Quon Algebra Research Articles

The inverse of a quantum bilinear form of the oriented braid arrangement

We follow here the results of Varchenko, who assigned to each weighted arrangement \(\mathcal{A}\) of hyperplanes in the \(n\)-dimensional real space a bilinear form, which he called the quantum bilinear form of the arrangement \(\mathcal{A}\). We briefly explain the quantum bilinear form of the oriented braid arrangement in the \(n\)-dimensional real space. The main concern of this paper is to compute the inverse of the matrix of the quantum bilinear form of the oriented braid arrangement in \(\mathbb{R}^n\), \({n\ge 2}\). To solve this problem, in [3] the authors used some special matrices and their factorizations in terms of simpler matrices. So, to simplify some matrix calculations, we first introduce a twisted group algebra \({\mathcal{A}(S_{n})}\) of the symmetric group \(S_{n}\) with coefficients in the polynomial ring in \(n^2\) commutative variables and then use a natural representation of some elements of the algebra \({\mathcal{A}(S_{n})}\) on the generic weight subspaces of the multiparametric quon algebra \({\mathcal{B}}\), which immediately gives the corresponding matrices of the quantum bilinear form.

A Multiparametric Quon Algebra

The quon algebra is an approach to particle statistics introduced by Greenberg to provide a theory in which the Pauli exclusion principle and Bose statistics are violated by a small amount. We generalize these models by introducing a deformation of the quon algebra generated by a collection of operators $$\mathtt {a}_i$$ , $$i \in {\mathbb {N}}^*$$ the set of positive integers, on an infinite dimensional module satisfying the $$q_{i,j}$$ -mutator relations $$\mathtt {a}_i \mathtt {a}_j^{\dag } - q_{i,j}\, \mathtt {a}_j^{\dag } \mathtt {a}_i = \delta _{i,j}$$ . The realizability of our model is proved by means of the Aguiar-Mahajan bilinear form on the chambers of hyperplane arrangements. We show that, for suitable values of $$q_{i,j}$$ , the module generated by the particle states obtained by applying combinations of $$\mathtt {a}_i$$ ’s and $$\mathtt {a}_i^{\dag }$$ ’s to a vacuum state $$|0\rangle $$ is an indefinite Hilbert module. Furthermore, we recover the extended Zagier’s conjecture established independently by Meljanac et al., and by Duchamp et al.

A Deformed Quon Algebra

Abstract The quon algebra is an approach to particle statistics in order to provide a theory in which the Pauli exclusion principle and Bose statistics are violated by a small amount. The quons are particles whose annihilation and creation operators obey the quon algebra which interpolates between fermions and bosons. In this paper we generalize these models by introducing a deformation of the quon algebra generated by a collection of operators a i,k, (i, k) ∈ ℕ* × [m], on an infinite dimensional vector space satisfying the deformed q-mutator relations a j , l a i , k † = q a i , k † a j , l + q β k , l δ i , j {a_j}_{,l}a_{i,k}^\dagger = qa_{i,k}^\dagger{a_{j,l}} + {q^{{\beta _{k,l}}}}{\delta _{i,j}} We prove the realizability of our model by showing that, for suitable values of q, the vector space generated by the particle states obtained by applying combinations of a i,k’s and a i , k † a_{i,k}^\dagger ‘s to a vacuum state |0〉 is a Hilbert space. The proof particularly needs the investigation of the new statistic cinv and representations of the colored permutation group.

Some factorizations in the twisted group algebra of symmetric groups

In this paper we will give a similar actorization as in [3, 4], where Svrtan and Meljanac examined certain matrix factorizations on Fock-like representation of a multiparametric quon algebra on the free associative algebra of noncommuting polynomials equipped with multiparametric partial derivatives. In order to replace these matrix factorizations (given from the right) by twisted algebra computation, we first consider the natural action of the symmetric group Sn on the polynomial ring Rn in n2 commuting variables Xab and also introduce a twisted group algebra (defined by the action of Sn on Rn) which we denote by A(Sn). Here we consider some factorizations given from the left because they will be more suitable in calculating the constants (= the elements which are annihilated by all multiparametric partial derivatives) in the free algebra of noncommuting polynomials.

Bosonic and k-fermionic coherent states for a class of polynomial Weyl–Heisenberg algebras

The aim of this paper is to construct coherent states à la Perelomov and à la Barut–Girardello for a polynomial Weyl–Heisenberg algebra. This generalized Weyl–Heisenberg algebra, denoted by , depends on r real parameters and is an extension of the one-parameter algebra (Daoud and Kibler 2010 J. Phys. A: Math. Theor. 43 115303) which covers the cases of the su(1, 1) algebra (for κ > 0), the su(2) algebra (for κ < 0) and the h4 ordinary Weyl–Heisenberg algebra (for κ = 0). For finite-dimensional representations of and , where is a truncation of order s of in the sense of Pegg–Barnett, a connection is established with k-fermionic algebras (or quon algebras). This connection makes it possible to use generalized Grassmann variables for constructing certain coherent states. Coherent states of the Perelomov type are derived for infinite-dimensional representations of and for finite-dimensional representations of and through a Fock–Bargmann analytical approach based on the use of complex (or bosonic) variables. The same approach is applied for deriving coherent states of the Barut–Girardello type in the case of infinite-dimensional representations of . In contrast, the construction of coherent states à la Barut–Girardello for finite-dimensional representations of and can be achieved solely at the price of replacing complex variables by generalized Grassmann (or k-fermionic) variables. Some of the results are applied to su(2), su(1, 1) and the harmonic oscillator (in a truncated or not truncated form).This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Coherent states: mathematical and physical aspects’.

Color Lie algebras and Lie algebras of order F

The notion of color algebras is generalized to the class of F-ary algebras, and corresponding decoloration theorems are established. This is used to give a construction of colored structures by means of tensor products with Cli ord-like algebras. It is, moreover, shown that color algebras admit realizations as q = 0 quon algebras.

Color Lie algebras and Lie algebras of order F

The notion of color algebras is generalized to the class of F-ary algebras, and corresponding decoloration theorems are established. This is used to give a construction of colored structures by means of tensor products with Clifford-like algebras. It is moreover shown that color algebras admit realisations as q=0 quon algebras.

Miscellaneous Applications of Quons

This paper deals with quon algebras or deformed oscillator algebras, for which the deformation parameter is a root of unity. We motivate why such algebras are interesting for fractional supersymmetric quantum mechanics, angular momentum theory and quantum information. More precisely, quon algebras are used for (i) a realization of a generalized Weyl-Heisenberg algebra from which it is possible to associate a fractional supersymmetric dynamical system, (ii) a polar decomposition of SU2 and (iii) a construction of mutually unbiased bases in Hilbert spaces of prime dimension. We also briefly discuss (symmetric informationally complete) positive operator valued measures in the spirit of (iii).

Representation Theory and Wigner-Racah Algebra of the SU(2) Group in a Noncanonical Basis

The Lie algebra su(2) of the classical group SU(2) is built from two commuting quon algebras for which the deformation parameter is a common root of unity. This construction leads to (i) a not very well-known polar decomposition of the generators J- and J+ of the SU(2) group, with J+ = J-† = HUr where H is Hermitean and Ur unitary, and (ii) an alternative to the {J2,Jz} quantization scheme, viz., the {J2,Ur} quantization scheme. The representation theory of the SU(2) group can be developed in this nonstandard scheme. The key ideas for developing the Wigner-Racah algebra of the SU(2) group in the {J2,Ur} scheme are given. In particular, some properties of the coupling and recoupling coefficients as well as the Wigner-Eckart theorem in the {J2,Ur} scheme are examined in great detail.

The two-level Lipkin model solution within the RPA approach for deformed fermions

We use a deformed algebra formalism, the so-called quon algebra, to investigate the role of deformation in an approximate treatment of a many-body system in a simple schematic model. A previously developed method is used in order to find the Hartree–Fock ground state energy and the RPA frequency as a function of the deformation parameter. We show that this dependence is very sensitive to small deviations of the algebra from the pure fermionic case and that the frequency is renormalized by those deviations. Also, we conclude that the well-known RPA collapse still occurs, although the critical value for the interaction constant now depends on the deformation parameter. Finally, we show that it is possible to make our results converge to the exact solution, allowing for a dependence of the deformation parameter on the two-body interaction strength of the model.

Compositeness effects in the Bose–Einstein condensation

Small deviations from purely bosonic behaviour of trapped atomic Bose–Einstein condensates are investigated with the help of the quon algebra, which interpolates between bosonic and fermionic statistics. A previously developed formalism is employed to obtain a generalized version of the Gross–Pitaeviskii equation. The depletion of the amount of condensed atoms for the case of repulsive forces between atoms in the trap can be accounted for by a universal fitting of the deformation parameter.

Some unusual algebraic structures and their applications in many-body problems

In this paper, we introduce the q-deformed and quon algebras formalism. Some applications of these algebraic structures are considered. A possible connection of the quon algebra with composite particle systems is discussed and perspectives on using such mathematical objects in the Bose-Einstein condensation is presented.

The energy operator for a model with a multiparametric infinite statistics

In this paper we consider an energy operator (a free Hamiltonian), in the second-quantized approach, for the multiparameter quon algebras: aia†j − qija†jai = δij, i, j ∊ I with (qij)i,j∊I any Hermitian matrix of deformation parameters. We obtain an elegant formula for normally ordered (sometimes called Wick-ordered) series expansions of number operators (which determine a free Hamiltonian). As a main result (see theorem 1) we prove that the number operators are given, with respect to a basis formed by 'generalized Lie elements', by certain normally ordered quadratic expressions with coefficients given precisely by the entries of the inverses of Gram matrices of multiparticle weight spaces. (This settles a conjecture by Meljanac S and Perica A (1994 J. Phys. A: Math. Gen. 27 4737–44).) These Gram matrices are Hermitian generalizations of the Varchenko matrices, associated with a quantum (symmetric) bilinear form of diagonal arrangements of hyperplanes. The solution of the inversion problem of such matrices in Meljanac S and Svrtan D (1996 Math. Commun. 1 1–24 (theorem 2.2.17)), leads to an effective formula for the number operators studied in this paper. The one-parameter case, in the monomial basis, was studied by Zagier, Stanciu and Møller.

Applications of the quon algebra: 3-D harmonic oscillator and the rotor model

In this work we present a method to build in a systematic way a many-body quon basis state. In particular, we show a closed expression for a given number N of quons, restricted to the permutational symmetric subspace, which belongs to the whole quonic space. The method is applied to two simple problems: the three-dimensional harmonic oscillator and the rotor model and compared to previous quantum algebra results. The differences obtained and possible future applications are also discussed.

Construction of bosons and fermions out of quons

The quon algebra describes particles, “quons”, that are neither fermions nor bosons, using a label q that parametrizes a smooth interpolation between bosons ( q=1) and fermions ( q=−1). Understanding the relation of quons on the one side and bosons or fermions on the other can shed light on the different properties of these two kinds of operators and the statistics which they carry. In particular, local bilinear observables can be constructed from bosons and fermions, but not from quons. In this Letter we construct bosons and fermions from quon operators. For bosons, our construction works for −1⩽ q⩽1. The case q=−1 is paradoxical, since that case makes a boson out of fermions, which would seem to be impossible. Nonetheless, when the limit q→−1 is taken from above, the construction works. For fermions, the analogous construction works for −1⩽ q⩽1, which includes the paradoxical case q=1.

Quons in relativistic theories must be bosons or fermions

The quon algebra describes particles, “quons”, that are neither fermions nor bosons using a label q that parameterizes a smooth interpolation between bosons ( q=+1) and fermions ( q=−1). We derive “conservation of statistics” relations for quons in relativistic theories, and show that in relativistic theories quons must be either bosons or fermions.

Quons as su(2) irreducible tensor operators

We prove that, for the quon algebra, which interpolates between the Bose and Fermi statistics and depends on a free parameter q, it is possible to build an su(2) irreducible representation. One of the consequences of this fact is that the quons couple via the same angular momentum coupling rules obeyed by ordinary bosons and fermions.

Quon Statistics for Composite Systems and a Limit on the Violation of the Pauli Principle for Nucleons and Quarks

The quon algebra describes particles, ``quons,'' that are neither fermions nor bosons. The parameter $q$ attached to a quon labels a smooth interpolation between bosons ( $q\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}+1$) and fermions ( $q\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{-}1$). Wigner and Ehrenfest and Oppenheimer showed that a composite system of identical bosons and fermions is a fermion if it contains an odd number of fermions and is a boson otherwise. We generalize this and show that ${q}_{\mathrm{composite}}{\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}q}_{\mathrm{constituent}}^{{n}^{2}}$ for a system of $n$ identical quons. Using this generalization, we find bounds on possible violations of the Pauli exclusion principle for nucleons and quarks based on such bounds for nuclei.

Boson mapping for a single j-shell within the quon statistics

The quon algebra, which interpolates between the Bose and Fermi algebras and depends on a free parameter q, is used to generate a Marumori mapping of the quadrupole-quadrupole Hamiltonian, for a single j-shell. This Hamiltonian is then diagonalized for a system composed of six fermions and its eigenvalues are compared with the ones obtained from the counterpart fermionic Hamiltonian and with another usual boson expansion method. The deformed mapping is shown to require a larger basis than the usual Marumori method for the diagonalization of the Hamiltonian. The low-lying levels of the spectrum obtained from the fermionic calculation in the SD subspace can be reproduced with our mapping for some selected j-shells.

Marumori expansion for a single j-shell within the quon algebra

The quon algebra, which interpolates between the Bose and Fermi algebras and depends on a free parameter q, is used to generate deformed Marumori boson expansions for the pair and quadrupole operators. Then we obtain a pairing plus quadrupole-quadrupole Hamiltonian, for a single j-shell, in terms of these deformed bosonic operators. The Hamiltonian is diagonalized and its eigenvalues are compared with the ones obtained from the counterpart fermionic Hamiltonian. The deformation parameter helps in achieving the correct energy levels, which cannot be encountered in practice with the usual non-deformed boson expansion.