Coherent States for Fractional Powers of the Harmonic Oscillator Hamiltonian

Inspired by special and general relativistic systems that can have Hamiltonians involving square roots or more general fractional powers, in this article, we address the question of how a suitable set of coherent states for such systems can be obtained. This becomes a relevant topic if the semiclass...

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Autores principales:	Kristina Giesel, Almut Vetter
Formato:	article
Lenguaje:	EN
Publicado:	MDPI AG 2021
Materias:	coherent states constrained systems operators involving fractional powers semiclassical analysis Elementary particle physics QC793-793.5
Acceso en línea:	https://doaj.org/article/5e7c57399246496d8dd47e16bde3bf79
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spelling	oai:doaj.org-article:5e7c57399246496d8dd47e16bde3bf792021-11-25T19:09:50ZCoherent States for Fractional Powers of the Harmonic Oscillator Hamiltonian10.3390/universe71104422218-1997https://doaj.org/article/5e7c57399246496d8dd47e16bde3bf792021-11-01T00:00:00Zhttps://www.mdpi.com/2218-1997/7/11/442https://doaj.org/toc/2218-1997Inspired by special and general relativistic systems that can have Hamiltonians involving square roots or more general fractional powers, in this article, we address the question of how a suitable set of coherent states for such systems can be obtained. This becomes a relevant topic if the semiclassical sector of a given quantum theory is to be analysed. As a simple setup, we consider the toy model of a deparametrised system with one constraint that involves a fractional power of the harmonic oscillator Hamiltonian operator, and we discuss two approaches to finding suitable coherent states for this system. In the first approach, we consider Dirac quantisation and group averaging, as have been used by Ashtekar et al., but only for integer powers of operators. Our generalisation to fractional powers yields in the case of the toy model a suitable set of coherent states. The second approach is inspired by coherent states based on a fractional Poisson distribution introduced by Laskin, which however turn out not to satisfy all properties to yield good semiclassical results for the operators considered here and in particular do not satisfy a resolution of identity as claimed. Therefore, we present a generalisation of the standard harmonic oscillator coherent states to states involving fractional labels, which approximate the fractional operators in our toy model semiclassically more accurately and satisfy a resolution of identity. In addition, motivated by the way the proof of the resolution of identity is performed, we consider these kind of coherent states also for the polymerised harmonic oscillator and discuss their semiclassical properties.Kristina GieselAlmut VetterMDPI AGarticlecoherent statesconstrained systemsoperators involving fractional powerssemiclassical analysisElementary particle physicsQC793-793.5ENUniverse, Vol 7, Iss 442, p 442 (2021)
institution	DOAJ
collection	DOAJ
language	EN
topic	coherent states constrained systems operators involving fractional powers semiclassical analysis Elementary particle physics QC793-793.5
spellingShingle	coherent states constrained systems operators involving fractional powers semiclassical analysis Elementary particle physics QC793-793.5 Kristina Giesel Almut Vetter Coherent States for Fractional Powers of the Harmonic Oscillator Hamiltonian
description	Inspired by special and general relativistic systems that can have Hamiltonians involving square roots or more general fractional powers, in this article, we address the question of how a suitable set of coherent states for such systems can be obtained. This becomes a relevant topic if the semiclassical sector of a given quantum theory is to be analysed. As a simple setup, we consider the toy model of a deparametrised system with one constraint that involves a fractional power of the harmonic oscillator Hamiltonian operator, and we discuss two approaches to finding suitable coherent states for this system. In the first approach, we consider Dirac quantisation and group averaging, as have been used by Ashtekar et al., but only for integer powers of operators. Our generalisation to fractional powers yields in the case of the toy model a suitable set of coherent states. The second approach is inspired by coherent states based on a fractional Poisson distribution introduced by Laskin, which however turn out not to satisfy all properties to yield good semiclassical results for the operators considered here and in particular do not satisfy a resolution of identity as claimed. Therefore, we present a generalisation of the standard harmonic oscillator coherent states to states involving fractional labels, which approximate the fractional operators in our toy model semiclassically more accurately and satisfy a resolution of identity. In addition, motivated by the way the proof of the resolution of identity is performed, we consider these kind of coherent states also for the polymerised harmonic oscillator and discuss their semiclassical properties.
format	article
author	Kristina Giesel Almut Vetter
author_facet	Kristina Giesel Almut Vetter
author_sort	Kristina Giesel
title	Coherent States for Fractional Powers of the Harmonic Oscillator Hamiltonian
title_short	Coherent States for Fractional Powers of the Harmonic Oscillator Hamiltonian
title_full	Coherent States for Fractional Powers of the Harmonic Oscillator Hamiltonian
title_fullStr	Coherent States for Fractional Powers of the Harmonic Oscillator Hamiltonian
title_full_unstemmed	Coherent States for Fractional Powers of the Harmonic Oscillator Hamiltonian
title_sort	coherent states for fractional powers of the harmonic oscillator hamiltonian
publisher	MDPI AG
publishDate	2021
url	https://doaj.org/article/5e7c57399246496d8dd47e16bde3bf79
work_keys_str_mv	AT kristinagiesel coherentstatesforfractionalpowersoftheharmonicoscillatorhamiltonian AT almutvetter coherentstatesforfractionalpowersoftheharmonicoscillatorhamiltonian
_version_	1718410218260398080

Coherent States for Fractional Powers of the Harmonic Oscillator Hamiltonian

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