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Linear pseudo-fermions

Fabio Bagarello

Published Jan 1, 2012• 37 citations
OpenAlex

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 {\em friendly}. We discuss some examples of our construction, again related to quasi-hermitian quantum mechanics, and the bi-coherent states for the system.

Related papers

Linear pseudo-fermions

F Bagarello

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’.

arXiv:1207.2874•Journal of Physics A Mathematical and Theoretical•Oct 23, 2012•37 citations

Pseudo-bosons, so far

Fabio Bagarello

In the past years several extensions of the canonical commutation relations have been proposed by different people in different contexts and some interesting physics and mathematics have been deduced. Here, we review some recent results on the so-called pseudo-bosons. They arise from a special deformation of the canonical commutation relation [a,a †]= ll, which is replaced by [a,b]=ll, with b not necessarily equal to a †. We start discussing some of their mathematical properties and then we discuss several examples.

•Jan 1, 2011•24 citations

Pseudo-bosons and bi-coherent states out of ℒ2(ℝ)

Fabio Bagarello

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].

Journal of Physics Conference Series•Oct 1, 2021•8 citations

Pseudo-bosons and bi-coherent states out of $\Lc^2(\mathbb{R})$

Fabio Bagarello

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}.

arXiv:2102.05614•Feb 10, 2021•0 citations

Deformed quons and bi-coherent states

Fabio Bagarello

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.

arXiv:1703.10021•Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences•Apr 1, 2017•26 citations

Construction of pseudobosons systems

Fabio Bagarello

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.

arXiv:1003.4620•Journal of Mathematical Physics•May 1, 2010•33 citations
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 the past years several extensions of the canonical commutation relations have been proposed by different people in different contexts and some interesting physics and mathematics have been deduced. Here, we review some recent results on the so-called pseudo-bosons. They arise from a special deformation of the canonical commutation relation [a,a †]= ll, which is replaced by [a,b]=ll, with b not necessarily equal to a †. We start discussing some of their mathematical properties and then we discuss several examples.
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}.
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.
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.