Coherent states for fractional powers of the harmonic oscillator Hamiltonian
Giesel K, Vetter A (2021)
Publication Type: Journal article, Original article
Publication year: 2021
Book Volume: Universe 2021, 7
Pages Range: 442
Journal Issue: 11
Open Access Link: https://www.mdpi.com/2218-1997/7/11/442
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.
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How to cite
Giesel, K., & Vetter, A. (2021). Coherent states for fractional powers of the harmonic oscillator Hamiltonian. Universe, Universe 2021, 7(11), 442. https://dx.doi.org/10.3390/universe7110442
Giesel, Kristina, and Almut Vetter. "Coherent states for fractional powers of the harmonic oscillator Hamiltonian." Universe Universe 2021, 7.11 (2021): 442.