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arXiv:1512.05076 · AIP conference proceedings

Generalized quantum statistics and Lie (super)algebras

Nedialka I. Stoilova

Published Jan 1, 2016• 0 citations
arXivHTMLPDF

Abstract

Generalized quantum statistics, such as paraboson and parafermion statistics, are characterized by triple relations which are related to Lie (super)algebras of type B. The correspondence of the Fock spaces of parabosons, parafermions as well as the Fock space of a system of parafermions and parabosons to irreducible representations of (super)algebras of type B will be pointed out. Example of generalized quantum statistics connected to the basic classical Lie superalgebra B(1|1) ≡ osp(3|2) with interesting physical properties, such as noncommutative coordinates, will be given. Therefore the article focuses on the question, addressed already in 1950 by Wigner: do the equation of motion determine the quantum mechanical commutation relation?

DOI
10.1063/1.4944182

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Generalized quantum statistics, such as paraboson and parafermion statistics, are characterized by triple relations which are related to Lie (super)algebras of type B. The correspondence of the Fock spaces of parabosons, parafermions as well as the Fock space of a system of parafermions and parabosons to irreducible representations of (super)algebras of type B will be pointed out. Example of generalized quantum statistics connected to the basic classical Lie superalgebra B(1|1) ≡ osp(3|2) with interesting physical properties, such as noncommutative coordinates, will be given. Therefore the article focuses on the question, addressed already in 1950 by Wigner: do the equation of motion determine the quantum mechanical commutation relation?
In this paper we reexamine the definition of parafermions and parabosons by means of Green's triple relations, and extend these relations by including a parity operator PPP which is also determined by means of triple relations. As a consequence, we are dealing with new algebraic structures. It is shown that the algebra underlying a set of nnn parafermions together with PPP is the orthogonal Lie algebra so(2n+2)so(2n+2)so(2n+2). The Fock spaces correspond to particular irreducible representations of so(2n+2)so(2n+2)so(2n+2), and the action of PPP in these spaces leads to interesting observations. Next, we show that the algebra underlying a set of nnn parabosons together with PPP is the orthosymplectic Lie superalgebra osp(2∣2n)osp(2|2n)osp(2∣2n). In this case, the Fock spaces correspond to certain irreducible infinite-dimensional representations of osp(2∣2n)osp(2|2n)osp(2∣2n). Both for parafermions and parabosons the spectrum of PPP is closely related to the so-called order of statistics ppp, introduced by Green.
In quantum mechanics, physical observables are represented by operators on a certain Hilbert space. The question of how such operators commute, has been a matter of discussion. In the standard perspective, the operators corresponding to the position and momentum of a system are assumed to satisfy the canonical commutation relations. It is known that these relations imply that the Hamilton and Heisenberg equations of motion are compatible as operator equations. However, Wigner showed that the inverse statement is not true. Therefore, it is a much weaker constraint to impose the compatibility of the equations of motion. For any physical system, this results in a set of compatibility conditions, which form the core of Wigner quantization. The key to finding operators that are subject to the compatibility conditions is provided by Lie superalgebras and their representations. Lie superalgebras can be defined as algebras generated by odd elements satisfying particular superbracket relations. Using these defining relations, Lie superalgebra generators can be found that obey the compatibility conditions. The Lie superalgebra elements act as operators on a vector space if we consider Lie superalgebra representations. Various physical systems are investigated in this thesis in the context of Wigner quantization. For each of these systems solutions are found in terms of Lie superalgebra generators, after which specific representations are examined. In such representations, the focus lies on determining particular physical properties of the system. Most of the studied systems are harmonic oscillator models. First, we examine a set-up of coupled harmonic oscillators, for which the interaction is represented by an interaction matrix. Then we focus on the angular momentum content of a 3N3N3N-dimensional harmonic oscillator. Finally, our attention goes to two one-dimensional systems, namely the free particle and the Berry-Keating-Connes Hamiltonian. The latter of these Hamiltonians is notorious for its possible connection with the Riemann hypothesis. All of the aforementioned Hamiltonians have been extensively investigated in the context of canonical quantization, so that our results can be compared to the well-known canonical case.
In the context of Lie (super)algebras of type A , generalized quantum statistics is considered. For sl ( n + 1) this is A-statistics, and for sl (1 |n ) this is A-superstatistics. The creation and annihilation operators, their triple relations, and their Fock spaces are given. We consider a ‘free’ Hamiltonian and some macroscopic properties are studied : the grand partition function, average number of particles, etc. Symmetric functions play a role in expressions for these thermodynamical quantities.
We consider the compatibility conditions for a N-particle D-dimensional Wigner quantum oscillator. These conditions can be rewritten as certain triple relations involving anticommutators, so it is natural to look for solutions in terms of Lie superalgebras. In the recent classification of “generalized quantum statistics ” for the basic classical Lie superalgebras [1], each such statistics is characterized by a set of creation and annihilation operators plus a set of triple relations. In the present letter, we investigate which cases of this classification also lead to solutions of the compatibility conditions. Our analysis yields some known solutions and several classes of new solutions.
We consider the compatibility conditions for an N -particle D -dimensional Wigner quantum oscillator. These conditions can be rewritten as certain triple relations involving anticommutators, so it is natural to look for solutions in terms of Lie superalgebras. In the recent classification of 'generalized quantum statistics' for the basic classical Lie superalgebras [1], each such statistics is characterized by a set of creation and annihilation operators plus a set of triple relations. In the present paper, we investigate which cases of this classification also lead to solutions of the compatibility conditions. Our analysis yields some known solutions and several classes of new solutions.