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Communications in Theoretical Physics

Non-Bosonic Excitations in q -Oscillator System for q a Root of Unity

Zhe Chang, Han-Ying Guo, Hong Yan

Published Dec 1, 1990• 9 citations
Publisher

Abstract

The systems of q -deformed oscillators of type two are studied in the case of q being the roots of unity. Non-bosonic excitations are found in (quantum mechanical) q -oscillators, and the classical mechanical counterparts and the saturation property are given. Especially when rank k is near 4, the excitations are found violating the Pauli exclusion principle, and possible physical interpretation is investigated.

DOI
10.1088/0253-6102/14/4/475

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math-phCross-listedarXiv:2512.00809•Dec 2, 2025
The systems of q -deformed oscillators of type two are studied in the case of q being the roots of unity. Non-bosonic excitations are found in (quantum mechanical) q -oscillators, and the classical mechanical counterparts and the saturation property are given. Especially when rank k is near 4, the excitations are found violating the Pauli exclusion principle, and possible physical interpretation is investigated.
The theory of Fermion oscillators has two essential ingredients: zero-point energy and Pauli exclusion principle. We devlop the theory of the statistical mechanics of generalized q-deformed Fermion oscillator algebra with inclusion principle (i.e., without the exclusion principle), which corresponds to ordinary fermions with Pauli exclusion principle in the classical limit q→1q \to 1q→1. Some of the remarkable properties of this theory play a crucial role in the understanding of the q-deformed Fermions. We show that if we insist on the weak exclusion principle, then the theory has the expected low temperature limit as well as the correct classical q-limit.
Quantum and classical mechanics of a system of q-deformed bosonic oscillators are considered. The q-deformed Heisenberg-Weyl algebra of creation and destruction operators is realized by differential operators in a space of functions of real commutative variables. The corresponding Hilbert space is constructed. In this approach, the deformation parameter turns out to be a function of the Planck constant, an oscillator frequency and a parameter with dimension of length. The Hamiltonian path integral is derived and its semiclassical approximation is investigated to obtain the corresponding q-deformed classical theory. The phase space spanned by the usual commutative coordinates is shown to be a cylinder in classical theory. The dimensional parameter introduced determines its radius. It is argued that the q-deformation can be associated with a special non-canonical transformation. The principle of least action for the classical q-deformed system is formulated. A representation of U q (m) in a space of functions on a phase space spanned by commutative coordinates is constructed.
The possibility of q-deformation of known exactly solvable quantum mechanical models is considered via the dynamical symmetry of the radial problem. The algebraic structure of the D-dimensional simple harmonic oscillator is analysed from this point of view. In the two-dimensional case, q-deformed bosons are introduced for the radial degrees of freedom, which together with the standard angular operators yield the so(2)(+)so q (2,1) dynamical algebra. The corresponding Hilbert space is identified and possible q-deformed Hamiltonians described.
When a boson interacts with another to form a composite system with SO(3) dynamic symmetry, it is shown that there exists the q-deformed bosonic excitation satisfying the q-deformed Heisenberg commutation relation in the quasi-classical limit that the angular momentum j for SO(3) is large, but not infinite. In second quantization this quasi-excitation is associated with the boson realization of SO(3) Lie algebra. Physically, the phenomena of q-deformed excitation can happen in many models of quantum dynamics, such as super emission from a system of many identical two-level atoms, the spin wave in Heisenberg chain, the high spin precession and the coherent output of Bose–Einstein atoms in a trap. Especially, in these models, the deformation parameter q is well defined intrinsically by a conservative quantity, such as the total atomic number and the angular momentum.
We introduce color Heisenberg-Lie (super)algebras graded by the abelian groups Z32Z_3^2Z32​, Z2p×Z32Z_2^p\times Z_3^2Z2p​×Z32​ for p=1,2,3p=1,2,3p=1,2,3, and investigate the properties of their associated multi-particle quantum paraoscillators. In the Rittenberg-Wyler's color Lie (super)algebras framework the above abelian groups are the simplest ones which induce mixed brackets interpolating commutators and anticommutators. These mixed brackets allow to accommodate two types of parastatistics: one based on the permutation group (beyond bosons and fermions in any space dimension) and an anyonic parastatistics based on the braid group. In both such cases the two broad classes of paraparticles are given by parabosons and parafermions. Mixed-bracket parafermions are created by nilpotent operators; they satisfy a generalized Pauli exclusion principle leading to roots-of-unity truncations in their multi-particle energy spectrum (braided Majorana qubits and their Gentile-type parastatistics are recovered in this color Lie superalgebra setting). Mixed-bracket parabosons do not admit truncations of the spectrum; the minimal detectable signature of their parastatistics is encoded in the measurable probability density of two indistinguishable parabosonic oscillators in a given energy eigenstate.