Oliver Winkler

Gauge field theory coherent states (GCS): II. Peakedness properties

In this article we apply the methods outlined in the previous paper of this series to the particular set of states obtained by choosing the complexifier to be a Laplace operator for each edge of a graph. The corresponding coherent state transform was introduced by Hall for one edge and generalized by Ashtekar, Lewandowski, Marolf, Mourao and Thiemann to arbitrary, finite, piecewise analytic graphs. However, both of these works were incomplete with respect to the following two issues : (a) The focus was on the unitarity of the transform and left the properties of the corresponding coherent states themselves untouched. (b) While these states depend in some sense on complexified connections, it remained unclear what the complexification was in terms of the coordinates of the underlying real phase space. In this paper we complement these results : First, we explicitly derive the com- plexification of the configuration space underlying these heat kernel coherent states and, secondly, prove that this family of states satisfies all the usual properties : i) Peakedness in the configuration, momentum and phase space (or Bargmann-Segal) representation. ii) Saturation of the unquenched Heisenberg uncertainty bound. iii) (Over)completeness. These states therefore comprise a candidate family for the semi-classical anal- ysis of canonical quantum gravity and quantum gauge theory coupled to quantum gravity. They also enable error-controlled approximations to difficult analytical cal- culations and therefore set a new starting point for numerical canonical quantum general relativity and gauge theory. The text is supplemented by an appendix which contains extensive graphics in order to give a feeling for the so far unknown peakedness properties of the states constructed.

Quantum resolution of black hole singularities

We study the classical and quantum theory of spherically symmetric spacetimes with scalar field coupling in general relativity. We utilize the canonical formalism of geometrodynamics adapted to the Painleve–Gullstrand coordinates, and present a new quantization of the resulting field theory. We give an explicit construction of operators that capture curvature properties of the spacetime and use these to show that the black hole curvature singularity is avoided in the quantum theory.

Manifestly gauge-invariant general relativistic perturbation theory: I. Foundations

Linear cosmological perturbation theory is pivotal to a theoretical understanding of current cosmological experimental data provided e.g. by cosmic microwave anisotropy probes. A key issue in that theory is to extract the gauge-invariant degrees of freedom which allow unambiguous comparison between theory and experiment. When one goes beyond first (linear) order, the task of writing the Einstein equations expanded to nth order in terms of quantities that are gauge-invariant up to terms of higher orders becomes highly non-trivial and cumbersome. This fact has prevented progress for instance on the issue of the stability of linear perturbation theory and is a subject of current debate in the literature. In this series of papers we circumvent these difficulties by passing to a manifestly gauge-invariant framework. In other words, we only perturb gauge-invariant, i.e. measurable quantities, rather than gauge variant ones. Thus, gauge invariance is preserved non-perturbatively while we construct the perturbation theory for the equations of motion for the gauge-invariant observables to all orders. In this first paper we develop the general framework which is based on a seminal paper due to Brown and Kuchař as well as the relational formalism due to Rovelli. In the second, companion, paper we apply our general theory to FRW cosmologies and derive the deviations from the standard treatment in linear order. As it turns out, these deviations are negligible in the late universe, thus our theory is in agreement with the standard treatment. However, the real strength of our formalism is that it admits a straightforward and unambiguous, gauge-invariant generalization to higher orders. This will also allow us to settle the stability issue in a future publication.

Manifestly gauge-invariant general relativistic perturbation theory: II. FRW background and first order

In our companion paper we identified a complete set of manifestly gauge-invariant observables for general relativity. This was possible by coupling the system of gravity and matter to pressureless dust which plays the role of a dynamically coupled observer. The evolution of those observables is governed by a physical Hamiltonian and we derived the corresponding equations of motion. Linear perturbation theory of those equations of motion around a general exact solution in terms of manifestly gauge invariant perturbations was then developed. In this paper we specialise our previous results to an FRW background which is also a solution of our modified equations of motion. We then compare the resulting equations with those derived in standard cosmological perturbation theory (SCPT). We exhibit the precise relation between our manifestly gauge-invariant perturbations and the linearly gauge-invariant variables in SCPT. We find that our equations of motion can be cast into SCPT form plus corrections. These corrections are the trace that the dust leaves on the system in terms of a conserved energy momentum current density. It turns out that these corrections decay, in fact, in the late universe they are negligible whatever the value of the conserved current. We conclude that the addition of dust which serves as a test observer medium, while implying modifications of Einsteins equations without dust, leads to acceptable agreement with known results, while having the advantage that one now talks about manifestly gauge-invariant, that is measurable, quantities, which can be used even in perturbation theory at higher orders.

Quantum black holes from null expansion operators

Using a recently developed quantization of spherically symmetric gravity coupled to a scalar field, we give a construction of null expansion operators that allow a definition of general, fully dynamical quantum black holes. These operators capture the intuitive idea that classical black holes are defined by the presence of trapped surfaces, that is, surfaces from which light cannot escape outward. They thus provide a mechanism for classifying quantum states of the system into those that describe quantum black holes and those that do not. We find that quantum horizons fluctuate, confirming long-held heuristic expectations. We also give explicit examples of quantum black hole states. The work sets a framework for addressing the puzzles of black hole physics in a fully quantized dynamical setting.

Semiclassical Quantum Gravity: Statistics of Combinatorial Riemannian Geometries

This paper is a contribution to the development of a framework, to be used in the context of semiclassical canonical quantum gravity, in which to frame questions about the correspondence between discrete spacetime structures at “quantum scales” and continuum, classical geometries at large scales. Such a correspondence can be meaningfully established when one has a “semiclassical” state in the underlying quantum gravity theory, and the uncertainties in the correspondence arise both from quantum fluctuations in this state and from the kinematical procedure of matching a smooth geometry to a discrete one. We focus on the latter type of uncertainty, and suggest the use of statistical geometry as a way to quantify it. With a cell complex as an example of discrete structure, we discuss how to construct quantities that define a smooth geometry, and how to estimate the associated uncertainties. We also comment briefly on how to combine our results with uncertainties in the underlying quantum state, and on their use when considering phenomenological aspects of quantum gravity.

Quantum Hamiltonian for gravitational collapse

Using a Hamiltonian formulation of the spherically symmetric gravity-scalar field theory adapted to flat spatial slicing, we give a construction of the reduced Hamiltonian operator. This Hamiltonian, together with the null expansion operators presented in an earlier work, form a framework for studying gravitational collapse in quantum gravity. We describe a setting for its numerical implementation, and discuss some conceptual issues associated with quantum dynamics in a partial gauge fixing.

HOW RED IS A QUANTUM BLACK HOLE

Radiating black holes pose a number of puzzles for semiclassical and quantum gravity. These include the transplanckian problem — the nearly infinite energies of Hawking particles created near the horizon, and the final state of evaporation. A definitive resolution of these questions likely requires robust inputs from quantum gravity. We argue that one such input is a mechanism for a quantum bound on curvature. We show how the same method leads to an upper limit on the redshift of a Hawking emitted particle, to a maximum temperature for a black hole, and to the prediction of a Planck scale remnant.

Discrete Hamiltonian evolution and quantum gravity

We study constrained Hamiltonian systems by utilizing general forms of time discretization. We show that for explicit discretizations, the requirement of preserving the canonical Poisson bracket under discrete evolution imposes strong conditions on both allowable discretizations and Hamiltonians. These conditions permit time discretizations for a limited class of Hamiltonians, which does not include homogeneous cosmological models. We also present two general classes of implicit discretizations which preserve Poisson brackets for any Hamiltonian. Both types of discretizations generically do not preserve first class constraint algebras. Using this observation, we show that time discretization provides a complicated time gauge fixing for quantum gravity models, which may be compared with the alternative procedure of gauge fixing before discretization.

Comparison of QG-induced dispersion with standard physics effects

One of the predictions of quantum gravity phenomenology is that, in situations where Planck-scale physics and the notion of a quantum spacetime are relevant, field propagation will be described by a modified set of laws. Descriptions of the underlying mechanism differ from model to model, but a general feature is that electromagnetic waves will have non-trivial dispersion relations. A physical phenomenon that offers the possibility of experimentally testing these ideas in the foreseeable future is the propagation of high-energy gamma rays from GRBs at cosmological distances. With the observation of non-standard dispersion relations within experimental reach, it is thus important to find out whether there are competing effects that could either mask or be mistaken for this one. In this letter, we consider possible effects from standard physics, due to electromagnetic interactions, classical as well as quantum, and coupling to classical geometry. Our results indicate that, for currently observed gamma-ray energies and estimates of cosmological parameter values, those effects are much smaller than the quantum gravity one if the latter is first order in the energy; some corrections are comparable in magnitude to the second-order quantum gravity ones, but they have a very different energy dependence.