Almut Vetter | Searcharxiv

arXiv2021

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 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 wants 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 for finding suitable coherent states for this system. In the first approach we consider Dirac quantisation and group averaging that 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

gr-qc↗

arXiv2019

Reduced Loop Quantization with four Klein-Gordon Scalar Fields as Reference Matter

We perform a reduced phase space quantization of gravity using four Klein-Gordon scalar fields as reference matter as an alternative to the Brown-Kuchar dust model in [1], where dust scalar fields are used. We compare our results to an earlier model by Domagala et. al. [2] where only one Klein-Gordon scalar field is considered as reference matter for the Hamiltonian constraint but the spatial diffeomorphism constraints are quantized using Dirac quantization. As a result we find that the choice of four conventional Klein-Gordon scalar fields as reference matter leads to a reduced dynamical model that cannot be quantized using loop quantum gravity techniques. However, we further discuss a slight generalization of the action for the four Klein-Gordon scalar fields and show that this leads to a model which can be quantized in the framework of loop quantum gravity. By comparison of the physical Hamiltonian operators obtained from the model by Domagala et. al. [2] and the one introduced in this work we are able to make a first step towards comparing Dirac and reduced phase space quantization in the context of the spatial diffeomorphism constraints.

gr-qc↗