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arXiv:q-alg/9510005 · International Journal of Modern Physics A

A UNIFIED VIEW OF MULTIMODE ALGEBRAS WITH FOCK-LIKE REPRESENTATIONS

Stjepan Meljanac, Marijan Mileković

Published Mar 30, 1996• 28 citations
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Abstract

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.

DOI
10.1142/s0217751x9600064x

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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.
We investigate some aspects of q Heisenberg algebra. We show how su(2) and su(1,1) generators can be constructed in terms of the q creation and annihilation operators. We also construct the coherent states for the q oscillator and show that they can be obtained by the action of a displacement operator on the vacuum. For the multimode case with q=0, corresponding to the infinite statistics of Greenberg [Phys. Rev. Lett. 64, 705 (1990)], we generalize our single-mode construction to obtain the corresponding coherent states. These states, which are eigenstates of the annihilation operators, interestingly, turn out to be degenerate owing to the noncommutativity of the displacement operators.
We formulate a theory of generalized Fock spaces which underlies the different forms of quantum statistics such as "infinite", Bose-Einstein and Fermi-Dirac statistics. Single-indexed systems as well as multi-indexed systems that cannot be mapped into single-indexed systems are studied. Our theory is based on a three-tiered structure consisting of Fock space, statistics and algebra. This general formalism not only unifies the various forms of statistics and algebras, but also allows us to construct many new forms of quantum statistics as well as many algebras of creation and destruction operators. Some of these are : new algebras for infinite statistics, q-statistics and its many avatars, a consistent algebra for fractional statistics, null statistics or statistics of frozen order, "doubly-infinite" statistics, many representations of orthostatistics, Hubbard statistics and its variations. Key Words : Fock space -- multioscillator algebras -- q-deformation -- quantum groups -- general...