Quasi-Particles Hamiltonian.pdf

a r X i v : h e p - t h / 0 4 0 9 2 7 8 v 2 2 0 O c t 2 0 0 4 Quasi-Particles Hamiltonian Jamila Douari ? Stellenbosch Institute for Advanced Study, Private Bag X1, Matieland, Stellenbosch, 7601, South Africa February 1, 2008 PACS: 03.65.Fd, 03.65.-w, 03.70.+k 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 [1] is recalled as a deformed Heisenberg algebra and a deformed Cλ-extended Heisenberg algebra. 1 Introduction The important result we got in the reference [1] is a symmetry describing anyons called quasi-particles [2] which could be seen as a unified symmetry based on the redefi- nition of the fundamental algebra underlying the non-commutative geometry [10, 11, 12], and the redefinition of annihilation and creation operators by imposing the existence of an excitation operator. In the particular case of statistical parameter ν ∈ [0, 1], we gave the anyonic algebra as a deformed Cλ-extended Heisenberg algebra, in which the Heisenberg algebra is extended by a polynomial in a hermitian operator denoted Ki and deformed in terms of statistical parameter ν. The obtained algebra is a deformed version of the extended Heisenberg algebra, discussed in the litterature [5, 6, 7, 9, 8]. The considerable interest to the quasi-particles living in 2-dimensional (2d) space with fractional spin, charge and statistics [2] is conditioned nowdays by their applications to the theory of planar physical phenomena such as fractional quntum Hall effect, high-Tc super- conductivity [3] and quantum computers [4]. One of the approaches describing anyons and realizing a quantization of its theory to reveal fractional statistics is the group-theoretical approach analogously to the case of integer and half-integer spin fields. In this letter, we recall the anyonic algebra and we give its representation. Th