$. However, because of the above constraint (eq(6.31)) which can be rewritten in the form : $$ \sum_\alpha n_{k\alpha} \ = \ 0 \quad {\rm or} \quad 1 \quad {\rm only} \eqno(6.33) $$ \noindent this sytem cannot be mapped into a single-indexed system. Algebra (b) can be regarded as the bosonic ``counterpoint'' of algebra (a). States are symmetric under simultanueous exchange of latin and greek indices for quanta with different latin indices $(k \ne m)$. But for quanta with identical latin indices $(k = m)$, there is no restriction on the symmetry with respect to the greek indices. In other words, there is infinite statistics in greek indices if the corresponding latin indices are identical. Whereas algebra (a) leads to more exclusive states than allowed by Pauli, algebra (b) leads to ``more inclusive'' states than allowed by Bose. For this reason we may call algebra (b) as the ``inclusive counterpoint'' to algebra (a). Such a restriction on the allowed states of a two-indexed system which distinguishes $k=m$ from $k\ne m$ cannot be mapped into a condition for a single-indexed system. Finally, algebra (c) leads to states that are symmetric for simultaneous exchange of latin and greek indices, but the stronger exclusion principle of eq (6.33) is also valid as in algebra (a). Hence, algebras (a) and (c) represent two forms of statistics which may be called antisymmetric and symmetric Hubbard statistics respectively both residing within the same reduced Fock space. On the other hand, algebra (b) and its statistics lie in a different reduced Fock space which is a Fock space with the new ``inclusion'' principle. A compilation of the algebras and statistics for two-indexed systems is given in Table IV. The $cc$ relations are not included since they can be shown to follow from the $cc^\dagger$ algebra, whenever they exist. \newpage %\end{document} %\documentstyle[12pt]{article} %\begin{document} \baselineskip=24pt \setcounter{page}{67} \noindent{\bf 7. Summary and Discussion} We have formulated a theory of generalized Fock spaces which is sufficiently general so as to encompass the well known Fock spaces and many newer ones. We have shown that such a theory can be constructed without introducing creation and annihilation operators. The only requirements for constructing a generalized Fock space are to specify the set of allowed states, and to make it an inner product space. By freeing the notion of the underlying state space from c and c$^{\dagger}$, we are able to define different forms of quantum statistics in a representation independent manner. Subsequently, one can construct c and c$^{\dagger}$ and their algebras in any desired representation. Our general formalism not only unifies the various forms of statistics and algebras proposed so far but also allows one to construct many new forms of quantum statistics as well as algebras of c and c$^{\dagger}$ in a systematic manner. Some of these are the following : \noindent (a) Many new algebras for infinite statistics \\ \noindent (b) Complex q-statistics and a number of cc$^{\dagger}$ algebras representing them \\ \noindent (c) A consistent algebra of c and c$^{\dagger}$ for ``fractional" statistics \\ \noindent (d) Null statistics or statistics of frozen order \\ \noindent (e) ``Doubly-infinite" statistics and its representations \\ \noindent (f) q-orthobose and q-orthofermi statistics \\ \noindent (g) A statistics for two-indexed systems with a new ``inclusion principle''. \\ \noindent (h) A symmetric version of Hubbard statistics. Our primary concept is that of generalized Fock space, of which many categories have been introduced in this paper. Next comes the notion of statistics which is defined by the type of symmetry or relationship among the state vectors residing in the particular type of Fock space. In a given Fock space, more than one type of symmetry can be postulated, the prime example of this being the symmetry, antisymmetry or q-symmetry in the bosonic and fermionic Fock spaces. For a given statistics, there can exist different representations of c and c$^{\dagger}$, leading to different $cc^{\dagger}$ relations. To summarize, a particular Fock space can admit different statistics, and a particular statistics can be represented by more than one $cc^{\dagger}$ algebra. But {\it {the important point is that various statistics and algebras residing in a given Fock space are all inter-related}}. These interconnections are given by generalized versions of the well-known Jordan-Wigner-Klein transformations. No such interconnections exist among statistics and algebras belonging to distinct Fock spaces. For the sake of clarity, the above-described logical order of concepts as well as their logical interconnections are presented in the form of flow charts or block diagrams in Figs.3 and 4. The single-indexed systems are considered in Fig.3. The Fock spaces of higher dimension are shown to the right of those of lower dimension . The Fock space of frozen order as well as the bosonic and fermionic Fock spaces have the lowest dimension $d = 1$ in any sector $\{n_g, n_h \ldots\}$. Next come the parafermionic and parabosonic Fock spaces which have d $>$ 1. At the extreme right, we have the super Fock space which has the largest dimension $d = s$ in each sector with s given by eq.(2.6). Null statistics and infinite statistics can be regarded as the opposite limiting cases of generalized statistics and hence these two forms of statistics along with their Fock spaces occupy the opposite ends of the diagram. Although not shown separately in Fig.3 because of lack of space, the bosonic and fermionic Fock spaces are distinct and each must be separately associated with the complete set of statistics and algebras shown. Same is true of the parabosonic and parafermionic Fock spaces. Further, there are two Fock spaces of frozen order, the bosonic and fermionic type. And finally, there exists another super Fock space with exclusion principle, which is not shown separately. Within the parafermionic and parabosonic Fock spaces many ``deformations" of parastatistics and many other representations and algebras apart from Green's trilinear algebra [35] are possible. These are indicated by the hanging arrows in Fig.3. Further, as shown by the dotted lines, there is enough room for many new varieties of Fock spaces and associated statistics and algebras. These possibilities may be pursued in the future. Coming to Fig.4 depicting the systems with two indices, here again Fock spaces of higher dimension generally lie to the right. Although shown together, the orthobosonic and orthofermionic Fock spaces must be regarded distinct. Here, one can envisage a richer harvest of new Fock spaces, statistics and algebras because of the two indices and this again is for the future. We now conclude with some general comments : \noindent 1. We must once again repeat and emphasize the point that most of the $q$-deformations on oscillators discussed in the literature amount to only a change of variable and hence must be regarded as different avatars of bosonic or fermionic systems. However, one must clearly distinguish those deformations such as the $q$-mutator algebra of Greenberg that require the construction of new types of Fock spaces. Obviously, Greenberg-type of deformations can never be reduced to change of variables living within the bosonic or fermionic Fock space. Some degree of confusion prevails in recent literature since this distinction is not kept in mind. (See for instance [7,32,41,42,47]). \vspace{.5cm} \noindent 2. In Sec.4, we have shown that algebras that are covariant under quantum groups are only a particular case of the more general class of algebras that can be derived from the formalism of generalized Fock spaces. This formalism is based on linear vector space and linear operators acting on this space; mathematically, no more sophistication is required. And yet it is capable of handling quantum-group related structures in a self-contained manner. It would seem that the basic concepts of quantum groups are contained in the theory of generalized Fock spaces and it must be possible to construct quantum group itself starting from this theory. \vspace{.5cm} \noindent 3. We have already referred to the desirability of covariance under unitary transformations that mix the indices as a requirement for the algebras of creation and annihilation operators. We shall call the algebras that satisfy this requirement as covariant algebras. This property stems from the superposition principle in quantum mechanics. Since the indices describe quantum states of a single particle, if we demand that, for any orthonormal set of quantum states obtained by superposition of the original set of quantum states, the algebra should retain the same form, then covariance under unitary transformations follows. Many of the algebras presented in this paper violate this requirement. Such noncovariant algebras probably cannot be used in a general context, as for instance, in constructing a quantum field theory that respects many of the known invariance principles such as translational or rotational invariance. Nevertheless, these algebras may be useful to describe specific systems in specific states such as those encountered in condensed matter physics. Some of these noncovariant algebras do have other nice properties, although these are motivated mainly from a mathematical point of view. This is the case of those algebras that are covariant under quantum groups. Among the algebras for single-indexed systems that have been discussed, Greenberg's $q$-mutator algebra is the only $q$-deformation that is covariant under unitary transformations, but then one has to pay the price of the enlarged Fock space. Every other known $q$-deformation leads to a noncovariant algebra. Greenberg's $q$-mutator algebra (including the case $q=0$ which is the standard representation), the canonical bosonic and fermionic algebras and Green's trilinear algebras for parabosons and parafermions are the covariant representatives living respectively in the super Fock space, bosonic and fermionic Fock spaces and the parabosonic and parafermionic Fock spaces. All the other algebras living in these three catagories of Fock spaces, although noncovariant, can be transformed to these covariant algebras through equations such as eq(4.32). This is not the case for the algebra of null statisitcs or the algebra of Boltzmann statistics with Pauli principle living respectively in the Fock space of frozen order and the super Fock space with Pauli principle. In these Fock spaces, covariant algebras do not exist. \vspace{.5cm} \noindent 4. Quantum mechanics is sometimes viewed as a deformation of classical mechanics since the commutator bracket of quantum mechanics can be related to the deformation of the classical Poisson bracket, the Planck's constant playing the role of the deformation parameter. Relying on similar reasoning it has been proposed that a deformation of canonical commutation relations will lead to fundamentally new mathematical or physical structures [48,49,50]. The analysis presented in this paper shows that nothing of this sort happens, if viewed within the framework of Fock space. The transition from classical to quantum mechanics requires the replacement of the notion of the phase space by that of the Hilbert space or Fock space. In contrast, we have seen that all the deformations of commutation relations can be formulated within the framework of Fock space. In fact most of the deformed structures proposed in the literature exist within the time-honoured bosonic and fermionic Fock spaces only. Even Greenberg's infinite statistics lives within a Fock space, although an enlarged one. \vspace{.5cm} \noindent 5. While remaining within the framework of quantum mechanics, the general theory of Fock spaces presented here throws light on the enlarged framework within which the familiar quantum field theory and statistical mechanics reside and hence may lead to newer forms of quantum field theory and statistical mechanics. This is infact the main motivation behind our work. Apart from earlier work on parastatistics [37], we may mention as examples of new forms of quantum field theories, Greenberg's construction [13,14,51] of a nonrelativistic quantum field theory based on infinite statistics and our construction [52] of a local relativistic quantum field theory based on orthostatistics. Many other forms of quantum field theories based on the generalized Fock spaces may be possible. Their formulation and study is an agenda for the future. \vspace{.5cm} \noindent 6. Although one may not be able to construct local relativistic quantum field theories corresponding to many of the newer forms of statistics and algebras, nonrelativistic quantum field theories based on these are still possible. Condensed matter physics is a rich field where applications of such theories may be relevant. In fact there is no reason why the quasiparticles encountered in condensed matter systems should be bosons or fermions only. We have shown that any of the generalized Fock spaces provides a perfectly valid quantum-mechanical framework for many-particle systems. Hence, quasiparticles living in a generalized Fock space offer an important field of study. \newpage %\end{document} %\documentstyle[12pt]{article} %\begin{document} \setcounter{page}{75} \baselineskip=24pt {\centerline{\bf Appendix A : Generalised Fock Spaces for Two-indexed Systems}} \bigskip Here we consider only those two-indexed systems in which $n_{g\alpha}$ do not exist. See Sec.6.1. \medskip \noindent{\underline {The state vectors, inner products and orthonormal sets}} $$ \langle n^{'}_g, n^{'}_h \ldots; n^{'}_{\alpha}, n^{'}_{\beta} \ldots ; \mu \vert n_g, n_h \ldots ; n_{\alpha}, n_{\beta} \ldots; \nu \rangle $$ $$ \quad = \quad \delta_{n^{'}_g n_g} \delta_{n'_h n_h} \, \ldots \delta_{n^{'}_{\alpha} n_{\alpha}} \delta_{n^{'}_{\beta} n_{\beta}} \ldots M_{\mu \nu} \eqno(A.1) $$ $$ \parallel n_g, n_h \ldots; n_{\alpha}, n_{\beta} \ldots ; \mu \gg = \sum_{\nu} \, X_{\nu \mu} \, \vert n_g, n_h \ldots; n_{\alpha}, n_{\beta} \ldots ; \nu \rangle \eqno(A.2) $$ $$ \ll n^{'}_g, n^{'}_h \ldots; n^{'}_{\alpha}, n^{'}_{\beta} \ldots ; \mu \parallel n_g, n_h \ldots ; n_{\alpha}, n_{\beta} \ldots; \nu \gg $$ $$ \quad = \quad \delta_{n^{'}_g n_g} \delta_{n^{'}_h n_h} \, \ldots \delta_{n^{'}_{\alpha} n_{\alpha}} \delta_{n^{'}_{\beta} n_{\beta}} \ldots \delta_{\mu \nu} \eqno(A.3) $$ $$ M^{-1} \, = \, XX^{\dagger} \eqno(A.4) $$ $$ I \, = \, \sum_{{\stackrel{n_g,n_h ..}{n_{\alpha}, n_{\beta}..}}} \, \sum_{\mu} \parallel n_g, n_h ..; n_{\alpha}, n_{\beta} .. ; \mu \gg \ \ll n_g, n_h .. ; n_{\alpha}, n_{\beta} \ldots; \mu \parallel \eqno(A.5) $$ $$ = \sum_{{\stackrel{n_g,n_h \ldots}{n_{\alpha}, n_{\beta}\ldots}}} \, \sum_{\lambda, \nu} \vert n_g, n_h \ldots ; n_{\alpha}, {n_\beta} \ldots ; \nu \rangle \, (M^{-1})_{\nu \lambda} \, \langle n_g \ldots ; n_{\alpha} \ldots ; \lambda \vert \eqno(A.6) $$ \vspace{0.2cm} \noindent {\underline {Projection Operators :}} $$ P (n_g,n_h \ldots; n_{\alpha}, n_{\beta} \ldots) \, $$ $$ = \, \sum_{\mu} \parallel n_g, n_h \ldots; n_{\alpha}, n_{\beta} \ldots ; \mu \gg \, \ll n_g, n_h \ldots ; n_{\alpha}, n_{\beta} \ldots; \mu \parallel \eqno(A.7) $$ $$ = \sum_{\lambda, \nu} \vert n_g, n_h \ldots ; n_{\alpha}, n_{\beta} \ldots ; \nu \rangle \, (M^{-1})_{\nu \lambda} \, \langle n_g \ldots ; n_{\alpha} \ldots ; \lambda \vert \eqno(A.8) $$ $$ I \, = \, \sum_{{\stackrel{n_g,n_h \ldots}{n_{\alpha}, n_{\beta}\ldots}}} \, P(n_g, n_h \ldots; n_{\alpha}, n_{\beta} \ldots) \eqno(A.9) $$ $$ P(n_g, n_h \ldots; n_{\alpha}, n_{\beta} \ldots) \, \parallel n^{'}_g, n^{'}_h \ldots; n^{'}_{\alpha}, n^{'}_{\beta} \ldots ; \mu \gg \, $$ $$ \quad = \quad \delta_{n_g n^{'}_g} \ldots \delta_{n_{\alpha} n^{'}_{\alpha}} \ldots \parallel n_g \ldots ; n_\alpha \ldots ; \mu \gg \eqno(A.10) $$ $$ P (n_g \ldots ; n_{\alpha} \ldots ) \, \vert n^{'}_g \ldots ; n^{'}_{\alpha} \ldots ; \mu \rangle \, $$ $$ \qquad = \quad \delta_{n_g n^{'}_g} \ldots \delta_{n_{\alpha}n^{'}_{\alpha}} \ldots \vert n_g \ldots ; n_{\alpha} \ldots ; \mu \rangle \eqno(A.11) $$ \noindent {\underline {Number operators}} $$ N_k \, = \, \sum_{{\stackrel{n_g \ldots n_k \ldots}{n_{\alpha} \ldots}}} \, n_k \, P (n_g \ldots n_k \ldots ; n_{\alpha} \ldots) \eqno(A.12) $$ $$ N_{\beta} \, = \, \sum_{{\stackrel{n_g \ldots}{n_{\alpha}, n_{\beta} \ldots}}} \, n_{\beta} \, P (n_g \ldots ; n_{\alpha}, n_{\beta} \ldots) \eqno(A.13) $$ $$ N_k \parallel n_g \ldots n_k \ldots ; n_{\alpha}, n_{\beta} \ldots ; \mu \gg $$ $$ \qquad = \quad n_k \parallel n_g \ldots n_k \ldots ; n_{\alpha}, n_{\beta} \ldots ; \mu \gg \eqno(A.14) $$ $$ N_k \vert n_g \ldots n_k \ldots ; n_{\alpha}, n_{\beta} \ldots ; \mu > = n_k \vert n_g \ldots n_k \ldots ; n_{\alpha}, n_{\beta} \ldots ; \mu \rangle \eqno(A.15) $$ $$ N_{\beta} \parallel n_g \ldots n_k \ldots ; n_{\alpha}, n_{\beta} \ldots ; \mu \gg $$ $$ \qquad = \quad n_{\beta} \parallel n_g \ldots n_k \ldots ; n_{\alpha}, n_{\beta} \ldots ; \mu \gg \eqno(A.16) $$ $$ N_{\beta} \vert n_g \ldots n_k \ldots ; n_{\alpha}, n_{\beta} \ldots ; \mu \rangle = n_{\beta} \vert n_g \ldots n_k \ldots ; n_{\alpha}, n_{\beta} \ldots ; \mu \rangle \eqno(A.17) $$ $$ [N_k, N_j] \, = \, [N_{\alpha}, N_{\beta}] \, = \, [N_k, N_{\alpha}] \, $$ $$ \qquad = \quad 0 \quad {\rm for \, any } \ k, j \ {\rm and \, any} \ \alpha, \beta \eqno(A.18) $$ \noindent Total number operator is $$ N = \sum_k N_k = \sum_\alpha N_\alpha \eqno(A.19) $$ \noindent {\underline {Creation and destruction operators}} : $$ c^{\dagger}_{j \beta} \, = \, \sum_{{\stackrel{n_g,n_j \ldots}{n_{\alpha}, n_{\beta} \ldots}}} \, \sum_{\mu^{'} \nu} \, A_{\mu^{'} \nu} \, \vert n_g \ldots (n_{j}+1) \ldots ; n_{\alpha}, n_{\beta} + 1 \ldots ; \mu^{'} \rangle \, $$ $$ \qquad \otimes \langle n_g \ldots n_j \ldots ; n_{\alpha},n_{\beta} \ldots \nu \vert \eqno(A.20) $$ $$ {[c^{\dagger}_{j \beta}, N_k]} = - c^{\dagger}_{j \beta} \delta_{jk} \eqno(A.21) $$ $$ {[c^{\dagger}_{j \beta}, N_{\alpha}]} = - c^{\dagger}_{j \beta} \delta_{\alpha \beta} \eqno(A.22) $$ \noindent For some particular $\mu$, $$ \vert 3_g, 2_h; 4_{\alpha}, 1_{\beta} ; \mu \rangle = \vert 1_{g \alpha} 1_{g \alpha} 1_{g \beta} 1_{h \alpha} 1_{h \alpha} \rangle \, = \, (c^{\dagger}_{g\alpha})^2 \, c^{\dagger}_{g \beta} \, (c^{\dagger}_{h\alpha})^2 \, \vert 0 \rangle \eqno(A.23) $$ $$ c^{\dagger}_{j\beta} \, \vert n_g \ldots \, n_{j \ldots } ; n_{\alpha}, n_{\beta} \ldots ; \lambda \rangle \, = \, \vert \underbrace{1_{j \beta,} ; n_g \cdots n_j \cdots ; n_{\alpha}, n_{\beta} \cdots ;}_ {n_g \cdots n_j +1 \cdots ; n_{\alpha}, n_{\beta} + 1} \lambda \rangle \eqno(A.24) $$ $$ c^{\dagger}_{j \sigma} \, \vert n_g \ldots n_j \ldots ; n_{\alpha} \ldots n_{\sigma} \ldots ; \lambda \rangle $$ $$ = \sum_{{\stackrel{n^{'}_g \ldots n^{'}_j \ldots}{n^{'}_{\alpha} \ldots n^{'}_{\sigma}\ldots}}} \, \sum_{\mu^{'} \nu} \, A_{\mu^{'} \nu} \, \vert n^{'}_g \ldots (n^{'}_{j}+1) \ldots ; n^{'}_{\alpha} \ldots n^{'}_{\sigma} +1 \ldots ; \mu^{'} \rangle \eqno(A.25) $$ $$ \otimes \langle n^{'}_g \ldots n^{'}_j \ldots ; n^{'}_{\alpha} \ldots n^{'}_{\sigma} \ldots ; \nu \vert n_g \ldots n_j \ldots ; n_{\alpha} \ldots n_{\sigma} \ldots ; \lambda \rangle $$ $$ = \sum_{\mu^{'} \nu} \, A_{\mu^{'} \nu} M_{\nu \lambda} \, \vert n_g \ldots (n_{j}+1) \ldots ; n_{\alpha} \ldots n_{\sigma} +1 \ldots ; \mu^{'} \rangle \eqno(A.26) $$ $$ \sum_{\nu} \, A_{\mu^{'}\nu} M_{\nu \lambda} \, = \, \delta_{\mu^{'} \lambda} \eqno(A.27) $$ $$ A \, = \, M^{-1} \eqno(A.28) $$ $$ c^{\dagger}_{j \sigma} \, = \, \sum_{{\stackrel{n_g \ldots n_j..}{n_{\alpha} .. n_{\sigma}..}}} \, \sum_{\lambda \nu} \, (M^{-1})_{\lambda \nu} \vert 1_{j\sigma} ; n_g \ldots n_j \ldots ; n_{\alpha} \ldots n_{\sigma} \ldots ; 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Lett. {\bf A 7} (1992) 3525. \end{enumerate} \end{document} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Please save this content as tables.tex and % do LaTeX the file to get the Tables I-IV % as well as the figure captions for Fig. 1-4 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \documentstyle[12pt]{article} \textheight=20cm \textwidth=15cm \setlength{\oddsidemargin}{-0.75in} \pagestyle{empty} \begin{document} \vfill \baselineskip=12pt \begin{center} \begin{tabular}{|c|l|l|} \hline & & \\ \multicolumn{1}{|c|}{\bf Statistics} & \multicolumn{1}{|c|}{\bf Representation} & \multicolumn{1}{|c|}{\bf Algebra} \\ & & \\ \hline & & \\ Boltzmann & Standard representation & $c_i c^{\dagger}_j = \delta_{ij}$ \\ & & \\ '' & q-mutator (with real q) & $c_i c^{\dagger}_j -qc^{\dagger}_j c_i = \delta_{ij}$ \\ & & \\ '' & Two-parameter algebra & \{ \begin{tabular}{l} $c_i c^{\dagger}_j - q_1 c^{\dagger}_jc_i$ \\ $-q_2 \delta_{ij} \Sigma_k c^{\dagger}_k c_k = \delta_{ij}$ \end{tabular} \\ & & \\ '' & q-mutator, transformed & $c_i c^{\dagger}_j - c^{\dagger}_j c_i = \delta_{ij} p^{2 \Sigma_{kR!P;W @,R Q(')O M;&P@,C4U(&1I=B!](&9OR!L:6YE=&\@?2!D968@+TT@>R!M;W9E=&\@?2!D968*+U @ M>R!M;W9E=&\@," q(')l:6ye="&\@R!P;W @," p($t@8w5r m " P(#4@+3$@ &-H("]R;6]V971O(&-V>" O R!P;W @,R Q(')O M;&P@,C4U(&1I=B!](&9O R!L:6YE=&\@?2!D968@+TT@>R!M;W9E=&\@?2!D968*+U @ M>R!M;W9E=&\@," q(')l:6ye="&\@R!P;W @," p($t@8w5r m " P(#4@+3$@ &-H("]R;6]V971O(&-V>" O