Parafermionic Oscillator Research Articles

Algebraic derivation of the energy eigenvalues for the quantum oscillator defined on the sphere and the hyperbolic plane

We give an algebraic derivation of eigenvalues of energy of a quantum harmonic oscillator on the surface of constant curvature, i.e., on the sphere or on the hyperbolic plane. We use the method proposed by Daskaloyannis [J. Math. Phys. 42, 1100–1119 (2001)] for fixing the energy eigenvalues of two-dimensional quadratically superintegrable systems by assuming that they are determined by the existence of a finite-dimensional representation of the polynomial algebra of motion integral operators. The tool for realizing representations is the deformed parafermionic oscillator. The eigenvalues of energy are calculated, and the result derived by us algebraically agrees with the known energy eigenvalues calculated by using classical analytical methods. This assertion, which is the main result of this article, is demonstrated by a detailed presentation. We also discuss the qualitative difference of the energy spectra on the sphere and on the hyperbolic plane.

Optical non-Hermitian para-Fermi oscillators

We present a proposal for the optical simulation of para-Fermi oscillators in arrays of coupled waveguides. We use a representation that arises as a deformation of the $su(2)$ algebra. This provides us with a set of chiral and a zero-energy-like normal modes. The latter is its own chiral pair and suggest the addition of controlled losses/gains following a pattern defined by parity. In these non-Hermitian para-Fermi oscillators, the analog of the zero-energy mode presents the largest effective loss/gain and it is possible to tune the system to show sequences of exceptional points and varying effective losses/gains. These arrays can be used for mode suppression or enhancement depending on the use of loss or gain, in that order. We compare our coupled mode theory predictions with finite element method simulations to good agreement.

Parastatistics and Uniquantization

A connection between the scheme of unitary quantization (uniquantization) and para-Fermi statistics of order 2 is considered. An appropriate generalization of the Green’s ansatz is suggested based on incorporation of the additional operator Ω which allows one to transform into the identity the bilinear and trilinear commutation relations of unitary quantization for the creation and annihilation operators of two different para-Fermi fields φа and φb. The way of incorporating para-Grassmann variables ξk into the general scheme of unitary quantization necessary for the definition of coherent states is suggested. For parastatistics of order 2, the new fact of existence of two alternative definitions of the coherent state for the para-Fermi oscillators is established.

Unitary Quantization and Para-Fermi Statistics of Order 2

A connection between a unitary quantization scheme and para-Fermi statistics of order 2 is considered. An appropriate extension of Green's ansatz is suggested. This extension allows one to transform bilinear and trilinear commutation relations for the annihilation and creation operators of two different para-Fermi fields $\phi_{a}$ and $\phi_{b}$ into identity. The way of incorporating para-Grassmann numbers $\xi_{k}$ into a general scheme of uniquantization is also offered. For parastatistics of order 2 a new fact is revealed, namely, the trilinear relations containing both the para-Grassmann variables $\xi_{k}$ and the field operators $a_{k}$, $b_{m}$ under a certain invertible mapping go over into the unitary equivalent relations, where commutators are replaced by anticommutators and vice versa. It is shown that the consequence of this circumstance is the existence of two alternative definitions of the coherent state for para-Fermi oscillators. The Klein transformation for Green's components of the operators $a_{k}$, $b_{m}$ is constructed in an explicit form that enables us to reduce the initial commutation rules for the components to the normal commutation relations of ordinary Fermi fields. A nontrivial connection between trilinear commutation relations of the unitary quantization scheme and so-called Lie-supertriple system is analysed. A brief discussion of the possibility of embedding the Duffin-Kemmer-Petiau theory into the unitary quantization scheme is provided.

Simulating para-Fermi oscillators

Quantum mechanics allows for a consistent formulation of particles that are neither bosons nor fermions. These para-particles are rather indiscernible in nature. Recently, we showed that strong coupling between a qubit and two field modes is required to simulate even order para-Bose oscillators. Here, we show that finite-dimensional representations of even order para-Fermi oscillators are feasible of quantum simulation under weak coupling. This opens the door to their potential implementation in different contemporaneous quantum electrodynamics platforms. We emphasize the intrinsic value of para-particles for the quantum state engineering of bichromatic field modes. In particular, we demonstrate that binomial two field mode states result from the evolution of para-Fermi vacuum states in the quantum simulation of these oscillators.

A Bose description of the 1-D para-Bose and para-Fermi oscillators

In the Fock space of two Bose operators all the irreducible representations of both the 1-D para-Bose and para-Fermi oscillators are constructed. Bose states of the form |p + k–1, kn > (|p – k, kn), n = 1,2, are shown to stand for states of k-parabosons (k-parafermions) of order p. For n = 1 or n = 2 the various subspaces may be visualized in the plane as either straight-lines or parabolae, respectively.

Quadratic Algebra Approach to an Exactly Solvable Position-Dependent Mass Schrödinger Equation in Two Dimensions

An exactly solvable position-dependent mass Schrodinger equation in two di- mensions, depicting a particle moving in a semi-infinite layer, is re-examined in the light of recent theories describing superintegrable two-dimensional systems with integrals of motion that are quadratic functions of the momenta. To get the energy spectrum a quadratic algebra approach is used together with a realization in terms of deformed parafermionic oscillator operators. In this process, the importance of supplementing algebraic considerations with a proper treatment of boundary conditions for selecting physical wavefunctions is stressed. Some new results for matrix elements are derived. This example emphasizes the interest of a quadratic algebra approach to position-dependent mass Schrodinger equations.

SUPERSYMMETRY OF PARAFERMIONS

We show that the single-mode parafermionic type systems possess supersymmetry, which is based on the symmetry of characteristic functions of the parafermions related to the generalized deformed oscillator of Daskaloyannis et al. The supersymmetry is realized in both unbroken and spontaneously broken phases. As in the case of parabosonic supersymmetry observed recently by one of the authors, the form of the associated superalgebra depends on the order of the parafermion and can be linear or nonlinear in the Hamiltonian. The list of supersymmetric parafermionic systems includes usual parafermions, finite-dimensional q-deformed oscillator, q-deformed parafermionic oscillator and parafermionic oscillator with internal Z2 structure.

Hyperfinite-dimensional representations of canonical commutation relation

This paper presents some methods of representing canonical commutation relations in terms of hyperfinite-dimensional matrices, which are constructed by nonstandard analysis. The first method uses representations of a nonstandard extension of the finite Heisenberg group, called hyperfinite Heisenberg group. The second is based on hyperfinite-dimensional representations of so(3). Then, the cases of infinite degree of freedom are argued in terms of the algebra of hyperfinite para-Fermi oscillators, which is mathematically equivalent to a hyperfinite-dimensional representation of so(n).

GENERALIZED DEFORMED su(2) ALGEBRAS, DEFORMED PARAFERMIONIC OSCILLATORS AND FINITE W-ALGEBRAS

Several physical systems (two identical particles in two dimensions, isotropic oscillator and Kepler system in a two-dimensional curved space) and mathematical structures (quadratic algebra QH(3), finite W-algebra [Formula: see text]) are shown to possess the structure of a generalized deformed su (2) algebra, the representation theory of which is known. Furthermore, the generalized deformed parafermionic oscillator is identified with the algebra of several physical systems (isotropic oscillator and Kepler system in two-dimensional curved space, Fokas-Lagerstrom, Smorodinsky-Winternitz and Holt potentials) and mathematical constructions (generalized deformed su (2) algebra, finite W-algebras [Formula: see text] and [Formula: see text]). The fact that the Holt potential is characterized by the [Formula: see text] symmetry is obtained as a by-product.

Discretization of the phase space for a q-deformed harmonic oscillator with q a root of unity

The “position” and “momentum” operators for the q-deformed oscillator with q being a root of unity are proved to have discrete eigenvalues which are roots of deformed Hermite polynomials. The Fourier transform connecting the “position” and “momentum” representations is also found. The phase space of this oscillator has a lattice structure, which is a non-uniformly distributed grid. Non-equidistant lattice structures also occur in the cases of the truncated harmonic oscillator and of the q-deformed parafermionic oscillator, while the parafermionic oscillator corresponds to a uniformly distributed grid.

Unified view of deformed single-mode oscillator algebras

A general framework for the deformation of the single-mode oscillators is presented and all deformed single-mode oscillators are unified. The extensions of the Aric-Coon, genon, the para-Bose and the para-Fermi oscillators are proposed. The generalized harmonic oscillator considered by Brzezinski et al. is rederived in a simple way.Some remarks on deformation of $SU(1,1)$ and supersymmetry are made.

Algebraic structure of parabose Fock space. I. The Green’s ansatz revisited

A description of the Fock space of a parabose oscillator of order p based on commutation relations bilinear in creation and annihilation operators rather than trilinear ones is developed. A new statement of the well-known Green’s ansatz is introduced to help understand the significance of the bilinear commutation relations. It is also applied to show how all parabose oscillators of even order (respectively, odd order) can be obtained as irreducible constituents of the Fock space associated with a Green’s ansatz of two (three) terms. Analogs of the form in which the parabose Green’s ansatz is presented are provided for one parafermi oscillator and for systems of many parabose and parafermi oscillators of the same order. Methods used in the paper allow various new results for q-deformed parabose oscillators to be derived.

PARASTATISTICAL SYSTEMS AT FINITE TEMPERATURES IN THERMOFIELD DYNAMICS

We consider para-Bose and para-Fermi oscillators within the framework of thermofield dynamics. For these systems, we construct the transformation relating the thermal vacuum state to the zero temperature vacuum. This construction makes use of a nonlinear realization of the single mode para-Bose (para-Fermi) algebra in terms of a single boson.

Q-analogues of the parabose and parafermi oscillators and representations of quantum algebras

Deformations of the one-dimensional parabose oscillators are presented. They are shown to provide Fock representations of the quantum sp q(2) algebra and ospq(1, 2) superalgebra. The representations of suq(2) are used to define the q-analogues of the parafermi oscillators.

Representations of parafermi rings

We consider the operator algebra corresponding to a system consisting of an arbitrary finite number v of parafermi oscillators with a view to obtaining all irrepresentations of the operator algebra defining it. It is shown that this algebra is isomorphic to the Lie algebra of the orthogonal group B v in 2 v + 1 dimensions. The known results of the representation theory of the orthogonal group are then used to find the representations of the ring of parafermi operators. In particular it is seen that the so-called Green ansatz for a parafermi ring arises in a natural way and that, in a certain sense, it furnishes the most general “solution” of the parafermi algebra. The case of a parafermi ring of order 2, which is the simplest non-trivial case, is considered in some detail.