
214
A.K. Mishra, G. Rajasekaran / Physics Letters A 188 (1994) 210-214
and orthofermi points via finite-dimensional matrices. The answer turns out to be in the affirmative, but one has to go beyond the bilinear algebra and include biquadratic terms in creation and destruction operators [ 8 ]. Finally, we would like to point out that the twoparameter algebra of eq. (2) must be regarded as a general representation of "infinite statistics". Except at the points B , F and G where the existence of cc relations lead to relationships between permuted states, for other values of ql and q2 any two states obtained by permutation are independent, and so we have infinite statistics or Boltzmann statistics implemented quantum mechanically. The algebra at ql = q2 = 0 gives the simplest representation of this statistics. In general, the explicit relationship between the algebras at different points in the (ql, q2 ) plane may be complicated. We have obtained this relationship in a closed form at least for one case, namely between the line ql = O, - 1 <~ q2 <~ oc and the p o i n t q l = q2 = 0 (see Eq. (15)).
References
[I] O.W. Greenberg, Phys. Rev. D 43 (199l) 4111. [2] D.1. Fivel, Phys. Rev. Lelt. 65 (1990) 3361; 69 (1992) 2020 (E). [3] D. Zagier, Commun. Malh. Phys. 147 (1992) 199. [4] A.K. Mishra and G. Rajasekaran, Pramana J. Phys. 40 (1993) 149. [5] O.W. Greenberg, Phys. Rev. Lett. 64 (1990) 705. [6] A. Khare, A.K. Mishra and G. Rajasekaran, lnt. J. Mod. Phys. A 8 (1993) 1245. [7] A.K. Mishra and G. Rajasekaran, Prarnana J. Phys. 36 (1991) 537; 37 (1991) 455 (E). [81 A.K. Mishra, G. Rajasekaran and S.K. Soni, to be published.
r, r~__,
PHYSICS LETTERS A
Physics Letters A 188 ( 1994 ) 210-214
1A ~St,VII'.R
" I n f i n i t e s t a t i s t i c s " a n d its t w o - p a r a m e t e r
q-deformation
A.K. Mishra 1, G. Rajasekaran Institute of Mathematical Sciences, C.I. T. Campus, Madras 600 113, India
Received 10 February 1994; accepted for publication 22 March 1994 Communicated by J.P. Vigier
Abstract
We generalize Greenberg's q-mutator algebra for infinite statistics to the most general form invariant under unitary transformations in the indices and study its consequences. Constraints on the parameter space arising from the positivity of the metric in the underlying Fock space are obtained.
Recently, Greenberg [ 1 ] studied the q-mutator algebra
coc~ - q,c~co = ~,,~,
(1t
where c and c t are annihilation and creation operators and ql is a real parameter. Fivel [2] and Zagier [3] proved that the underlying Fock space has a positive metric for - 1 ~< qt ~< 1. In this note, we generalize the above to a two-parameter algebra CaCtB--qlCtBCc~-q2t~c~#ZC~C,,, = ~,#,
c,C'~
where q2 is another real p a r a m e t e r and study the consequences. This algebra is invariant under the unitary transformation U~,,,~co,
utu
the parameters, addition of a third term p5,~ }-~. c:.c~ (where p is a real parameter) does not lead to a more general algebra. A more detailed discussion of this point is given in Ref. [4]. We first study some special points and regions in the (qL, q2) plane represented by Fig. 1. The points B and F respectively correspond to Bose or Fermi statistics. What about C and G? At C Eq. (2) becomes
(2)
7
c"' = Z
is easy to obtain a linear relation between ~ , q,c~ and ~ . c~q, and hence, barring some exceptional values of
= UU* = 1.
d~,fl Z
C,!C:, = O',,/s.
~'4)
Let us define the total number operator ?v = Z c ~ c : , , 7
(3)
which satisfies In fact, one can show that under certain conditions Eq. (2) is the most general bilinear algebra of c¢, and c~ invariant under this unitary transformation. In particular, from any cc t algebra such as Eq. (1) or (2) it E-mail: graj @imsc.ernet.in.
IN, c,,] = - c ,
6)
as can be verified using Eq. (4). In Fock space, 1 + A is a positive-definite diagonal operator. So, one is allowed to define new annihilation and creation operators by
0375-9601/94/$07.00 (~) 1994 Elsevier Science B.V. All rights reserved SSD1 0375-9601 ( 9 4 ) 0 0 2 2 5 - E
A.K. Mishra, G. Rajasekaran /Physics Letters A 188 (1994) 210-214
211
Although N can still be defined by Eq. (5), the transformation corresponding to Eq. (7) will now involve the operator 1 - N which is not positive definite and hence one cannot obtain a and a S from c and c t. So, it is best to leave Eq. (10) as it stands. Further, as will be explained below, there exists a cc relation at G,
C
tk c~cp = 0 ,
,,<
I?,
for a l l a a n d f l .
(11)
As a consequence of Eq. (1 1 ), the eigenvalues of N are restricted to 0 and 1 only. The algebra defined by Eqs. (10) and (11) has been used to construct a generalization of supersymmetric quantum mechanics named orthosupersymmetric quantum mechanics [6]. Let us now consider the algebra on the line GC (ql = 0),
d
cnc~ - q2~n# Z
C~Cr = ~nt~.
(12)
Y
Fig. 1. The (ql,q2) plane. The region of positivity in the two and three particle sectors with the index taking only two values is the triangular region lying above and bounded by the straight lines GF and GB both extended to infinity.
The n-particle inner product for this case is given by . . . c
c
c,
cl
. . . to)
= [1(I + q 2 ) ( 1 + q 2 + q ~ ) . . . co = (1 + N)l/2ao,
c~ = a~(l + N) 1/2
(7a) × (1 + q2 + q~ + ... + q ; - ' ) ]
or
an = (1 +
x ~o~6#u~~. . . . .
N)-I/2cn,
a~ = c~(l + N ) -1/2,
(7b)
so that Eq. (4) can be cast into the form
a~,a~ = & p .
(8)
Eq. (8) is in some sense the simplest algebra of creation and destruction operators and it was first proposed by Greenberg [5] as an example of "infinite statistics". Thus, Eq. (4) provides a different algebraic representation for the infinite statistics. One also notes that N is a simple quadratic expression of c and c t whereas, in terms of a and a t, it is known to be an infinite series [ 5 ]:
N = Z atar + Z a t a ~ a ' a ' + . . . .
The q2-dependent factor on the right-hand side of Eq. (13) is positive for qz > - I . We thus see that Fock space with positive-definite metric exists all along the line ql = 0, q2 > -- I, i.e. the infinite line GC starting with G and extending beyond C to q2 ~ +c~ (Fig. 1 ). For q2 = - 1, the right-hand side of (13) vanishes except for n = 1, which is consistent with Eq. ( 11 ). For the above values o f q2, c,~ct~is a positive operator and so is I + q2 ~ r c~cr because of the identity
C~C~ = 1 + q2 E
ctrcr,
(14)
which follows from (12). Now we can define \ -1/2
(9)
an=
l + q2 Z
c~cr ) Y
At the point G, we have
(13)
cn
(15)
/
and by repeating the earlier argument, we can again convert Eq. (12) to
ana~ = 8ha.
(8)
A.K. Mishra, G. Rajasekaran / Physics Letters A 188 (1994) 210-214
212
Thus, the whole line qt = 0, - 1 < q2 < oc is mapped on to the single point ql = 0, q2 = 0 and this generalizes the earlier mapping of the point q~ = 0, q2 = 1 to the point q~ = 0, q2 = 0. Within the framework of Fock space there exists [4] a systematic method to derive cc relations, if any, from the cc t relation such as Eq. (2). Using this method it is straightforward to prove that cc relations exist only at B, F and G and nowhere else in the (ql,q2) plane. We get Eq. (11) for the point G and the relations c,~cBJzc~c,~ = 0,
c~c~ - (q, 5- q2)c~ca - q 2 c t G = 1, CaCta -- (ql 5- q2)ctOco -- q2ctaCo = 1,
For q2 = 0, qx = ±1, this is a system of two independent bosonic or fermionic oscillators while for q~ = 0, q2 = :t: 1, they represent a new type of oscillators which may be called "orthobosonic" or "orthofermionic" oscillators [7 ]. An arbitrary state-vector of this system can be defined by (18)
For the dual state-vector, we shall specify the order of the indices by ( m , h, p, s . . . . I =- (01 • • - c~dc~,com.
( 19 )
The inner products can be calculated directly from the algebra (17). A general result of interest is (mtm)
!1), il),
]2), Ill), ]11), ]2),
13), 121), II11), 112), i12), II11),
!21), 13).
(21}
Let us now consider the positivity of the inner products. We have, for the two-particle sector, (212) = (212) = l + q] + q2,
{22)
(illll)
/Jl[lI~
=
q~
1
+q2
). {23,
The matrix (23) has eigenvectors
with corresponding eigenvalues 1 + q~ + q2 and l + q: q~. Hence positivity of the two-particle sector allows the infinite triangular region bounded by the straight linesGF(!+ql+q2 = 0)andGB(l+q2-q~ = O) (see Fig. l ). For the three-particle sector, we have the innerproducts (313) = (3{3) = (l+ql+q2)[l+ql+q2+{qj+q2)
2]
(24)
and a 3 × 3 matrix of inner products, ((2112i) (1]1121) (i212i)
(211111) (1]111il) (i2llil)
(2ili2)) (1111]2) (i2Ii2)
,
where
(2il2i) = (l+q2)(l+ql+q2+qd+qlq2)Wq(q2,
(25)
~lilllll) = (1 + q2)(l + q2 + q~_} + q~(ql + q2),
(26)
12q12)
= (ml/n)
----- (1 + ql + q2)(l + q2 + q2 5- qlq2),
= (1 + q ) ( 1 + q 5- q 2 ) . . . ( 1 + q + . . . + q m - J ) ,
(20) where q = q~ + q2.
[0),
(16)
the upper and lower signs being for the points F and B respectively. What about the positivity for the other regions in the (ql, q2 ) plane? To study this problem, we consider the simpler situation when the index a ranges over only two values which we shall call a and 6. So, we have c~,, co and their Hermitian conjugates satisfying the algebra
Im, h , p , g , . . . ) ~ (c~)m(c~t)n(ca~)P(cfo)~...[O).
Up to the three-particle level, we have the following set of states,
(27)
(2111~) = (111121) = qj(l + q~ + 2q2 + q~q2 + 2q 2)
(28)
A.K. Mishra, G. Rajasekaran / Physics Letters A 188 (1994) 210-214
(21112) = (i212i) = q~(1 + ql + q2),
(29)
(111112) = (121111) = q l ( 1 + q~ + q 2 ) ( l
+ q2).
(30)
We have another 3 × 3 matrix of inner products defined by exchanging barred and unbarred indices in the above matrix, but the values o f the inner products are identical to those in ( 2 5 ) - ( 3 0 ) . The determinant D o f this matrix turns out to be of the form D=(l+ql+q2)(l+q2-ql)f(ql,q2),
(31)
where f (ql, q2 ) is a polynomial which does not have any zeroes inside the triangular region bounded by G F and GB (Fig. 1 ). Thus, D vanishes only along the two lines G F and GB of Fig. 1. Also, the factor 1 + ql + q2 + (ql + q 2 ) 2 occurring in (24) does not have any zero in this region. Hence, we conclude that the positivity of the norm in the three-particle sector is preserved over the whole region for which the norm in the two-particle sector was positive. However positivity for the n-particle sector remains an open question except for the lines ql = 0, q2 >/ - 1 and q2 = 0, - 1 ~< ql ~< 1 where positivity has been proved. It is possible to get representations of the creation and destruction operators in terms of the above state vectors and their duals. The results, up to the threeparticle level, are
+ 1i)(1i I + d [ ( 1 + q2)[il) - q ~ l l i ) ] ( l i l l
+ d[(1 + q2)lil)-
+ q2)]li)-qllil)](ill[
ql[li)](El[
+ (1 + ql + q2)-~[2)(121 + . . . .
(33)
where d = (1 + q l + q 2 ) - t ( 1 - q l
+ q 2 ) -1.
for rn >I 2.
(35)
All the matrix elements in ( 2 4 ) - ( 3 0 ) are zero. There are only four nonvanishing states : [0), I1), Ii) and I 1 i) with Ill) = - [ 1 i ) . Hence,
= 40)
(36)
ca = IO)(il + II>(ill.
(37)
co
Alternatively, co and ca can be represented by 4 x 4 matrices. Orthofermi [4,7]: ql = O, q2 = - 1 , f o r m t> 2.
(38)
(32)
ca = 10)(il + 1i)(2l + (1 + qt + q2)-~12)(31 + I1)(il I + d [ ( l
(mlm) = (rhlrh) = 0
Now all the matrix elements in ( 2 2 ) - ( 3 0 ) vanish and there are only three nonvanishing states : 10), [1), [i). Hence,
+ d [ ( l + q2)lli) - q~lil)](2il + (1 + ql + q2)-~12)(121 + . . . .
The representations o f ca gwen in (32) and (33) appear to become singular for 1 + q~ + q2 ----- 0 and 1 - q~ + q2 = 0. As shown earlier these are the straight lines bounding the triangular region of positive definite norm for the two- and three-particle sectors (Fig. 1 ). The singular terms drop out, because the state vectors rA) or their duals (B] which occur in the form IA)(BI in these singular terms become null vectors on the boundaries. In general, Eqs. (32) and (33) contain an infinite number of terms on the right-hand side ; alternatively, one may represent c, and c~ by infinite-dimensional matrices. However, for two special cases, we have only a finite number of terms in (32) and (33) and c, and ca become finite-dimensional matrices. These are the Fermi-Dirac and orthofermi cases [4,7]. Fermi-Dirac: qt = - 1 , q2 = 0,
( m l m ) = (rh]gt) = 0
c,~ = I0)(1[ + 11)(2[ + (1 + ql + q2)-112)(3[
213
(34)
c~ and ca* can be obtained by taking the Hermitian conjugate of (32) and (33).
co = J0)(ll,
(39)
ca = 10)(i[.
(40)
Alternatively, ca and co can be represented by 3 × 3 matrices. Although at the two points F and G (in Fig. 1 ), co and ca are represented by finite-dimensional matrices, finite-dimensional representation of the algebra (17) is not possible along the line F G connecting these points. One may ask whether there exists an algebra which can interpolate between the Fermi-Dirac
214
A.K. Mishra, G. Rajasekaran / Physics Letters A 188 (1994) 210-214
and orthofermi points via finite-dimensional matrices. The answer turns out to be in the affirmative, but one has to go beyond the bilinear algebra and include biquadratic terms in creation and destruction operators [ 8 ]. Finally, we would like to point out that the twoparameter algebra of eq. (2) must be regarded as a general representation of "infinite statistics". Except at the points B , F and G where the existence of cc relations lead to relationships between permuted states, for other values of ql and q2 any two states obtained by permutation are independent, and so we have infinite statistics or Boltzmann statistics implemented quantum mechanically. The algebra at ql = q2 = 0 gives the simplest representation of this statistics. In general, the explicit relationship between the algebras at different points in the (ql, q2 ) plane may be complicated. We have obtained this relationship in a closed form at least for one case, namely between the line ql = O, - 1 <~ q2 <~ oc and the p o i n t q l = q2 = 0 (see Eq. (15)).
References
[I] O.W. Greenberg, Phys. Rev. D 43 (199l) 4111. [2] D.1. Fivel, Phys. Rev. Lelt. 65 (1990) 3361; 69 (1992) 2020 (E). [3] D. Zagier, Commun. Malh. Phys. 147 (1992) 199. [4] A.K. Mishra and G. Rajasekaran, Pramana J. Phys. 40 (1993) 149. [5] O.W. Greenberg, Phys. Rev. Lett. 64 (1990) 705. [6] A. Khare, A.K. Mishra and G. Rajasekaran, lnt. J. Mod. Phys. A 8 (1993) 1245. [7] A.K. Mishra and G. Rajasekaran, Prarnana J. Phys. 36 (1991) 537; 37 (1991) 455 (E). [81 A.K. Mishra, G. Rajasekaran and S.K. Soni, to be published.