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Coherent States for Fractional Powers of the Harmonic Oscillator Hamiltonian

Department of Physics, Institute for Quantum Gravity, Theoretical Physics III, FAU Erlangen-Nürnberg, Staudtstr. 7, 91058 Erlangen, Germany
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Academic Editors: Sergei D. Odintsov and Stefano Bellucci
Received: 27 September 2021 / Revised: 5 November 2021 / Accepted: 10 November 2021 / Published: 16 November 2021
(This article belongs to the Special Issue Cosmological Models, Quantum Theories and Astrophysical Observations)
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. View Full-Text
Keywords: coherent states; constrained systems; operators involving fractional powers; semiclassical analysis coherent states; constrained systems; operators involving fractional powers; semiclassical analysis
MDPI and ACS Style

Giesel, K.; Vetter, A. Coherent States for Fractional Powers of the Harmonic Oscillator Hamiltonian. Universe 2021, 7, 442. https://0-doi-org.brum.beds.ac.uk/10.3390/universe7110442

AMA Style

Giesel K, Vetter A. Coherent States for Fractional Powers of the Harmonic Oscillator Hamiltonian. Universe. 2021; 7(11):442. https://0-doi-org.brum.beds.ac.uk/10.3390/universe7110442

Chicago/Turabian Style

Giesel, Kristina, and Almut Vetter. 2021. "Coherent States for Fractional Powers of the Harmonic Oscillator Hamiltonian" Universe 7, no. 11: 442. https://0-doi-org.brum.beds.ac.uk/10.3390/universe7110442

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