A general formula for the canonical partition function for a system obeying
any statistics based on the permutation group is derived. The formula expresses
the canonical partition function in terms of sums of Schur functions. The only
hitherto known result due to Suranyi [ Phys. Rev. Lett. {\bf 65}, 2329 (1990)]
for parasystems of order two is shown to arise as a special case of our general
formula. Our results also yield all the relevant information about the
structure of the Fock spaces for parasystems.Comment: 9 pages, No figures, Revte

A.M.L. Messiah

F.D.M. Haldane

H.S. Green

I.G. Macdonald

J.B. Hartle

O.W. Greenberg

P. Suranyi

R.H. Stolt

S. Chaturvedi

S. Doplicher

Y. Ohnuki

Y.-S. Wu

English

arXiv

Crossref

Canonical partition functions for parastatistical systems of any order

A general formula for the canonical partition function for a parastatistical system of any order is derived. The formula expresses the canonical partition functions for these in terms of sums of Schur functions.The only hitherto known result due to Suranyi [Phys. Rev. Lett. 65, 2329 (1990)] for parasystems of order two is obtained as a special case. Our results apply not only to parastatistics but to all statistics that can be defined on the basis of the permutation group, including those for which no simple definition in terms of the algebra of creation and annihilation operators is possible

Chaturvedi, S.

Publications of the IAS Fellows

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Canonical Partition Functions for Parastatistical Systems of any
order
S. Chaturvedi
School of Physics
University of Hyderabad
Hyderabad - 500 046 (INDIA)
Abstract
A general formula for the canonical partition function for a system obeying
any statistics based on the permutation group is derived. The formula ex-
presses the canonical partition function in terms of sums of Schur functions.
The only hitherto known result due to Suranyi [ Phys. Rev. Lett. 65, 2329
(1990)] for parasystems of order two is shown to arise as a special case of our
general formula. Our results also yield all the relevant information about the
structure of the Fock spaces for parasystems.
PACS No: 05.30.-d,03.65 Bz, 05.70 Ce
19 September 1995
1
There exist two approaches to parastatistics. The first is the field theoretic approach1−3
based on a generalization of the creation-annihilation operator algebra for bosons and
fermions. This is the way in which parastatistics was first introduced by Green.1 In the
second approach pioneered by Messiah and Greenberg4 and further investigated by Hartle
Stolt and Taylor,5 parastatistics arises in the quantum mechanical description of an assembly
of N -identical particles. Here the permutation group SN plays a central role. Of the two, for
calculational purposes, the first seems to have found greater favour with the workers in this
field.6,7 Thus, for instance, using this approach Suranyi7 has given the canonical partition
function for a parabose and parafermi gas of order p = 2. The calculation involves a clever
use of the simplifications which occur in the para algebra when the order of the parastatis-
tics is two. In this letter we show that the quantum mechanical approach to parastatistics
when combined with the machinery of symmetric functions8 yields a powerful method which
enables one to answer, with great facility, all questions pertaining to statistics based on the
permutation group. In particular, we give the canonical partition function for a parabose or
parafermi system of an arbitrary order.
To set the notation we begin with a brief summary of some familiar results. This is
also necessitated by the fact that it is an in-depth appreciation and examination of what is
all too familiar along with the intution gained in the process which leads us, in one stroke
to the desired results. Consider a Hilbert space H built by an N -fold tensor product of a
Hilbert space H of dim M . We shall assume that M ≥ N . Let 1, 2, 3, · · · ,M denote the
basis vectors of H . The MN basis vectors of H correspond to each term in the product
(1 + 2 + · · ·+M)(1 + 2 + · · ·+M) · · · (1 + 2 + · · ·+M)
N factors
. (1)
One may consider two decompositions of this set of MN states.
1. Decomposition based on occupation numbers:
Here one groups together states which have the same number of 1’s, 2’s · · · etc., regardless
of their location in the product. Each such group is characterized by a set of occupation
2
numbers which give the number of times 1, 2, · · ·M occur in the states in that group. All
relevant aspects of this decomposition are encapsulated in the following decomposition of
the symmetric function (x1 + · · ·+ xM )
N ,
Z infN (x1, · · · , xM) ≡ (x1 + x2 · · ·+ xM)
N =
∑
λ
|λ|=N
N !
λ1! · · ·λM !
mλ(x1, · · · , xM) . (2)
Here λ ≡ (λ1, λ2, · · · , λM), λ1 ≥ λ2 ≥ λ3 · · · ≥ λM is a partition of N (indicated by
|λ| = N) and mλ(x1, · · · , xM) denotes the monomial symmetric function
8 corresponding to
the partition λ
mλ(x1, · · · , xM) =
∑
xλ11 x
λ2
2 · · ·x
λM
M . (3)
The sum on the R.H.S. of (3) is over all distinct permutations of (λ1, · · · , λM). Setting
x1 = x2 = xM = 1 in (2) we obtain
MN =
∑
λ
N !
λ1 · · ·λM !
mλ(1, · · · , 1) , (4)
which tells us that each partition λ = (λ1, · · · , λM) corresponds to mλ(1, · · · , 1) sets of
occupation numbers obtained by distinct permutations of λi’s and each such set contains
N !/λ1! · · ·λM ! states. The number mλ(1, · · · , 1) is given by
mλ(1, 1, · · · , 1) =
M !
m1!m2! · · ·
, (5)
where mi’s denote the number of times λi’s occur in the given partition λ.
Note also that with the identification
x1 = e
−βǫ1 , x2 = e
−βǫ2, · · · , xM = e
−βǫM , (6)
where ǫ1, ǫ2 · · · , ǫM are taken to denote the “single particle energies” corresponding to the
states 1, · · · ,M , the symmetric function Z infN (x1, · · · , xM) represents the partition function
of the system under consideration. This fact is made more explicitly by rewriting (2) as
Z infN (x1, · · · , xM) = (x1 + · · ·+ xM )
N =
∑
ni
Σni=N
N !
n1!n2! · · ·nM !
xn11 x
n2
2 · · ·x
nM
M . (7)
(The symbol ZN was introduced with a view to emphasizing this point. The superscript
“inf” indicates that we are dealing with the infinite statistics9)
3
1. Decomposition based on the permutation group:
In this decomposition we regard the MN states as the carrier space for an MN dimen-
sional representation of the permutation group SN . This reducible representation can be
decomposed into the irreducible representations of SN which, as is well known, are in one
to one correspondence with the partitions of N . In this case, the relation analogous to (2)
which summarises all relevant features of this decomposition is
Z infN (x1, · · · , xM ) ≡ (x1 + · · ·+ xM)
N =
∑
λ
|λ|=N
n(λ)Sλ(x1, · · · , xM) , (8)
where Sλ(x1, · · · , xM ) are the Schur functions
8
Sλ(x1, · · · , xM) =
det(x
λj+M−j)
i
det(xM−ji )
; 1 ≤ i, j ≤M , (9)
and n(λ) denotes the dimension of the irreducible representation λ of SN . Setting x1 = x2 =
· · · = xM = 1 in (8) we get
MN =
∑
λ
|λ|=N
n(λ)Sλ(1, · · · , 1) , (10)
which tells us that Sλ(1, · · · , 1) is the number of times the irreducible representation λ occurs
in this decomposition.
So far we had been dealing with H. Following refs. 4 and 5 we now construct out of it
a generalized ray space Hphy by
• (a) admitting only those operators on H which are permutation symmetric,
• (b) identifying those states in H which have the same expectation values for all per-
mutation symmetric operators.
These assumptions, together with the Schur’s Lemma, imply that all states belonging to
an irreducible representation λ of SN count as one state of Hphy. This immediately tells us
that the symmetric function of degree N appropriate to Hphy is simply obtained by setting
n(λ) = 1 in (8).
4
ZHSTN (x1, · · · , xM) =
∑
λ
|λ|=N
Sλ(x1, · · · , xM) . (11)
This is the key result of this work. (Here we use the superscripts HST to denote Hartle Stolt
and Taylor in honour of their contributions to parastatistics). This symmetric function
captures all aspects of Hphy much the same way as (x1 + · · ·+ xM)
N does for H.
So far no restrictions have been put on λ — the sum on the R.H.S. of (11) is over all
partitions of N . We shall refer to this statistics as HST statistics. Parabose case of order p
arises when we retrict the sum in (11) to only those partitions of N whose length l(λ) (the
number of the non-zero λi’s) is less than equal to ≤ p. In terms of Young tableaux, this
amounts to retaining only those irreducible representations pf SN in which the number of
boxes in the first column is ≤ p. The appropriate symmetric function for this case is
ZP.BN (x1, · · · , xM ; p) =
∑
λ |λ|=N
l(λ)≤p
Sλ(x1, · · · , xM ) . (12)
Similarly parafermi case of order p arises when we restrict λ in (11) to those partitions whose
conjugate partition λ′ is of length ≤ p. In terms of Young tableaux this implies retaining
only those irreducible representations of SN in which the number of boxes in the first row
is ≤ p. The symmetric function appropriate to this case is
ZPFN (x1, · · · , xM ; p) =
∑
λ
|λ|=N
l(λ′)≤p
Sλ(x1, · · · , xM ) . (13)
Likewise, for the (p, q) statistics the corresponding symmetric function Z
(p,q)
N (x1, · · · , xM) is
obtained by restricting the sum in (11) to those partitions for which l(λ) ≤ p and l(λ′) ≤ q.
It may be noted that if p, q ≥ N , all these cases reduce to HST.
The symmetric functions above contain all the relevant information about the appropriate
Hphy. For instance the dimension of Hphys is obtained by setting x1 = x2 = · · ·xM =
1. The appropriate canonical partition function is obtained by setting x1 = e
−βǫ1, x2 =
e−βǫ2, · · · , xM = e
−βǫM . Some formulae which prove to be extremely useful in carrying out
the sums in (12) and (13) with restrictions on the lengths of the partitions are as follows.8
5
Sλ(x1, · · · , xM) = det(hλi − i+ j) ; 1 ≤ i, j ≤ l(λ) , (14)
Sλ(x1, · · · , xM) = det(eλ′
i
− i+ j) ; 1 ≤ i, j ≤ l(λ′) . (15)
Here the complete symmetric functions hr(x1, · · · , xM ) and the elementary symmetric
functions er(x1, · · · , xM) are defined as follows
hr(x1, · · · , xM ) =
∑
λ
|λ|=r
mλ(x1, · · · , xM) , (16)
er(x1, · · · , xM ) =
∑
i1<i2<···<ir
xi1xi2 · · ·xir . (17)
Using these formulae one can express the appropriate symmetric functions in terms of either
h’s or e’s. As an illustration, let us consider the Bose case. Here, since l(λ) ≤ 1, we have
only one term on the R.H.S. of (13) corresponding to λ = (N, 0, 0 · · · , 0). Using (14) we
obtain
ZBN (x1, · · · , xM) = hN (x1, · · · , xM) . (18)
Similarly, for the Fermi case, one has
ZFN(x1, · · · , xM) = eN(x1, · · · , xM ) . (19)
Consider parabose of order 2. Using (12) and (14) we obtain
ZPBN (x1, · · · , xM ; 2) = hN +
∑
λ1+λ2=N
λ1≥λ2
det


hλ1 hλ1+1
hλ2−1 hλ2

 , (20)
which on simplification leads to
= h2P (x1, · · · , xM) if N = 2P ,
ZPBN (x1, · · · , xM ; 2) (21)
= hP+1(x1, · · · , xM)hP (x1, · · · , xM) if N = 2P + 1 .
The result for parafermi of order 2 is obtained by replacing h’s by e’s. Thus we obtain the
the results due to Suranyi which arise as a special case of (12) and (13). One can carry out
similar calculations for any order p.
6
For the HST case a number of interesting formulae can be derived from the following
result for the Schur functions8
∑
N
∑
λ
|λ|=N
Sλ(x1, · · · , xM) =
∏
i
1
(1− xi)
∏
i<j
1
(1− xixj)
. (22)
The R.H.S. of (22), with appropriate identifications of the xi’s, can be seen to be the grand
canonical partition function for the HST statistics. Further, setting x1 = · · · = xM = t and
reading off the coefficient of tN in the resulting expression on the R.H.S. one obtains
dim(Hphys) =
∑
λ
|λ|=N
Sλ(1, · · · , 1)
=
[N/2]∑
S=0


M +N − 2S − 1
N − 2S




M(M − 1)/2 + S − 1
S

 . (23)
This formula is interesting in its own right as it gives the number of irreducible representa-
tions of SN which occur in the decomposition of theM
N dimensional reducible representation
of SN discussed above.
Finally, having derived the symmetric function appropriate to Hphy (with or without
restrictions on the irreducible representations of the permutation group) we can immediately
obtain all the relevant information regarding its occupation number decomposition. All that
needs to be done is to expand the appropriate symmetric functions in terms of the monomial
symmetric functionsmλ(x1, · · · , xM). Now since Sλ(x1, · · · , xM) andmλ(x1 · · ·xM); |λ| = N ,
serve as bases for symmetric polynomials in x1, · · · , xM of degree N , we can expand one in
terms of the other
Sχ(x1, · · · , xM) =
∑
λ
|λ|=N
Kχλmλ(x1, · · · , xM) , (24)
where Kxλ are Kostka-Foulkes numbers.
10 Substituting this in (11) we obtain
ZHSTN (x1, · · · , xM) =
∑
λ
(
∑
χ
Kχλ)mλ(x1, · · · , xM) . (25)
(For parabose or parafermi or (p, q) case one has to put appropriate restrictions on the
partitions χ in (26)). The coefficients
∑
χKχλ immediately give us the number of states
7
corresponding to the occupation numbers λ = (λ1, · · · , λM) or any distinct permutation
thereof.
To conclude, by combining the approach propounded in refs. 4 and 5, with the theory of
symmetric functions, we have been able to obtain partition functions for all statistics based
on the permutation group. Detailed analyses of the thermodynamic properties derivable
from these results would be published elsewhere.
Acknowledgements: The author wishes to express his deep gratitude towards Prof. V.
Srinivasan for his continuous encouragement and guidance. He is also grateful to Prof. C.
Musili, Dr. P.K. Panigrahi, Prof. A.K. Kapoor and Prof. R. Jagannathan for valuable
discussions and advice.
8
References
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Berlin, 1982); S.N. Biswas in Statistical Physics, eds. N. Mukunda, A.K. Rajagopal
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and A.M.L. Messiah, B138, 1155 (1965).
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9


Canonical Partition Functions for Parastatistical Systems of any order

Canonical Partition Functions for Parastatistical Systems of any order

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