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Macroscopic properties of A-statistics and A-superstatistics

T. D. Palev, Joris Van Der Jeugt

Published Nov 30, 2003• 0 citations
Publisher

Abstract

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.

DOI
10.1201/9781482269079-76

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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.
A -statistics is defined in the context of the Lie algebra sl ( n + 1). Some thermal properties of A -statistics are investigated under the assumption that the particles interact only via statistical interaction imposed by the Pauli principle of A -statistics. Apart from the general case, three particular examples are studied in more detail: (a) the particles have one and the same energy and chemical potential; (b) equidistant energy spectrum; (c) two species of particles with one and the same energy and chemical potential within each class. The grand partition functions and the average number of particles are among the thermodynamical quantities written down explicitly.
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?
Beginning with the assumption that the quantum state of a many-particle system is a functional on the internal space of the particles, a unified quantum statistics, which may interpolate smoothly between Bose-Einstein and Fermi-Dirac statistics, is proposed. The commutation relations for such particle creation and annihilation operators are derived, and statistical partition function and thermodynamical properties of an ideal gas of the particles are investigated. Besides the Bose-Einstein condensation, another kind of particle condensation may appear.
The recent discoveries of new forms of quantum statistics require a close look at the under-lying Fock space structure. This exercise becomes all the more important in order to provide a general classification scheme for various forms of statistics, and establish interconnections among them whenever it is possible. We formulate a theory of generalized Fock spaces, which has a three tired structure consisting of Fock space, statistics and algebra. This general formalism unifies various forms of statistics and algebras, which were earlier considered to describe different systems. Besides, the formalism allows us to construct many new kinds of quantum statistics and the associated algebras of creation and destruction operators. Some of these are: orthostatistics, null statistics or statistics of frozen order, quantum group based statistics and its many avatars, and ‘doubly-infinite’ statistics. The emergence of new forms of quantum statistics for particles interacting with singular potential is also highlighted.
A new kind of quantum statistics which interpolates between Bose and Fermi statistics is proposed beginning with the assumption that the quantum state of a many-particle system is a functional on the internal space of the particles. The quantum commutation relations for such particle creation and annihilation operators are derived, and statistical partition function and thermodynamical properties of an ideal gas of the particles are investigated. The application of this quantum statistics for the ensemble of extremal black holes are discussed.