Heisenberg quantization for the systems of identical particles and the Pauli exclusion prin.pdf

a r X i v : h e p - t h / 0 2 0 8 0 0 7 v 1 1 A u g 2 0 0 2 Heisenberg quantization for the systems of identical particles and the Pauli exclusion principle in noncommutative spaces S. A. Alavi High Energy Physics Division, Department of Physics, University of Helsinki and Helsinki Institute of Physics, FIN-00014 Helsinki, Finland. On leave of absence from : Department of Physics, Ferdowsi University of Mashhad, Mashhad, P. O. Box 1436, Iran E-mail: Ali.Alavi@Helsinki.fi s?alialavi@ 1 Abstract. We study the Heisenberg quantization for the systems of identi- cal particles in noncommtative spaces. We get fermions and bosons as a special cases of our argument, in the same way as commutative case and therefore we conclude that the Pauli exclusion principle is also valid in noncommutative spaces. 1 Introduction. Recently there have been notable studies on the formulation and possible experimental consequences of extensions of the standard (usual) quantum me- chanics in the noncommutative spaces [1-13]. Many physical problems have been studied in the framework of the noncommutative quantum mechanics (NCQM), see e.g. [1-4]. NCQM is formulated in the same way as the standard quantum mechanics SQM (quantum mechanics in commutative spaces), that is in terms of the same dy- namical variables represented by operators in a Hilbert space and a state vector that evolves according to the Schroedinger equation : i d dt |ψ = Hnc|ψ . (1) 1 we have taken in to account h? = 1 and Hnc ≡ Hθ denotes the Hamiltonian for a given system in the noncommutative space. In the literatures two approaches have been considered for constructing the NCQM : a) Hθ = H , so that the only difference between SQM and NCQM is the presence of a nonzero θ in the commutator of the position operators : [x?i, x?j ] = iθij [x?i, p?i] = iδij [p?i, p?j ] = 0. (2) b) By deriving the Hamiltonian from the Moyal analogue of the standard Schroedinger equation : i ? ?t ψ(x, t) = H(p = 1 i ?, x) ? ψ(x, t) ≡ Hθψ(x, t),