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arXiv:hep-th/0409278

Quasi-Particles Hamiltonian

Jamila Douari

Published Sep 27, 2004• 0 citations
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Abstract

The anyonic Hamiltonian is quantum mechanically given and the bosonic and the fermionic Hamiltonians are found as extremes by discussing the cases of the statistical parameter $ν$ and the dimension of space. The anyonic algebra \cite{upa} is recalled as a deformed Heisenberg algebra and a deformed $C_λ$-extended Heisenberg algebra.

DOI
10.48550/arxiv.hep-th/0409278

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The anyonic Hamiltonian is quantum mechanically given and the bosonic and the fermionic Hamiltonians are found as extremes by discussing the cases of the statistical parameter ννν and the dimension of space. The anyonic algebra \cite{upa} is recalled as a deformed Heisenberg algebra and a deformed CλC_λCλ​-extended Heisenberg algebra.
We discuss the connection between anyons (particles with fractional statistics) and deformed Lie algebras (quantum groups). After a brief review of the main properties of anyons, we present the details of the anyonic realization of all deformed classical Lie algebras in terms of anyonic oscillators. The deformation parameter of the quantum groups is directly related to the statistics parameter of the anyons. Such a realization is a direct generalization of the Schwinger construction in terms of fermions and is based on a sort of bosonization formula which yields the generators of the deformed algebra in terms of the undeformed ones. The entire procedure is well defined on two-dimensional lattices, but it can be consistently reduced also to one-dimensional chains.
All classical Lie algebras can be realized a la Schwinger (1965) in terms of fermionic oscillators. We show that the same can be done for their q-deformed counterparts by simply replacing the fermionic oscillators with anyonic ones defined on a two-dimensional lattice. The deformation parameter q is a phase related to the anyonic statistical parameter. A crucial role in this construction is played by a sort of bosonization formula which gives the generators of the quantum algebras in terms of the undeformed ones. The entire procedure works even on one-dimensional chains; in such a case q can also be real.
Quantum algebras (q(q(q algebras) are used to describe interactions between fermions and bosons. Particularly, the concept of a suq(2){\mathrm{su}}_{q}(2)suq​(2) dynamical symmetry is invoked in order to reproduce the ground state properties of systems of fermions and bosons interacting via schematic forces. The structure of the proposed suq(2){\mathrm{su}}_{q}(2)suq​(2) Hamiltonians, and the meaning of the corresponding deformation parameters, are discussed.
qqq-Deformed Classical Lie Algebras and their Anyonic Realization
All classical Lie algebras can be realized \\`a la Schwinger in terms of fermionic oscillators. We show that the same can be done for their qqq-deformed counterparts by simply replacing the fermionic oscillators with anyonic ones defined on a two dimensional lattice. The deformation parameter qqq is a phase related to the anyonic statistical parameter. A crucial r\\^ole in this construction is played by a sort of bosonization formula which gives the generators of the quantum algebras in terms of the underformed ones. The entire procedure works even on one dimensional chains; in such a case qqq can also be real.