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arXiv:1207.2874 · Journal of Physics A Mathematical and Theoretical

Linear pseudo-fermions

F Bagarello

Published Oct 23, 2012• 37 citations
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Abstract

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’.

DOI
10.1088/1751-8113/45/44/444002

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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.
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We discuss how a q-mutation relation can be deformed replacing a pair of conjugate operators with two other and unrelated operators, as it is done in the construction of pseudo-fermions, pseudo-bosons and truncated pseudo-bosons. This deformation involves interesting mathematical problems and suggests possible applications to pseudo-hermitian quantum mechanics. We construct bi-coherent states associated to D -pseudo-quons, and we show that they share many of their properties with ordinary coherent states. In particular, we find conditions for these states to exist, to be eigenstates of suitable annihilation operators and to give rise to a resolution of the identity. Two examples are discussed in details, one connected to an unbounded similarity map, and the other to a bounded map.
Nonlinear pseudo-fermions of degree n ( n -pseudo-fermions) are introduced as (pseudo) particles with creation and annihilation operators b and a , b ≠ a †, obeying the simple nonlinear anticommutation relation ab + b n a n = 1. The ( n + 1)-order nilpotency of these operators follows from the existence of unique (up to a bi-normalization factor) a -vacuum. Supposing appropriate ( n + 1)-order nilpotent para-Grassmann variables and integration rules the sets of n -pseudo-fermion number states, 'right' and 'left' ladder operator bi-overcomplete sets of coherent states are constructed. Explicit examples of n -pseudo-fermion ladder operators are provided, and the relation of pseudo-fermions to finite level pseudo-Hermitian systems is briefly considered. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to 'Quantum physics with non-Hermitian operators'.
Abstract In this paper we continue our analysis on deformed canonical commutation relations and on their related pseudo-bosons and bi-coherent states. In particular, we show how to extend the original approach outside the Hilbert space ℒ 2 ( ℝ ) , leaving untouched the possibility of defining eigenstates of certain number-like operators, manifestly non self-adjoint, but opening to the possibility that these states are not square-integrable. We also extend this possibility to bi-coherent states, and we discuss in many details an example based on a couple of superpotentials first introduced in [30]. The results deduced here belong to the same distributional approach to pseudo-bosons first proposed in [29].
In this paper we continue our analysis on deformed canonical commutation relations and on their related pseudo-bosons and bi-coherent states. In particular, we show how to extend the original approach outside the Hilbert space $\Lc^2(\mathbb{R})$, leaving untouched the possibility of defining eigenstates of certain number-like operators, manifestly non self-adjoint, but opening to the possibility that these states are not square-integrable. We also extend this possibility to bi-coherent states, and we discuss in many details an example based on a couple of superpotentials first introduced in \cite{bag2010jmp}. The results deduced here belong to the same distributional approach to pseudo-bosons first proposed in \cite{bag2020JPA}.