A Unified View of Multimode Algebras With Fock-Like Representations

Abstract

A unified view of general multimode oscillator algebras with Fock-like representations is presented.It extends a previous analysis of the single-mode oscillator algebras.The expansion of the a i a † j operators is extended to include all normally ordered terms in creation and annihilation operators and we analyze their action on Focklike states.We restrict ourselves to the algebras compatible with number operators.The connection between these algebras and generalized statistics is analyzed.We demonstrate our approach by considering the algebras obtainable from the generalized Jordan-Wigner transformation, the para-Bose and para-Fermi algebras, the Govorkov "paraquantization" algebra and generalized quon algebra.Recently,much attention has been devoted to the study of quantum groups and algebras 1 , noncommutative spaces and geometries 2 ,generalized notions of symmetries as well as to their diverse applications in physics 3 . These approaches are in close relationship with study of deformed oscillators,algebras and their Fock representations. Single-mode oscillator algebras were studied by a number of authors 4 .A unified view of single-mode oscillator algebras was proposed by Bonatsos and Daskaloyannis 5 and Meljanac et al. 6 . Multimode oscillator algebras are much more complicated and only partial results exist in the literature 7 .Particularly,the R-matrix approach to multimode algebras was followed in Refs.(8).Some of multimode algebras,but not all, can be obtained from an ordinary Bose algebra with equal number of oscillators 9 . On the other hand, there is an old additional physical motivation to study multimode oscillator algebras,connected with the problem of generalized statistics,different from Bose and Fermi statistics 10 .The first consistent example of it was para-Bose and para-Fermi statistics 11,12 and, recently, a new paraquantization 13 satisfying trilinear commutation relations between annihilation and creation operators. These types of statistics are characterized by a discrete parameter called the order of parastatistics p ∈ N,interpolating between Bose and Fermi statistics.Recently,a new interpolation,namely infinite quon statistics characterized by a continuous parameter, has been proposed and analyzed 14,15 .Its multiparameter extension was studied in Refs.(16).Parastatistics can be applied in spaces with an arbitrary number of dimensions and those statistics with a continuous parameter could be relevant to lower dimensions.Specially,in the (2 + 1) dimensional space,another

Covariant realizations of kappa-deformed space

“…A realization of the NC coordinatesx µ in terms of ordinary commutative coordinates and their derivatives was found using the Bargman representation and the methods developed in [29], [30], [34], [35]. The goal of this section is to find covariant realizations of the algebra generated by the NC coordinatesx µ , the rotation generators determined by the commutation relations forx µ , M µν and D µ .…”

“…In particular, we use the methods for constructing deformed creation and annihilation operators in terms of ordinary bosonic multimode oscillators, i.e. a kind of bosonization [29], [30], [35].…”

Calogero Model(s) and Deformed Oscillators

“…Apart from its particular realization, S N -invariant Heisenberg algebra is basically a multimode oscillator algebra with permutational invariance. We developed earlier a general techniques for analyzing such a class of algebras [27,31]. Information about structure of the Fock space is encoded either in (i) action of annihilation operators on monomial states in Fock space, a i (a †n 1 1 · · · a †n N N |0 ) or in (ii) Gram matrices of scalar products in Fock space,…”

Fun and frustration with Calogero model

“…Apart from its particular realization, S N -invariant Heisenberg algebra is basically a multimode oscillator algebra. We developed earlier a general technique for analyzing such a class of algebras [8]. Information about structure of the Fock space is encoded in (i) action of annihilation operators on monomial states in Fock space, a i a †n1 1 · · · a †nN N |0 and (ii) Gram matrices of scalar product in Fock space, 0 |a iN · · · a i1 a † j1 · · · a † jN |0 .…”