Quon Algebra Research Articles

Exact Dyson 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 paramenter q, is used to generate a deformed Dyson boson expansion of the quadrupole operator. Then we obtain a quadrupole-quadrupole hamiltonian, for a single j-shell, in terms of this deformed bosonic operator. The hamiltonian is diagonalized and its eigenvalues are compared with the ones obtained from the fermionic quadrupole-quadrupole hamiltonian. The deformation parameter helps in achieving the correct excitation energy levels, what cannot be encountered in practice with the usual non-deformed Dyson expansion.

Noncommutative symmetric functions III : Deformations of Cauchy and convolution algebras

[in "Special Issue : Lie Computations", G. Jacob, V. Koseleff, Eds.] This paper discusses various deformations of free associative algebras and of their convolution algebras. Our main examples are deformations of noncommutative symmetric functions related to families of idempotents in descent algebras, and a simple q-analogue of the shuffle product, which has unexpected connections with quantum groups, hyperplane arrangements, and certain questions in mathematical physics (the quon algebra, generalized Brownian motion).

A UNIFIED VIEW OF MULTIMODE ALGEBRAS WITH FOCK-LIKE REPRESENTATIONS

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 [Formula: see text] operators is extended to include all normally ordered terms in creation and annihilation operators, and we analyze their action on Fock-like 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.

Q-Deformed determinants and a unified realization of boson, fermion and quon algebras

A new mathematical concept, theq-deformation of the determinant is proposed. By using theq-deformed Slater determinant, a unified realization of boson, fermion and quon algebras is presented. The quon algebra is characterized by the qumutation relationa j a ° -qa δ a j = δ jk , which interpolates between Fermi anticommutation and Bose commutation relations. It turns out that for allq∈(−1, 1), the Fock-like spaces are the same, that is the direct sum of all tensor product powers of the single particle spaceh, . This state vector space describes systems of identical particles obeying Boltzmann statistics.

Generalized statistics and dynamics in curved spacetime

We consider the generalized quon algebra in a dynamically evolving curved spacetime. We find that particle statistics remains unchanged in time as the universe expands. Besides the Bose and Fermi statistics, the para-Bose, para-Fermi and anyonic like statistics are also allowed.

Non-standard covariant deformed oscillator algebras

It is shown that deformed oscillator algebras with a non-standard behaviour, similar to that observed for the quon algebra, arise quite naturally when imposing covariance properties under the transformations of some quantum group. By non-standard behaviour, we mean that there may not be enough rules available to order creation operators and that some Fock states with positive squared norm for generic q-values disappear for q to 1. Two examples are given for the quantum groups Oq(3,R) and SUq(2)*SUq(2) approximately=Oq(4,R), respectively.

Number operators in a general quon algebra

The existence of the number operators Nk in a general quon algebra is proved and a general construction of Nk is presented. A new general simple structure of Nk is proved when mod qij mod <1, For all i, j in S. General recurrent relations for the corresponding coefficients are constructed and solved in special cases. These solutions are discussed and a conjecture for a general solution is proposed. Cases in which some or all mod qij mod =1 are discussed.

Particles with small violations of Fermi or Bose statistics.

I discuss the statistics of quons (pronounced to rhyme with muons), particles whose annihilation and creation operators obey the $q$-deformed commutation relation (the quon algebra or $q$-mutator) which interpolates between fermions and bosons. Topics discussed include representations of the quon algebra, proof of the $\mathrm{TCP}$ theorem and clustering, violation of the usual locality properties, and experimental constraints on violations of the Pauli exclusion principle (i.e., Fermi statistics) and of Bose statistics.