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Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences

A pseudo-bosonic Klein–Gordon field with finite two-point function

Fabio Bagarello

Published Mar 1, 2026• 0 citations
Publisher

Abstract

Abstract We introduce a class of pseudo-bosonic Klein–Gordon fields (KGFs) in 1+1 dimensions, and we discuss some of their properties. This approach originates from non-Hermitian quantum mechanics and deformed canonical commutation relations (CCRs). We show that, within this class of fields, there exists a specific subclass with the quite interesting property of having finite equal space-time two-point function, contrarily to what happens for ordinary KGFs. This, in our opinion, is a relevant aspect of our proposal, which is a good motivation to undertake a deeper analysis of this (and related) quantum fields.

DOI
10.1098/rspa.2025.0994

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Abstract We introduce a class of pseudo-bosonic Klein–Gordon fields (KGFs) in 1+1 dimensions, and we discuss some of their properties. This approach originates from non-Hermitian quantum mechanics and deformed canonical commutation relations (CCRs). We show that, within this class of fields, there exists a specific subclass with the quite interesting property of having finite equal space-time two-point function, contrarily to what happens for ordinary KGFs. This, in our opinion, is a relevant aspect of our proposal, which is a good motivation to undertake a deeper analysis of this (and related) quantum fields.
We introduce a class of pseudo-bosonic Klein-Gordon fields in 1+1 dimensions and we discuss some of their properties. This work originates from non Hermitian quantum mechanics and deformed canonical commutation relations. We show that, within this class of fields, there exist a specific subclass with the interesting feature of having finite equal space-time two-points function, contrarily to what happens for {\em standard} Klein-Gordon fields. This, in our opinion, is a relevant aspect of our proposal which is a good motivation to undertake a deeper analysis of this (and related) quantum fields.
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 The purpose of this article is to construct an explicit relation between the field operators in Quantum Field Theory and the relevant operators in Quantum Mechanics for a system of N identical particles, which are the symmetrised functions of the canonical operators of position and momentum, thus providing a clear relation between Quantum Field Theory and Quantum Mechanics. This is achieved in the context of the non-interacting Klein–Gordon field. Though this procedure may not be extendible to interacting field theories, since it relies crucially on particle number conservation, we find it nevertheless important that such an explicit relation can be found at least for free fields. It also comes out that whatever statistics the field operators obey (either commuting or anticommuting), the position and momentum operators obey commutation relations. The construction of position operators raises the issue of localizability of particles in Relativistic Quantum Mechanics, as the position operator for a single particle turns out to be the Newton–Wigner position operator. We make some clarifications on the interpretation of Newton–Wigner localized states and we consider the transformation properties of position operators under Lorentz transformations, showing that they do not transform as tensors, rather in a manner that preserves the canonical commutation relations. From a complex Klein–Gordon field, position and momentum operators can be constructed for both particles and antiparticles.
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].