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arXiv:q-alg/9712017 · Modern Physics Letters A

ON PARASTATISTICS DEFINED AS TRIPLE OPERATOR ALGEBRAS

STJEPAN MELJANAC, MARKO STOJIĆ, MARIJAN MILEKOVIĆ

Published Apr 30, 1998• 7 citations
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Abstract

Parastatistics, defined as triple operator algebras represented on Fock space, are unified in a simple way using the transition number operators. They are expressed as a normal ordered expansion of creation and annihilation operators. We discuss several examples of parastatistics, particularly Okubo's and Palev's parastatistics connected to many-body Wigner quantum systems and relate them to the notion of extended Haldane statistics.

DOI
10.1142/s0217732398001078

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Parastatistics, defined as triple operator algebras represented on Fock space, are unified in a simple way using the transition number operators. They are expressed as a normal ordered expansion of creation and annihilation operators. We discuss several examples of parastatistics, particularly Okubo's and Palev's parastatistics connected to many-body Wigner quantum systems and relate them to the notion of extended Haldane statistics.
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 general formula for the canonical partition function for a parastatistical system of any order is derived. The formula expresses the canonical partition functions for these in terms of sums of Schur functions.The only hitherto known result due to Suranyi [Phys. Rev. Lett. 65, 2329 (1990)] for parasystems of order two is obtained as a special case. Our results apply not only to parastatistics but to all statistics that can be defined on the basis of the permutation group, including those for which no simple definition in terms of the algebra of creation and annihilation operators is possible. \textcopyright{} 1996 The American Physical Society.
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.
A new formulation of the many-electron correlation problem is presented based on the use of parastatistics. It is shown that second-order para-Fermi creation and annihilation operators, which correspond to the creation and annihilation of spin-averaged paraparticles, occur naturally for the spin-independent many-electron problem. Moreover, the spin-independent many-electron Hamiltonian is directly expressible in terms of the parafield operators. The structure of the general para-Fermi algebra is also investigated from the viewpoint of the pseudo-orthogonal group O(2n+1, 1). Finally, an explicit matrix representation for the para-Fermi algebra of order 2, which enables one to handle even particle-number-nonconserving operators, is obtained in a canonical U(n)-adapted basis.