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Exclusion statistics,operator algebras and Fock space representations

We study exclusion statistics within the second quantized approach. We consider operator algebras with positive definite Fock space and restrict them in a such a way that certain state vectors in Fock space are forbidden ab initio.We describe three characteristic examples of such exclusion, namely exclusion on the base space which is characterized by states with specific constraint on quantum numbers belonging to base space M (e.g. Calogero-Sutherland type of exclusion statistics), exclusion in the single-oscillator Fock space, where some states in single oscillator Fock space are forbidden (e.g. the Gentile realization of exclusion statistics) and a combination of these two exclusions (e.g. Green's realization of para-Fermi statistics). For these types of exclusions we discuss extended Haldane statistics parameters g, recently introduced by two of us in Mod.Phys.Lett.A 11, 3081 (1996), and associated counting rules. Within these three types of exclusions in Fock space the original Haldane exclusion statistics cannot be realized.


Abstract
We study exclusion statistics within the second quantized approach. We consider operator algebras with positive definite Fock space and restrict them in a such a way that certain state vectors in Fock space are forbidden ab initio. We describe three characteristic examples of such exclusion, namely exclusion on the base space which is characterized by states with specific constraint on quantum numbers belonging to base space M (e.g. Calogero-Sutherland type of exclusion statistics), exclusion in the single-oscillator Fock space , where some states in single oscillator Fock space are forbidden (e.g. the Gentile realization of exclusion statistics) and a combination of these two exclusions (e.g. Green's realization of para-Fermi statistics). For these types of exclusions we discuss extended Haldane statistics parameters g, recently introduced by two of us in Mod.Phys.Lett. A 11, 3081 (1996), and associated counting rules. Within these three types of exclusions in Fock space the original Haldane exclusion statistics cannot be realized.

Introduction
Statistics plays a fundamental role in the description of macroscopic or thermodynamic phenomena in quantum many-body systems. It is well known that the stability of matter, built out of protons and electrons, depends crucially on their fermion nature. Also, the Bose -Einstein condensation is essentially responsible for the existence of such effects as superfluidity or superconductivity.
Attempts to generalize these conventional (i.e. Bose and Fermi ) statistics date back to Gentile's [1] and Green's [2] works on parastatistics in the 1940s and 1950s. Since then a number of papers have been devoted to this topic, culminating in recent years in the discovery of the fractional quantum Hall effect (FQHE) [3], theory of anyon superconductivity [4] and the Haldane generalization of the Pauli exclusion principle [5].
In principle, there are two distinct approaches to generalized statistics. The starting point of the first approach is some symmetry principle, such as symmetric group (e.g. parastatistics [2]) , braid group (e.g. anyon statistics [4]) or quantum groups (e.g. quon statistics [6]). It can also be characterized by either an operator algebra of creation and annihilation operators with Fock-like representations (second quantization) or monodromy properties of the multiparticle wave function (first quantization). For example, Green's parastatistics [2] is based on trilinear commutation relations for particle creation and annihilation operators. For the para-Bose case of order p exactly those representations of the symmetric group S N with at most p rows in the corresponding Young pattern occur, which means that at most p particles can be in an antisymmetric state. On the contrary, for the para-Fermi case exactly those representations of the S N with at most p columns occur, which means that at most p particles can be in a symmetric state. Both cases of parastatistics have some kind of exclusion built in by their very definition, i.e. both a priori exclude certain IRREP's of the symmetric group.
The second approach is based on the state-counting procedure. It is characterized by some Hilbert space of quantum states, generally without a direct connection with creation and annihilation operators acting on Fock-like space. This class includes the recently suggested Haldane generalization of the Pauli exclusion principle, interpolating between Bose and Fermi statistics [5]. The monodromy properties of wave functions are not used to define Haldane statistics and the definition of statistics is independent of the dimension of space. Instead, the Haldane statistics of a particle [5] is determined by the statistics parameter g , which for the case of one species of particles is defined as 1) where N is the number of particles and d N is the dimension of the one-particle Hilbert space obtained by keeping the boundary conditions and quantum numbers of (N − 1 ) particles fixed. For bosons, g = 0 and for fermions, the Pauli principle implies g = 1.
Alternatively, Wu has defined the exclusion statistics by the interpolating counting formula [7], which gives the number of all independent N-particle states distributed over M quantum states described by M independent oscillators Various aspects of this novel statistics have been investigated [8] and the systems exhibiting it have been described, including 1D spinons [5] with g = 1 2 , FQHE quasiparticles [9] and anyonic systems [10] (especially anyons in the lowest Landau level in a strong magnetic field). In addition, the Haldane concept of statistics applies to the integrable models of the Calogero-Sutherland type [11,12]. There is some evidence that it can also be helpfull in understanding the low-temperature physics of 1D Luttinger liquids [13] and the models which exibits the Mott metal-insulator transition [14]. Although Haldane statistics is defined in an arbitrary number of space dimensions,it is evident that most of these examples are essentially 1-D systems. Also, there are still some poorly understood or unanswered questions such as what is microscopic realization of Haldane statistics or its algebraic and grouptheoretical characterization. Furthermore, is there any connection between Green's parastatistics and Haldane statistics, since Green's parastatistics also interpolates between bosons and fermions and generalizes the Pauli exclusion principle.
There have been several attempts to use operator methods to realize Haldane exclusion statistics algebraically [15] but the counting rule D(M,N,g), calculated for these algebras ( basically of the Gentile type ) differs from the Haldane-Wu counting rule.
In a previous paper [16] we found that any operator algebra of creation and annihilation operators with a Fock-like representations could be described in terms of extended Haldane statistics parameters. Using this result, we described para-Fermi and para-Bose statistics as exclusion statistics of the Haldane type and calculated a few extended statistics parameters.
In this paper we continue to study exclusion statistics within the second quantized approach. We consider various operator algebras with positive definite Fock spaces, which have the exclusion principle built in them by the very definition and lead to exclusion statistics. Since there are many ways to perform such exclusion, we describe three characteristic ones, namely exclusion on the base space, the Gentile type of exclusion and a combination of these two exclusions. In Section 2 we collect the basic notions of multimode operator algebras [16,17] which are necessary for  in which we give several examples of exclusions mentioned previously.
Inspired by exclusion statistics in Calogero-Sutherland model [12], in Section 3 we define restricted multimode oscillator algebra of quonic type, which depends on the parameter λ = p q and obeys generalized exclusion principle. We calculate extended Haldane statistics parameters and discuss corresponding counting rules for different values of p and q. We recover Haldane-Wu counting formula for q = 1 and integer values of p. We also briefly mention Fermi and Bose -like exclusion algebras. In Section 4 we discuss Gentile -type algebras, with common feature that restriction is placed on the single -oscillator Fock space. We find that average value of extended Haldane statistics parameters and counting rules differs from the original Haldane parameters and counting rules, in agreement with results of Chen et al. in Ref.[15].
In Section 5 we discuss Greeen's and Palev's parastatistics as types of exclusion statistics. Finally, in Section 6 we briefly summarize the main results of the paper.

Definition of the multimode oscillator algebras 2.1 Fock space
In this section we briefly review the definition of general multimode oscilator algebras possessing Fock-like representations and well defined number operators [16,17].
We start with Hermitian conjugated pairs of annihilation and creation operators, i |i ∈ M}, defined on some base space M. We build a Fock-like space starting from the unique vacuum state |0 , such that 0|0 = 1, a i |0 = 0, , ∀i ∈ M.
An arbitrary multiparticle state can be described as a linear combination of monomial state vectors (a † i 1 · · · a † in |0 ) ,and the corresponding Fock space F n is given as The annihilation operators a i act on the space F n in such a way that and so on [16,17].

Algebra
We define the algebra of creation and annihilation operators as a normally ordered ( Wick ordered ) expansion Γ ij (a † , a) ≡ a i a † j ( no symmetry principle is assumed ): where C's are scalar coefficients. Notice that there is no need to define any relation between the creation (e.g. Γ ij (a † , a † )) or annihilation (e.g. Γ ij (a, a)) operators as they appear implicitly as norm zero vectors in Fock space. ( For the treatment of the class of Wick ordered multimode oscillator algebras of the form a i a † j = δ ij 1 + k,l C kl ij a † l a k , see [18].) We also demand that the algebra (2.

Matrix of inner products A (N ) and statistics
For an N-particle state (a † Here , n 1 , n 2 ...n M are eigenvalues of the number of their scalar products [16,17]. As we have already stated, the appearance of null-vectors implies corresponding relations between monomials in a † i and reduces the number of linearly independent states in π(a † i 1 · · · a † i N |0 >).
The number of linearly independent states is now given by the rank of the matrix The set of d i 1 ,···i N for all possible i 1 , · · · i N = 1, 2, · · · M and all integers N completely characterizes the statistics and the thermodynamic properties of a free system with the corresponding Fock space. ( Note that the statistics,i.e. the set d i 1 ,···i N do not uniquely determine the algebra given by Eq. (2.3).) Now, we would like to connect the set d i 1 ,···i N with the notion of Haldane generalized exclusion statistics . Following Haldane's idea [5] , we define the dimension of the one-particle subspace of Fock space keeping the (N − 1 ) quantum numbers i 1 , · · · i N −1 inside the N-particle states, fixed : The number of all independent N-particle states distributed over M quantum states described by M independent oscillators (i = 1, 2 · · · M) is given by The next step is to define the analog of the Haldane statistics parameter g.
Recall that Haldane introduced the statistics parameter g through the change of the single-particle Hilbert space dimension d n ,Eq.(1.1). In a similar way we define extended Haldane statistics parameters [16] g i 1 ,···i N−1 ;j 1 ···j k through the change of Note that Eq.(2.6) implies that extended Haldane statistics parameters can be any rational numbers. Examples of calculation of the matrix A (N ) and extended Haldane statistics parameters g i 1 ,···i N−1 ;j 1 ···j k for parastatistics are given in Ref. [16].
In the next sections, which constitute the core of the paper, we discuss extended Haldane statistics parameters for various types of generalized exclusion statistics.

Restricted algebras and projected Fock spaces
As we have seen, one can define extended Haldane parameters for any algebra. To study exclusion statistics within the second quantized approach, we start with an operator algebra and its positive definite Fock space representation. Then we restrict the algebra in a such way that certain state vectors in Fock space are forbidden, while the rest of Fock space remains unchanged. There are many ways to perform such exclusions. Here we do not pretend to give a complete list of all possible exclusions, but we describe and analyze three main classes of exclusion statistics : • (1) Exclusion on the base space M ( example: the Calogero-Sutherland type of exclusion statistics) which is characterized by states with a specific constraint on positions, momenta or other quantum numbers belonging to the base space (lattice) M. Generally, we can write (no summation over repeated indices) where Θ jk is 0 or 1 , depending on whether the simultaneous appearance of j, k is, respectively, forbidden or allowed. The creation operators of the above Θ -restricted algebra, acting on the vacuum |0 >, formally reproduces all the states of the initial Fock space (2.1) of the algebra Γ ij , Eq.(2.3). However, owing to the appearance of Θ jk in the above restricted algebra, monomial states, which do not obey the Θ jk -restriction, will have zero norm and effectively disappear from the Fock space leading to the projected Fock space. This can be easily seen on two particle states (see, e.g. Examples 3.4 and 3.5).
Then we can restrict the algebra by a † a = φ(N)θ(N), where θ(N) is 0 or 1, depending on whether the given N -particle (excitation) state is forbidden or allowed. The simplest case [22] is θ(n) = 1, n ≤ p and θ(n) = 0, n > p.
• (3) [2,23]). Generally, where Φ(N a ) is a function of the number operators N a of the a-th particle.
In some cases it is not needed to project states out of Fock space, since the algebra itself incorporates such exclusions, for example a i a † j = Γ ij Φ(N), where Φ(N) is a functional of the total number operator such that Φ(n) > 0, n ≤ p and Φ(p + 1) = 0.

Calogero-Sutherland type of fractional statistics
Let us start with the dynamical C-S model in 1-D, a well known example of exclusion fractional statistics [11,12]. The Hamiltonian for N particles on a ring of length L is given by whereh 2 2m = 1 and d(x) = L 2π sin( πx L ) and λ ≥ 0. The spectrum of this Hamiltonian is simple and can be expresed in terms of pseudomomenta k j , j = 1, 2, ...N, in the following form : , implying fractional statistics. The ground-state energy is for n 2 = n 3 = .. = n N = 0 and reads E(k 0 We are inspired and motivated by the relation k i+1 −k i = κ(λ+n i+1 ) to construct the algebra of creation and annihilation operators characterized by λ > 0, λ ∈ R + .To do this , we start with the quon algebra [6] of creation and annihilation operators a i , a † i on the real line satisfying If |q| < 1, the corresponding Fock space is positive definite and for a generic Nparticle state with mutually different indices there are (N!) linearly independent states [19].
Without loss of generality, we can restrict ourselves to the choice q = 0 [6,24].
Then the restricted algebra, of type (1) (i.e. exclusion on the base space M), corresponding to the algebra a i a † j = δ ij , becomes i 1 ···i N = ∞ and the extended statistics parameters g i 1 ···i N ;j 1 ···j k are not well defined. However, it is possible to define these parameters in the following way. We choose the N-particle state (a † i 1 · · · a † i N |0 >) and then take the sufficiently large cut-off from the left and from the right, which includes the given N-particle state. So, we obtain a finite segment or a finite lattice with M sites,

M >> N.
We shall discuss several cases, depending on values of λ.The case when λ = p/q (p, q ∈ N) is rational , is relatively simple. In this case Ha [12] suggested normalization of the pseudo-momenta such that the neighbour momenta satisfy i α+1 − i α = p + n α q, n α ∈ N 0 and M reduces to an infinite lattice, M = Z, with Fermi oscillators placed on each site. For N-particle state (a † i 1 · · · a † i N |0 >) the number of blocked oscillators is Specially, if p = q = 1 only N-oscillators are blocked. Note that the case p = q = 1 does not correspond to the standard Fermi oscillators, but the group of p oscillators behaves like ordinary Fermi oscillator. If p = 1, q = 1, then N oscillators are blocked but internal oscillators inside the neighbours are strongly correlated.
For the closest N-particle states n 1 = n 2 = · · · = n N −1 = 0 and for the finite lattice with M sites {1, 2, · · · M}, we obtain the dimension of the one -particle subspace, Then, it is easy to find extended Haldane statistics parameters, Eq.(2.6) Hereafter, [x] ± denotes the minimal (+)/ maximal (-) integer greater / smaller than a given number x, respectively.. We observe that g N →N +j depends on M and p, q as well, and not just on the ratio λ = p/q . Moreover, there does not exist a limit value when M → ∞. However, one can define the average value of the statistics parameterḡ for M, M + 1, · · · M + q − 1, since g is periodic in M with period q. We assume that j is always "on the right" and for M >> Np, M >> q we find: This follows from the identity: Similarly, one can find g i 1 ,i 2 ,···i N ;j for arbitrary N-particle states. They depend on M , p, q and n 1 , n 2 , · · · n N −1 . The average valueḡ depends generally on p, q and n N .
Simple consideration [12] gives, for general p, q the Haldane statistics parameter which is variable. The above consideration gives, for p = 1, g Hald = 1 q and for p = q, g Hald = 1. Hence, only if p = 1, λ = 1 q , the corresponding C-S model has the Haldane statistics parameter g Hald = 1 q andḡ = g Hald . Moreover, two statistical models with the same g Hald (for example, g Hald = 1) are not the same. Namely, the counting rule D(M, N; p, q) depends also on both p and q (not only on the ratio λ = p/q ): The above equation follows from the identity [25] α−1 i=0 n + i n = α n + α n − n n + α n + 1 = n + α n + 1 , with α ∈ N and n ∈ N 0 .
If p = q, 12) and only if q = 1, Eq. (3.10) coincides with an ad hoc interpolation formula by Haldane and Wu [5]. The simple interpolation of the above equation is obtained by Θ jk = 1 if k − j ≥ p and Θ jk = 0, k − j < p ∈ N, i.e. that a single particle blocks p-units (for fermions, p = 1). We point out that the case p = 0, q = 1 makes sense and reproduces the Bose statistics D(M, N; 0, 1) = M + N − 1 N , but the case p = 0 and q = 1 corresponds to a generalized Bose statistics since We note that for fractional values of λ = p q , the counting rule, Eq.(3.10), is completely different, even asimptotically, from the Haldane-Wu formula .
If the coupling constant λ is an irrational positive number, then the Ha lattice construction and the counting formula , Eq.(3.10), cannot be applied. In this case it is more appropriate to define (3.14) where λ is the occupation width of one-particle state, and the counting rule Here we have assumed that the first particle can occupy M-states and that the whole N-particle state is smaller than M. It is interesting to note that if λ ∈ N 0 , the last equation coincides with Eq. (3.12) and with the Haldane-Wu formula. However, if λ is not an integer, then Eq. (3.15) differs from both Eq. (3.12) and  has the advantage to be well defined for any real λ ≥ 0 and if λ = p q , then D(M,N) depends only on λ. In this case, one can define an effective

Remark
The case p = q = 1 (p = 0, q = 1) corresponds to non-standard fermions (bosons) since the operators a † i do not satisfy the commutation relations for ordinary fermions ( bosons ), although the statistical properties are the same as for ordinary fermions ( bosons ) with g = 1 ( g = 0 ).
In the following subsections we briefly mention possible generalizations of exclusion fractional statistics by constructing projected Fock spaces.

Fermi-like exclusion statistics
Let us define a monotonic series (or a finite set) Then, we easily generalize the condition, Eq. (3.4), to This restriction leads to the N-particle states (The energy dependence on i α i.e. the dispersion relation is not specified). All other states are null-states. The meaning of these restrictions is that only ordered states survive and that distances between neighbours are "quantized" according to the rule i α+1 − i α ∈ X. This rule generalizes the Pauli exclusion principle and we call the corresponding statistics X-type restricted Fermi statistics. The statistics parameters g and the counting rules D(M,N;X,F) can be found using results of Ref. [25]. The special case of this statistics is the C-S type of fractional statistics (Section 3.1).

Bose-like exclusion statistics
This is in principle the same kind of exclusions as in subsection (3.2), but with the only difference that in a given state there may be an arbitrarily large number of particles (excitations), i.e. the Θ projector satisfies The corresponding allowed states are All other states are null-states. The above restrictions lead to the X-restricted Bose

Restricted Fermi algebra
One can start from the permutation invariant Fermi algebra a i a † j = δ ij − a † j a i and restrict it in different ways. For example: where The creation (annihilation) operators anti-commute as ordinary fermions, whereas the operators satisfying Eq. (3.4) have no commutation relations at all. The algebra (3.17) is different from the algebra (3.4), but their corresponding statistics are the same. The counting rule is given by Eq. (3.12).
The opposite example is which implies that many-particle states satisfy the condition N ≤ p+1 and all other states are forbidden.

Restricted Bose algebra
Starting with the Bose algebra a i a † j = δ ij + a † j a i , we can restrict it to a permutation invariant form where Θ jk is given, for example, as in subsection (3.4).
In the first example (Θ jk = 1, |k − j| ≥ p), the creation (annihilation) operators commute and the corresponding statistics are the same as for the quon-projected construction, Eqs. (3.4). The second example (Θ jk = 1, |k − j| ≤ p) is equivalent with any of (p + 1)-neighbouring Bose oscillators inside the initial lattice.
Generally, one can start with any algebra and restrict it in the following way and proceed as in the above examples. We also note that the restricted Fermi and Bose statistics can be defined not only on a line, but also on a circle or a lattice with periodic boundary conditions. A special kind of this type of statistics with is given in Ref. [26].

Remarks
The construction of the C-S type fractional statistics and generalization proposed in subsections (3.2) and (3.3) ( but not (3.4) and (3.5) ) relies crucially on the oriented 1-D space (lattice). However, we point out that this obstacle can be evaded and we suggest some interesting physical speculations.
If the creation (annihilation) operators are defined on M 1 ⊗ M 2 , where M 1 is Ddimensional and M 2 is 1-dimensional space, then one can apply the construction described in subsections (3.2) and (3.3). For example, there can be ordinary bosons and fermions in the M 1 direction but fractional in the M 2 direction: where Θ βγ is described in subsection (3.4). The whole space is not isotropic i.e.
there is a preferable direction M 2 . This is one of the main assumptions for the appearance of generalized statistics. where n α denotes the number of oscillators with α-particles (excitations) . Let N i 1 ≥ N i 2 ≥ · · · ≥ N in , then n α is number of boxes (oscillators) with α particles, if "i" is added to the (m − 1) filling and g N →N +i = ∆n m = 0 (4.2) if "i" is added to n α , α ≤ m − 2. We find Note that the average value is different from 1/m, even if we perform averaging over different N (except for the case of the single oscillator M = 1).
Generally, the Gentile-type statistics can be defined by As example, we consider restricted Bose oscillator 5) with Fock space spanned by |0 >, a † |0 >, ...(a † ) m |0 >. This oscillator coincides with the truncated Bose oscillator with cutoff [22] aa † = (1 + a † a) − (N + 1) δ N,m (4.6) or aa † = Θ(m − N + 1). The counting rule is General properties of the Gentile-type algebra is that (i) extended Haldane parameters are not constant (ii) the average value of extended Haldane statistics parameters differs from 1 m , m ∈ N except for a single oscillator for which g n→n+1 = δ n,m andḡ = 1 m for n ≤ m (iii) the counting rule differs from the Haldane-Wu formula for m = 1 and (iv) thermodynamic properties are different from the Haldane-Wu thermodynamics [8]. These results are in agreement with the results obtained by Chen et al. in Ref.[15].

Karabali -Nair realization of Gentile statistics
All algebras with the Gentile-type statistics satisfy (a i ) m = 0, but (a i ) m+1 = 0 for every i = 1, 2, · · · M, and the states a † i a † j |0 > and a † j a † i |0 >, i = j, describe the same physical state. Karabali and Nair constructed a special type of Gentile statistics in one dimension which is also of anyonic type. The corresponding statistics has all properties of Gentile statistics and differs from the original Haldane statistics.

Remarks
If the boxes are filled with small number of particles, after adding a few new particles , the system behaves like a Bose system. Contrary to this, if all boxes are filled with (m − 1) particles the system behaves like a Fermi system. For Gentile statistics for large number of states is g N →N +i = 0 and for some N-particle states g N →N +i > 0.
Finally, let us mention that besides "local" restrictions described in this paper, there are "global" restrictions on Fock space. Examples are Green's para-Bose and para-Fermi statistics [2] of order p , which can be realized through projections of complete quon Fock space [26] and parastatistics in which only states with N ≤ N 0 particles are allowed [23], regardless of their local structure.

Exclusions on the base space and single -oscillator Fock space
The construction of exclusion statistics performed in the preeceding sections can be combined to include restrictions between neighbours, as well as cutoff of single oscillators.We present two examples of such exclusions, which includes parastatistics introduced by Palev [23,28]. Consider the algebra if n + k ≤ p. If n + k = p + 1, then g n→n+k = (M −n+1) (p−n+1) , n = 1, 2, · · · p is fractional but g is not constant any more. Hence, this is not an example for the original Haldane statistics for which the statistics parameter is g = const. Moreover, the above statistics is also not the statistics of the Karabali-Nair type , where a p i = 0, a p+1 i = 0, and for any N ≤ Mp N-particle state is allowed, since we already have a 2 i = 0 and N ≤ p.
The second example is the Bose counterpart of the algebra (5.1 ), namely: Therefore, the above statistics is characterized by the Haldane statistics parameter if n + k ≤ p. If n + k = p + 1, then g n→n+k = M (p−n+1) , n = 1, 2, · · · p, is fractional but not constant . Hence, this is not an example for the original Haldane exclusion statistics. The above statistics is also not of the Karabali-Nair type , since a p i = 0, a p+1 i = 0 but N ≤ p. This would be equivalent only for the single-mode oscillator, Let us mention that Green's para-Fermi statistics [2] of order p ∈ N is also example of this kind of exclusion statistics since at most p particles can occupy a given quantum state, a p+1 i = 0. For a single oscillator a p+1 = 0, the extended statistical parameters are g i→j = 0 for j ≤ p and g i→p+1 = 1 p+1−i . In a recent papers [16,28], we have discussed these algebras and statistics in more details.

Summary
In previous papers [16,28], we defined the extended Haldane statistics parameters g, ( see Eq.(2.6)) and the counting rules D(M, N; Γ) ( see Eq.(2.5)), for the generalized statistics formulated in the second quantized approach.
In this paper, we have proposed and further analyzed three types of exclusion statistics, namely, the Calogero-Sutherland (C-S) type, the Gentile type and a combination of these two types of exclusion statistics.
We have started with the multimode oscillator algebra Γ ij , Eq.(2.3), with positive definite Fock space. Introducing the appropriately defined step -function Θ jk , we have restricted the algebra Γ ij in such a way that certain states in the Fock space of the algebra Γ ij are forbidden, i.e. they have zero norms by construction.
The realization of the C-S type of exclusion statistics relies on the quon algebra, Eq. (3.3), and on the restriction between neighbour oscillators ( placed on the 1D lattice ), induced by the step-function, Eq. (3.4). For this type of statistics, we have calculated extended statistics parameters g for the finite lattice, Eq.(3.7), and the infinite lattice, Eq. (3.9). We have also defined and calculated the average value of the extended statistics parameters, Eq. (3.8). Furthermore, we have calculated counting rule, Eq. (3.10), and discussed its dependence on the parameters p and q, Eqs. (3.11-3.13). In the subsections (3.2-3.5), we have briefly described the possible generalization of the above procedure.
As an example of the Gentile type of exclusion statistics, we have considered the Bose-like algebra with (a † ) m+1 = 0 and the step-function Θ(m − N), Eq. (4.5). This is a restriction in the single-oscillator Fock space. We have found that the extended statistics parameters are not constant and that the counting rule differs from the Haldane-Wu formula. We have also mentioned the Karabali-Nair realization of the Gentile statistics. Finally, we have described the combination of these two exclusions. As an example, we discussed parastatistics, Eqs. (5.1,5.2). We have found that the extended statistics parameters are fractional but not constant .
None of the examples of exclusion presented here, include the original Haldane proposal [5]. As we stressed before [16], it seems that the original Haldane statistics cannot be realized in the above sense, i.e. one cannot define the underlying operator algebra of creation and annihilation operators with positive Fock space and satsfying Haldane's requirements ( fractional and constant g ), except for free bosons and fermions. One should recall that the Haldane fractional exclusion statistics arises because the system is an interacting system and particles are topological excitations of a condensed matter state, rather than real particles which can exist outside the finite region of condensed matter. Hence, our analysis confirms Haldane statement that the techniques of the sceond-quantized many-body theory cannot be applied to this type of exclusion statistics.