arXiv subjects

Kristina Giesel

Publications and source records attributed to Kristina Giesel.

At least 37 records · Page 2Linked to original sources

arXiv2023

An open scattering model in polymerized quantum mechanics

We derive a quantum master equation in the context of a polymerized open quantum mechanical system for the scattering of a Brownian particle in an ideal gas environment. The model is formulated in a top-down approach by choosing a Hamiltonian with a coupling between the system and environment that is generally associated with spatial decoherence. We extend the existing work on such models by using a non-standard representation of the canonical commutation relations, inspired by the quantization procedure applied in loop quantum gravity, which yields a model in which position operators are replaced by holonomies. The derivation of the master equation in a top-down approach opens up the possibility to investigate in detail whether the assumptions, usually used in such models in order to obtain a tractable form of the dissipator, hold also in the polymerized case or whether they need to be dropped or modified. Furthermore, we discuss some physical properties of the master equation associated to effective equations for the expectation values of the fundamental operators and compare our results to the already existing models of collisional decoherence.

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arXiv2022

Spherical symmetric gravitational collapse of a dust cloud: polymerized dynamics in reduced phase space

Based on the effective dynamics in the $\bar μ$ scheme of the spherical symmetry reduced model in the reduced phase space formulation of loop quantum gravity (LQG), we investigate the gravitational collapse of a homogeneous dust cloud, with Gaussian dust serving as both the reference field and the source of the gravitational collapse. The effective dynamics from the considered model for a homogeneous dust cloud reduces precisely to the effective dynamics of loop quantum cosmology (LQC) with extrinsic curvature based K-quantization, indicating that the LQC effective dynamics lives as a subsector of the model presented here. In both the marginally bound and the bound cases of the collapse in effective dynamics, the singularity is resolved and replaced by a bounce. Though quantum geometric modification from spatial curvature is not directly included in the K-quantization it does affect the qualitative dynamics of the collapsing dust cloud in the sense that on the one hand for the marginally bound case, the dust cloud bounces once at fixed maximum energy density and on the other hand for the bound case, the dust cloud undergoes infinite cycles of contraction and expansion at energy densities dependent on the dust mass. Finally, the mass threshold for the formation of a trapped surface in each case is found and the matching conditions between the interior collapsing spacetime and an effective exterior static solution are discussed.

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arXiv2022

A gravitationally induced decoherence model using Ashtekar variables

We consider the coupling of a scalar field to linearised gravity and derive a relativistic gravitationally induced decoherence model using Ashtekar variables. The model is formulated at the gauge invariant level using suitable geometrical clocks in the relational formalism, broadening existing gauge invariant formulations of decoherence models. For the construction of the Dirac observables we extend the known observable map by a kind of dual map where the role of clocks and constraints is interchanged. We also discuss a second choice of geometrical clocks existing in the ADM literature. Then we apply a reduced phase space quantisation on Fock space and derive the final master equation choosing a Gibbs state for the gravitational environment and using the projection operator technique. The resulting master equation is not automatically of Lindblad type, a starting point sometimes assumed for phenomenological models, but still involves a residual time dependence at the level of the effective operators in the master equation due to the form of the correlation functions that we express in terms of thermal Wightman functions. Furthermore, we discuss why in the model analysed here the application of a second Markov approximation in order to obtain a set of time independent effective system operators is less straightforward than in some of the quantum mechanical models.

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arXiv2022

On consistent gauge fixing conditions in polymerized gravitational systems

For classical gravitational systems the lapse function and the shift vector are usually determined by imposing appropriate gauge fixing conditions and then demanding their preservation with respect to the dynamics generated by a canonical Hamiltonian. Effective descriptions encoding quantum geometric effects motivated by loop quantum gravity for symmetry reduced models are often captured by polymerization of connection (or related) variables in gauge fixing conditions as well as constraints. Usually, one chooses the same form of polymerization in both cases. A pertinent question is if the dynamical stability of the effective gauge fixing conditions under the effective dynamics generated by the polymerized canonical Hamiltonian is provided by the lapse function and the shift vector obtained from the polymerization of their classical counterparts. If this is the case, then we say that gauge fixing and polymerization commute. In this manuscript we investigate these issues and obtain consistency conditions for the commutativity of gauge fixing and polymerization. Our analysis shows that such a commutativity occurs in rather special situations and reveals pitfalls in making seemingly well motivated choices which turn out to be inconsistent with the effective dynamics. We illustrate these findings via examples of symmetry reduced models in the loop quantization of the Schwarzschild interior and Lema\^ıtre-Tolman-Bondi (LTB) spacetimes and report the non-commutativity of gauge fixing and polymerization, and inherent limitations of some choices made in the literature with a consistent effective dynamics.

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arXiv2021

Coherent States on the Circle: Semiclassical Matrix Elements in the Context of Kummer Functions and the Zak transformation

We extend former results for coherent states on the circle in the literature in two ways. On the one hand, we show that expectation values of fractional powers of momentum operators can be computed exactly analytically by means of Kummer's confluent hypergeometric functions. Earlier, these expectation values have only been obtained by using suitable estimates. On the other hand, we consider the Zak transformation not only to map harmonic oscillator coherent states to coherent states on the circle as it has been discussed before, but we also use the properties of the Zak transformation to derive a relation between matrix elements with respect to coherent states in L2(R) and L2(S1). This provides an alternative way for computing semiclassical matrix elements for coherent states on the circle. In certain aspects, this method simplifies the semiclassical computations in particular if one is only interested in the classical limit, that is the zeroth order term in the semiclassical expansion.

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arXiv2021

Analysing (cosmological) singularity avoidance in loop quantum gravity using U$(1)^3$ coherent states and Kummer's functions

Using a new procedure based on Kummer's Confluent Hypergeometric Functions, we investigate the question of singularity avoidance in loop quantum gravity (LQG) in the context of U$(1)^3$ complexifier coherent states and compare obtained results with already existing ones. Our analysis focuses on the dynamical operators, denoted by q(r), whose products are the analogue of the inverse scale factor in LQG and also play a pivotal role for other dynamical operators such as matter Hamiltonians or the Hamiltonian constraint. For graphs of cubic topology and linear powers in q(r), we obtain the correct classical limit and demonstrate how higher order corrections can be computed with this method. This extends already existing techniques in the way how the involved fractional powers are handled. We also extend already existing formalisms to graphs with higher-valent vertices. For generic graphs and products of q(r), using estimates becomes inevitable and we investigate upper bounds for these semiclassical expectation values. Compared to existing results, our method allows to keep fractional powers involved in q(r) throughout the computations, which have been estimated by integer powers elsewhere. Similar to former results, we find a non-zero upper bound for the inverse scale factor at the initial singularity. Additionally, our findings provide some insights into properties and related implications of the results that arise when using estimates and can be used to look for improved estimates.

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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

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arXiv2021

Non-singular quantum gravitational dynamics of an LTB dust shell model: the role of quantization prescriptions

We study some consequences of the loop quantization of the outermost dust shell in the Lemaître-Tolman-Bondi spacetime with a homogeneous dust density using different quantization strategies motivated by loop quantum gravity. Prior work has dealt with loop quantizing this model by employing holonomies and the triads, following the procedure in standard loop quantum cosmology. In this work we compare this quantization with the one in which holonomies and gauge-covariant fluxes are used. While both of the quantization schemes resolve the central singularity, they lead to different mass gaps at which a trapped surface forms. This trapped surface which is matched to an exterior generalized Vaidya spacetime disappears when the density of the dust shell is in the Planck regime. We find that the quantization based on holonomies and gauge-covariant fluxes generically results in an asymmetric evolution of the dust shell in which the effective mass associated with the white hole as seen by an external observer is $2/π$ of the one for the black hole. This effective difference in masses results from difference in the classical limits in pre- and post-bounce regimes in the two quantizations. This distinctive feature rules out formation of any black hole-white hole twin in presence of gauge-covariant flux modifications which is in contrast to the quantization using holonomies and triads where the gravitational collapse always leads to a black hole-white hole twins. In another striking difference, for the quantization based on holonomies and gauge-covariant fluxes there can be situations in which during a non-singular collapse only a black hole forms without a white hole.

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arXiv2021

Relating dust reference models to conventional systems in manifestly gauge invariant perturbation theory

Models with dust reference fields in relational formalism have proved useful in understanding the construction of gauge-invariant perturbation theory to arbitrary orders in the canonical framework. These reference fields modify the dynamical equations for perturbation equations. However, important questions remain open on the relation with conventional perturbation theories of inflaton coupled to gravity and of multi-fluid systems, and on understanding modifications in terms of physical degrees of freedom. These gaps are filled in this manuscript for Brown-Kuchař and Gaussian dust models, both of which involve three scalar physical degrees of freedom. We establish a relationship of these models with conventional inflationary and multi-fluid system of inflation and ordinary dust by introducing a set of gauge invariant variables on the reduced phase space of the dust reference models. We find the modifications due to dust clocks to Bardeen equation in the longitudinal gauge and Mukhanov-Sasaki equation in the spatially-flat gauge, in terms of physical degrees of freedom. This results in a closed system of equations for all the degrees of freedom needed to explore the evolution of the scalar perturbations. Our analysis shows for the first time that even for two-fluid systems, there is a natural choice of the set of gauge invariant variables for each chosen gauge which not only offers a direct physical interpretation but also results in simplifications to the dynamical equations.

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arXiv2020

Towards a reduced phase space quantization in loop quantum cosmology with an inflationary potential

We explore a reduced phase space quantization of loop quantum cosmology (LQC) for a spatially flat FLRW universe filled with reference fields and an inflaton field in a Starobinsky inflationary potential. We consider three separate cases in which the reference fields are taken to be the Gaussian dust, the Brown-Kuchař dust, and massless Klein-Gordon scalar reference fields respectively. This is a "two-fluid" model in which reference fields act as global clocks providing a physical time in an inflationary spacetime, and allow bypassing various technical hurdles in conventional quantum cosmological models. The reduced phase space is obtained in terms of the Dirac observables of the gravitational as well as the inflaton degrees of freedom. The physical Hamiltonians of the two dust models take the same form but turn out to be quite different from that of the Klein-Gordon reference field which reflects an aspect of the multiple choice problem of time. Loop quantization is implemented using the so-called $\bar μ$ scheme and the Schrödinger equations involving the physical Hamiltonian operators generating the evolution in the physical time in the dust and massless Klein-Gordon models are obtained. These turn out to be quantum difference equations with same non-singular structure as for other models in LQC. We study some phenomenological implications of the quantization using the effective dynamics resulting from the reduced phase space quantization including the resolution of the big bang singularity via a quantum bounce, and effects of the different reference fields on e-foldings in both the pre-inflationary and the slow-roll inflationary phases. We find that different clocks, even when starting with a small but same energy density, can leave tiny but different imprints on the inflationary dynamics. (abridged)

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arXiv2020

Mukhanov-Sasaki equation in manifestly gauge-invariant linearized cosmological perturbation theory with dust reference fields

The aim of this article is to understand the role of dust reference fields, often also called clocks, on cosmological perturbations around a flat FLRW universe. We derive the Mukhanov-Sasaki (MS) equation for the Brown-Kuchar (BK) and Gaussian dust models, which both consider four dust fields as reference fields. The reduced phase space of Dirac observables, that is the gauge-invariant part of the theory, is constructed by means of an observable map applied to all elementary phase space variables of the coupled system, consisting of gravity, a massive scalar field and the dust degrees of freedom. The evolution of these observables is governed by a so called physical Hamiltonian which can be derived once the set of reference fields are chosen and differs for each model. First, the reduced phase space as well as the corresponding equations of motion are derived for full general relativity. Then from this, the gauge-invariant version of the equations of motion for the background are derived which contain a fingerprint of the dust reference fields. Afterwards we study linear cosmological perturbations around a FLRW metric using the scalar-vector-tensor decomposition and derive the equation of motion for the MS variable in this formalism for a chosen set of variables on the reduced phase space and expressed in terms of Dirac observables. The MS equation involves additional contributions that can be understood as back reactions from the dust reference fields. These additional dust contributions to the MS equation were absent if the dust energy and momentum density as well as their perturbations are vanishing. The nature of the correction terms suggest that BK and Gaussian dust reference fields contribute differently. Using numerical simulations we study the behavior of the dust contributions to the MS equation during inflation.

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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.

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arXiv2019

Dynamical Properties of the Mukhanov-Sasaki Hamiltonian in the context of adiabatic vacua and the Lewis-Riesenfeld invariant

We use the method of the Lewis-Riesenfeld invariant to analyze the dynamical properties of the Mukhanov-Sasaki Hamiltonian and, following this approach, investigate whether we can obtain possible candidates for initial states in the context of inflation considering a quasi-de Sitter spacetime. Our main interest lies in the question to which extent these already well-established methods at the classical and quantum level for finitely many degrees of freedom can be generalized to field theory. As our results show, a straightforward generalization does in general not lead to a unitary operator on Fock space that implements the corresponding time-dependent canonical transformation associated with the Lewis-Riesenfeld invariant. The action of this operator can be rewritten as a time-dependent Bogoliubov transformation and we show that its generalization to Fock space has to be chosen appropriately in order that the Shale-Stinespring condition is not violated, where we also compare our results to already existing ones in the literature. Furthermore, our analysis relates the Ermakov differential equation that plays the role of an auxiliary equation, whose solution is necessary to construct the Lewis-Riesenfeld invariant, as well as the corresponding time-dependent canonical transformation to the defining differential equation for adiabatic vacua. Therefore, a given solution of the Ermakov equation directly yields a full solution to the differential equation for adiabatic vacua involving no truncation at some adiabatic order. As a consequence, we can interpret our result obtained here as a kind of non-squeezed Bunch-Davies mode, where the term non-squeezed refers to a possible residual squeezing that can be involved in the unitary operator for certain choices of the Bogoliubov map.

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arXiv2019

Dynamics of Dirac observables in canonical cosmological perturbation theory

The relational formalism based on geometrical clocks and Dirac observables in linearized canonical cosmological perturbation theory is used to introduce an efficient method to find evolution equations for gauge invariant variables. Our method generalizes an existing technique by Pons, Salisbury and Sundermeyer [1, 2] to relate the evolution of gauge invariant observables with the one of gauge variant quantities, and is applied as a demonstration for the longitudinal and spatially flat gauges. Gauge invariant evolution equations for the Bardeen potential and the Mukhanov-Sasaki variable are derived in the extended ADM phase space. Our method establishes a full agreement at the dynamical level between the canonical and conventional cosmological perturbation theory at the linear order using Dirac observables.

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arXiv2018

Gauge invariant variables for cosmological perturbation theory using geometrical clocks

Using the extended ADM-phase space formulation in the canonical framework we analyze the relationship between various gauge choices made in cosmological perturbation theory and the choice of geometrical clocks in the relational formalism. We show that various gauge invariant variables obtained in the conventional analysis of cosmological perturbation theory correspond to Dirac observables tied to a specific choice of geometrical clocks. As examples, we show that the Bardeen potentials and the Mukhanov-Sasaki variable emerge naturally in our analysis as observables when gauge fixing conditions are determined via clocks in the Hamiltonian framework. Similarly other gauge invariant variables for various gauges can be systematically obtained. We demonstrate this by analyzing five common gauge choices: longitudinal, spatially flat, uniform field, synchronous and comoving gauge. For all these, we apply the observable map in the context of the relational formalism and obtain the corresponding Dirac observables associated with these choices of clocks. At the linear order, our analysis generalizes the existing results in canonical cosmological perturbation theory twofold. On the one hand we can include also gauges that can only be analyzed in the context of the extended ADM-phase space and furthermore, we obtain a set of natural gauge invariant variables, namely the Dirac observables, for each considered choice of gauge conditions. Our analysis provides insights on which clocks should be used to extract the relevant natural physical observables both at the classical and quantum level. We also discuss how to generalize our analysis in a straightforward way to higher orders in the perturbation theory to understand gauge conditions and the construction of gauge invariant quantities beyond linear order.

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arXiv2018

Gauge invariant canonical cosmological perturbation theory with geometrical clocks in extended phase space - a review and applications

The theory of cosmological perturbations is a well elaborated field. To deal with the diffeomorphism invariance of general relativity one generally introduces combinations of the metric and matter perturbations which are gauge invariant up to the considered order in the perturbations. For linear cosmological perturbations one works with the so-called Bardeen potentials widely used in this context. However, there exists no common procedure to construct gauge invariant quantities also for higher order perturbations. Usually, one has to find new gauge invariant quantities independently for each order in perturbation theory. With the relational formalism introduced by Rovelli and further developed by Dittrich and Thiemann, it is in principle possible to calculate manifestly gauge invariant quantities, that is quantities that are gauge invariant up to arbitrary order once one has chosen a set of so-called reference fields, often also called clock fields. This article contains a review of the relational formalism and its application to canonical general relativity following the work of Garcia, Pons, Sundermeyer and Salisbury. As the starting point for our application of this formalism to cosmological perturbation theory, we also review the Hamiltonian formulation of the linearized theory for perturbations around FLRW spacetimes. The main aim of our work will be to identify clock fields in the context of the relational formalism that can be used to reconstruct quantities like the Bardeen potential as well as the Mukhanov-Sasaki variable. This requires a careful analysis of the canonical formulation in the extended ADM-phase space where lapse and shift are treated as dynamical variables. The actual construction of such observables and further investigations thereof will be carried out in our companion paper.

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arXiv2017

The kinematical Setup of Quantum Geometry: A Brief Review

In this article we present a brief introduction to the kinematical setup that underlies the quantization used in loop quantum gravity. This review has been published as a chapter in the monograph "Loop Quantum Gravity: The First 30 Years", edited by Abhay Ashtekar and Jorge Pullin, that was recently published in the series "100 Years of General Relativity".

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arXiv2015

Manifestly Gauge Invariant Perturbations of Scalar-Tensor Theories of Gravity

The general relativistic perturbations of scalar-tensor theories (STT) of gravity are studied in a manifestly gauge invariant Hamiltonian formalism. After the derivation of the Hamiltonian equations of motion in this framework, the gauge invariant formalism is used to compute the evolution equations of linear perturbations around a general relativistic spacetime background in the Jordan frame. These equations are then specialized to the case of a flat FRW cosmological background. Furthermore, the equivalence between the Jordan frame and the Einstein frame of STT in the manifestly gauge invariant Hamiltonian formalism is analyzed, and it is shown that also in this framework they can be related by a conformal transformation. Finally, the obtained evolution equations for the linear perturbations in our formalism are compared with those in the standard cosmological perturbation theory. It turns out that the perturbation equations in the two different formalisms coincide with each other in a suitable limit.

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