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arXiv:1003.4620 · Journal of Mathematical Physics

Construction of pseudobosons systems

Fabio Bagarello

Published May 1, 2010• 33 citations
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Abstract

In a recent paper we have considered an explicit model of a PT-symmetric system based on a modification of the canonical commutation relation. We have introduced the so-called pseudobosons, and the role of Riesz bases in this context has been analyzed in detail. In this paper we consider a general construction of pseudobosons based on an explicit coordinate representation, extending what is usually done in ordinary supersymmetric quantum mechanics. We also discuss an example arising from a linear modification of standard creation and annihilation operators, and we analyze its connection with coherent states.

DOI
10.1063/1.3397408

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In a recent paper we have considered an explicit model of a PT-symmetric system based on a modification of the canonical commutation relation. We have introduced the so-called pseudobosons, and the role of Riesz bases in this context has been analyzed in detail. In this paper we consider a general construction of pseudobosons based on an explicit coordinate representation, extending what is usually done in ordinary supersymmetric quantum mechanics. We also discuss an example arising from a linear modification of standard creation and annihilation operators, and we analyze its connection with coherent states.
In a recent series of papers we have analyzed a certain deformation of the canonical commutation relations producing an interesting functional structure which has been proved to have some connections with physics, and in particular with quasi-hermitian quantum mechanics. Here we repeat a similar analysis starting with the canonical anticommutation relations. We will show that in this case most of the assumptions needed in the former situation are automatically satisfied, making our construction rather {\em friendly}. We discuss some examples of our construction, again related to quasi-hermitian quantum mechanics, and the bi-coherent states for the system.
In a recent series of papers, we have analyzed a certain deformation of the canonical commutation relations producing an interesting functional structure which has been proved to have some connections with physics, and in particular with quasi-Hermitian quantum mechanics. Here we repeat a similar analysis starting with the canonical anticommutation relations. We will show that in this case, most of the assumptions needed in the former situation are automatically satisfied, making our construction rather friendly . We discuss some examples of our construction, again related to quasi-Hermitian quantum mechanics, and the bi-coherent states for the system. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Quantum physics with non-Hermitian operators’.
In a series of recent papers, one of us has analyzed in some details a class of elementary excitations called pseudo-bosons. They arise from a special deformation of the canonical commutation relation [a, a†] = 11, which is replaced by [a, b] = 11, with b not necessarily equal to a†. Here, after a two-dimensional extension of the general framework, we apply the theory to a generalized version of the two-dimensional Hamiltonian describing Landau levels. Moreover, for this system, we discuss coherent states and we deduce a resolution of the identity. We also consider a different class of examples arising from a classical system, i.e., a damped harmonic oscillator.
It is well known that Parafermi and Parabose statistics are natural extensions of the usual Fermi and Bose ones, enhancing trilinear (anti)commutation relations instead of bilinear ones. Due to this generalization, positive parameters appear: the so-called orders of paraquantization p (= 1, 2, 3, ...) and h sub 0 (= 1/2, 1, 3/2, ...), respectively, the first value leading in each case to the usual statistics. The superpostion of the parabosonic and parafermionic operators gives rise to parasupermultiplets for which mixed trilinear relations have already been studied leading to two (nonequivalent) sets: the relative Parabose and the relative Parafermi ones. For the specific values p = 1 = 2h sub 0, these sets reduce to the well known supersymmetry. Coherent states associated with this last model have been recently put in evidence through the annihilation operator point of view and the group theoretical approach or displacement operator context. We propose to realize the corresponding studies within the new context p = 2 = 2h sub 0, being then directly extended to any order of paraquantization.
We study in this series the group theoretical structure of Fermion many-body systems arising from the canonical anticommutation relation of the annihilation-creation operators. Owing to the canonical anticommutation relation, a Fermion system with N single particle states has at least six Lie algebras of Fermion operators, a U(N), an SO(2N), an SO(2N+1), an SO(2N+2) and two U(N+1) Lie algebras. There are also two Clifford algebras of 2N and 2N+1 dimensions. The Fermion space is shown to belong to the spinor representations of the SO(2N), SO(2N+1) and SO(2N+2) groups. The canonical transformations generated by the U(N), SO(2N) and SO(2N+1) Lie algebras are characterized as the transformations to induce the linear U(N), SO(2N) and SO(2N+1) transformations for the Clifford algebras. The independent (quasi-) particle type wave functions of three kinds including the Hartree-Fock and Hartree-Bogoliubov wave functions are constructed by means of the canonical transformations and their relationship is studied. We derive three exact generator coordinate representations for state vectors in which the generator coordinates are the U(N), SO(2N) or SO(2N+1) group and the generating functions are the independent (quasi-) particle type wave functions. We characterize the structures of state vectors in the generator coordinate representations and study the relationship of the three representations.