Using a Hamiltonian formulation of the spherically symmetric gravity-scalar field theory adapted ... more 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.
We give a Hamiltonian analysis of the asymptotically flat spherically symmetric system of gravity... more We give a Hamiltonian analysis of the asymptotically flat spherically symmetric system of gravity coupled to a scalar field. This 1+1 dimensional field theory may be viewed as the "standard model" for studying black hole physics. Our analysis is adapted to the flat slice Painleve-Gullstrand coordinates. We give a Hamiltonian action principle for this system, which yields an asymptotic mass formula. We then perform a time gauge fixing that gives a Hamiltonian as the integral of a local density. The Hamiltonian takes a relatively simple form compared to earlier work in Schwarzschild gauge, and therefore provides a setting amenable to full quantisation.
In a metric variable based Hamiltonian quantization, we give a prescription for constructing semi... more In a metric variable based Hamiltonian quantization, we give a prescription for constructing semiclassical matter-geometry states for homogeneous and isotropic cosmological models. These "collective" states arise as infinite linear combinations of fundamental excitations in an unconventional "polymer" quantization. They satisfy a number of properties characteristic of semiclassicality, such as peaking on classical phase space configurations. We describe how these states can be used to determine quantum corrections to the classical evolution equations, and to compute the initial state of the universe by a backward time evolution.
Classical and Quantum Gravity - CLASS QUANTUM GRAVITY, 2004
We study constrained Hamiltonian systems by utilizing general forms of time discretization. We sh... more 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.
We apply a singularity resolution technique utilized in loop quantum gravity to the polymer repre... more We apply a singularity resolution technique utilized in loop quantum gravity to the polymer representation of quantum mechanics on R with the singular −1/|x| potential. On an equispaced lattice, the resulting eigenvalue problem is identical to a finite difference approximation of the Schrödinger equation. We find numerically that the antisymmetric sector has an energy spectrum that converges to the usual Coulomb spectrum as the lattice spacing is reduced. For the symmetric sector, in contrast, the effect of the lattice spacing is similar to that of a continuum self-adjointness boundary condition at x = 0, and its effect on the ground state is significant even if the spacing is much below the Bohr radius. Boundary conditions at the singularity thus have a significant effect on the polymer quantization spectrum even after the singularity has been regularized. * Electronic address: husain@math.unb.ca † Electronic address: jorma.louko@nottingham.ac.uk ‡ Electronic address: owinkler@perimeterinstitute.ca
We examine the singularity resolution issue in quantum gravity by studying a new quantization of ... more We examine the singularity resolution issue in quantum gravity by studying a new quantization of standard Friedmann-Robertson-Walker geometrodynamics. The quantization procedure is inspired by the loop quantum gravity programme, and is based on an alternative to the Schrödinger representation normally used in metric variable quantum cosmology. We show that in this representation for quantum geometrodynamics there exists a densely defined inverse scale factor operator, and that the Hamiltonian constraint acts as a difference operator on the basis states. We find that the cosmological singularity is avoided in the quantum dynamics. We discuss these results with a view to identifying the criteria that constitute "singularity resolution" in quantum gravity.
We present a short overview of some recent work on coherent states aimed at providing tools for a... more We present a short overview of some recent work on coherent states aimed at providing tools for a semiclassical analysis of quantum general relativity and other quantum gauge field theories
We calculate the power spectrum of vacuum fluctuations of a generic scalar field in a quantum cos... more We calculate the power spectrum of vacuum fluctuations of a generic scalar field in a quantum cosmological setting that is manifestly singularity-free. The power spectrum is given in terms of the usual scale invariant spectrum plus scale dependent corrections. These are induced by welldefined quantum fluctuations of de Sitter spacetime, resulting in a modified dispersion relation for the scalar field. The leading correction turns out to be proportional to the ratio of Hubble scale and Planck Mass. The maximal relative change in the spectrum is on the ten percent level and might be observable with future CMB experiments. PACS numbers: 98.70.Vc, 98.80.Cq, 98.80.Qc A fundamental description of the physics during the Planck epoch remains elusive. Important issues like the big bang singularity, the unknown laws that govern the dynamics of all physical degrees of freedom during that period and the correct initial conditions are not yet fully understood.
One of the predictions of quantum gravity phenomenology is that, in situations where Planck-scale... more 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 GRB's 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 with the second-order quantum gravity ones, but they have a very different energy dependence.
We propose a framework, to be used in the context of semiclassical quantum gravity, in which to f... more We propose a framework, to be used in the context of semiclassical quantum gravity, in which to frame questions about the correspondence between discrete, non-geometrical spacetime structures at ``quantum scales" and continuum, classical geometries at large scales. The main ideas of the approach are based on statistical geometry, and are general enough to be applied independently of the dynamics of the theory, or even the specific variables used. We illustrate the application of these ideas using as basic structure a cell complex with additional variables attached to its edges and cells, along the lines of some approaches to canonical quantum gravity, and discuss how to construct quantities that define a smooth geometry, and how to estimate the associated uncertainties. These statistical uncertainties are of a different nature from the uncertainties of the underlying quantum state, and need to be taken into account together when considering phenomenological aspects of quantum gravity.
We describe recent progress on the problem of gravitational collapse in quantum gravity. The mode... more We describe recent progress on the problem of gravitational collapse in quantum gravity. The model we study is the gravity-scalar field theory in spherical symmetry, which is the original setting for Hawking's semiclassical work on black-hole radiation. We present an approach to full quantization of this model in a Hamiltonian framework, with a view to understanding the late time behaviour of collapsing matter. We give operator realizations of curvature and null expansion variables. The first of these operators has a bounded spectrum indicating curvature singularity avoidance; the second provides a "quantum trapping" criterion for detecting whether a quantum state describes a black hole. It leads to the physical result that the boundary of a quantum black hole is subject to quantum fluctuations. We describe an approach to quantum dynamics, whereby an initial state can be evolved, and tracked to the point of quantum trapping and beyond.
We address the "inverse problem" for discrete geometry, which consists in determining whether, gi... more We address the "inverse problem" for discrete geometry, which consists in determining whether, given a discrete structure of a type that does not in general imply geometrical information or even a topology, one can associate with it a unique manifold in an appropriate sense, and constructing the manifold when it exists. This problem arises in a variety of approaches to quantum gravity that assume a discrete structure at the fundamental level; the present work is motivated by the semiclassical sector of loop quantum gravity, so we will take the discrete structure to be a graph and the manifold to be a spatial slice in spacetime. We identify a class of graphs, those whose vertices have a fixed valence, for which such a construction can be specified. We define a procedure designed to produce a cell complex from a graph and show that, for graphs with which it can be carried out to completion, the resulting cell complex is in fact a PL-manifold. Graphs of our class for which the procedure cannot be completed either do not arise as edge graphs of manifold cell decompositions, or can be seen as cell decompositions of manifolds with structure at small scales (in terms of the cell spacing). We also comment briefly on how one can extend our procedure to more general graphs.
In the preceding paper of this series of articles we established peakedness proper- ties of a fam... more In the preceding paper of this series of articles we established peakedness proper- ties of a family of coherent states that were introduced by Hall for any compact gauge group and were later generalized to gauge field theory by Ashtekar, Lewandowski, Marolf, Mourao and Thiemann. In this paper we establish the "Ehrenfest Property" of these states which are labelled by a point (A, E), a connection and an electric field, in the classical phase space. By this we mean that i) The expectation value of all elementary quantum operators ˆ O with respect to the coherent state with label (A, E) is given to zeroth order inh by the value of the corresponding classical function O evaluated at the phase space point (A, E) and ii) The expectation value of the commutator between two elementary quantum operators ( ˆ O1, ˆ O2)/(ih) divided by ih with respect to the coherent state with label (A, E) is given to zeroth order inh by the value of the Poisson bracket between the corresponding classical functions {O1, O2} evaluated at the phase space point (A, E). These results can be extended to all polynomials of elementary operators and to a certain non-polynomial function of the elementary operators associated with the volume operator of quantum general relativity. It follows that the infinitesimal quantum dynamics of quantum general relativity is to zeroth order inh indeed given by classical general relativity.
This paper is a contribution to the development of a framework, to be used in the context of semi... more 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.
These results can be extended to all polynomials of elementary operators and to a certain non-pol... more These results can be extended to all polynomials of elementary operators and to a certain non-polynomial function of the elementary operators associated with the volume operator of quantum general relativity. It follows that the infinitesimal quantum dynamics of quantum general relativity is to zeroth order in $\hbar$ indeed given by classical general relativity.
In this article we apply the methods outlined in the previous paper of this series to the particu... more 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, Mour\~ao 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 resolve these issues, in particular, we 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 analysis of canonical quantum gravity and quantum gauge theory coupled to quantum gravity, enable error-controlled approximations and set a new starting point for {\it 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.
Linear cosmological perturbation theory is pivotal to a theoretical understanding of current cosm... more 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 n'th 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 Kuchar as well as the realtional 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 generalisation to higher orders. This will also allow us to settle the stability issue in a future publication.
In our companion paper we identified a complete set of manifestly gauge-invariant observables for... more 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 Einstein's 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.
Radiating black holes pose a number of puzzles for semiclassical and quantum gravity. These inclu... more 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 quantum bound on curvature. We show how this 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.
Using a Hamiltonian formulation of the spherically symmetric gravity-scalar field theory adapted ... more 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.
We give a Hamiltonian analysis of the asymptotically flat spherically symmetric system of gravity... more We give a Hamiltonian analysis of the asymptotically flat spherically symmetric system of gravity coupled to a scalar field. This 1+1 dimensional field theory may be viewed as the "standard model" for studying black hole physics. Our analysis is adapted to the flat slice Painleve-Gullstrand coordinates. We give a Hamiltonian action principle for this system, which yields an asymptotic mass formula. We then perform a time gauge fixing that gives a Hamiltonian as the integral of a local density. The Hamiltonian takes a relatively simple form compared to earlier work in Schwarzschild gauge, and therefore provides a setting amenable to full quantisation.
In a metric variable based Hamiltonian quantization, we give a prescription for constructing semi... more In a metric variable based Hamiltonian quantization, we give a prescription for constructing semiclassical matter-geometry states for homogeneous and isotropic cosmological models. These "collective" states arise as infinite linear combinations of fundamental excitations in an unconventional "polymer" quantization. They satisfy a number of properties characteristic of semiclassicality, such as peaking on classical phase space configurations. We describe how these states can be used to determine quantum corrections to the classical evolution equations, and to compute the initial state of the universe by a backward time evolution.
Classical and Quantum Gravity - CLASS QUANTUM GRAVITY, 2004
We study constrained Hamiltonian systems by utilizing general forms of time discretization. We sh... more 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.
We apply a singularity resolution technique utilized in loop quantum gravity to the polymer repre... more We apply a singularity resolution technique utilized in loop quantum gravity to the polymer representation of quantum mechanics on R with the singular −1/|x| potential. On an equispaced lattice, the resulting eigenvalue problem is identical to a finite difference approximation of the Schrödinger equation. We find numerically that the antisymmetric sector has an energy spectrum that converges to the usual Coulomb spectrum as the lattice spacing is reduced. For the symmetric sector, in contrast, the effect of the lattice spacing is similar to that of a continuum self-adjointness boundary condition at x = 0, and its effect on the ground state is significant even if the spacing is much below the Bohr radius. Boundary conditions at the singularity thus have a significant effect on the polymer quantization spectrum even after the singularity has been regularized. * Electronic address: husain@math.unb.ca † Electronic address: jorma.louko@nottingham.ac.uk ‡ Electronic address: owinkler@perimeterinstitute.ca
We examine the singularity resolution issue in quantum gravity by studying a new quantization of ... more We examine the singularity resolution issue in quantum gravity by studying a new quantization of standard Friedmann-Robertson-Walker geometrodynamics. The quantization procedure is inspired by the loop quantum gravity programme, and is based on an alternative to the Schrödinger representation normally used in metric variable quantum cosmology. We show that in this representation for quantum geometrodynamics there exists a densely defined inverse scale factor operator, and that the Hamiltonian constraint acts as a difference operator on the basis states. We find that the cosmological singularity is avoided in the quantum dynamics. We discuss these results with a view to identifying the criteria that constitute "singularity resolution" in quantum gravity.
We present a short overview of some recent work on coherent states aimed at providing tools for a... more We present a short overview of some recent work on coherent states aimed at providing tools for a semiclassical analysis of quantum general relativity and other quantum gauge field theories
We calculate the power spectrum of vacuum fluctuations of a generic scalar field in a quantum cos... more We calculate the power spectrum of vacuum fluctuations of a generic scalar field in a quantum cosmological setting that is manifestly singularity-free. The power spectrum is given in terms of the usual scale invariant spectrum plus scale dependent corrections. These are induced by welldefined quantum fluctuations of de Sitter spacetime, resulting in a modified dispersion relation for the scalar field. The leading correction turns out to be proportional to the ratio of Hubble scale and Planck Mass. The maximal relative change in the spectrum is on the ten percent level and might be observable with future CMB experiments. PACS numbers: 98.70.Vc, 98.80.Cq, 98.80.Qc A fundamental description of the physics during the Planck epoch remains elusive. Important issues like the big bang singularity, the unknown laws that govern the dynamics of all physical degrees of freedom during that period and the correct initial conditions are not yet fully understood.
One of the predictions of quantum gravity phenomenology is that, in situations where Planck-scale... more 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 GRB's 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 with the second-order quantum gravity ones, but they have a very different energy dependence.
We propose a framework, to be used in the context of semiclassical quantum gravity, in which to f... more We propose a framework, to be used in the context of semiclassical quantum gravity, in which to frame questions about the correspondence between discrete, non-geometrical spacetime structures at ``quantum scales" and continuum, classical geometries at large scales. The main ideas of the approach are based on statistical geometry, and are general enough to be applied independently of the dynamics of the theory, or even the specific variables used. We illustrate the application of these ideas using as basic structure a cell complex with additional variables attached to its edges and cells, along the lines of some approaches to canonical quantum gravity, and discuss how to construct quantities that define a smooth geometry, and how to estimate the associated uncertainties. These statistical uncertainties are of a different nature from the uncertainties of the underlying quantum state, and need to be taken into account together when considering phenomenological aspects of quantum gravity.
We describe recent progress on the problem of gravitational collapse in quantum gravity. The mode... more We describe recent progress on the problem of gravitational collapse in quantum gravity. The model we study is the gravity-scalar field theory in spherical symmetry, which is the original setting for Hawking's semiclassical work on black-hole radiation. We present an approach to full quantization of this model in a Hamiltonian framework, with a view to understanding the late time behaviour of collapsing matter. We give operator realizations of curvature and null expansion variables. The first of these operators has a bounded spectrum indicating curvature singularity avoidance; the second provides a "quantum trapping" criterion for detecting whether a quantum state describes a black hole. It leads to the physical result that the boundary of a quantum black hole is subject to quantum fluctuations. We describe an approach to quantum dynamics, whereby an initial state can be evolved, and tracked to the point of quantum trapping and beyond.
We address the "inverse problem" for discrete geometry, which consists in determining whether, gi... more We address the "inverse problem" for discrete geometry, which consists in determining whether, given a discrete structure of a type that does not in general imply geometrical information or even a topology, one can associate with it a unique manifold in an appropriate sense, and constructing the manifold when it exists. This problem arises in a variety of approaches to quantum gravity that assume a discrete structure at the fundamental level; the present work is motivated by the semiclassical sector of loop quantum gravity, so we will take the discrete structure to be a graph and the manifold to be a spatial slice in spacetime. We identify a class of graphs, those whose vertices have a fixed valence, for which such a construction can be specified. We define a procedure designed to produce a cell complex from a graph and show that, for graphs with which it can be carried out to completion, the resulting cell complex is in fact a PL-manifold. Graphs of our class for which the procedure cannot be completed either do not arise as edge graphs of manifold cell decompositions, or can be seen as cell decompositions of manifolds with structure at small scales (in terms of the cell spacing). We also comment briefly on how one can extend our procedure to more general graphs.
In the preceding paper of this series of articles we established peakedness proper- ties of a fam... more In the preceding paper of this series of articles we established peakedness proper- ties of a family of coherent states that were introduced by Hall for any compact gauge group and were later generalized to gauge field theory by Ashtekar, Lewandowski, Marolf, Mourao and Thiemann. In this paper we establish the "Ehrenfest Property" of these states which are labelled by a point (A, E), a connection and an electric field, in the classical phase space. By this we mean that i) The expectation value of all elementary quantum operators ˆ O with respect to the coherent state with label (A, E) is given to zeroth order inh by the value of the corresponding classical function O evaluated at the phase space point (A, E) and ii) The expectation value of the commutator between two elementary quantum operators ( ˆ O1, ˆ O2)/(ih) divided by ih with respect to the coherent state with label (A, E) is given to zeroth order inh by the value of the Poisson bracket between the corresponding classical functions {O1, O2} evaluated at the phase space point (A, E). These results can be extended to all polynomials of elementary operators and to a certain non-polynomial function of the elementary operators associated with the volume operator of quantum general relativity. It follows that the infinitesimal quantum dynamics of quantum general relativity is to zeroth order inh indeed given by classical general relativity.
This paper is a contribution to the development of a framework, to be used in the context of semi... more 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.
These results can be extended to all polynomials of elementary operators and to a certain non-pol... more These results can be extended to all polynomials of elementary operators and to a certain non-polynomial function of the elementary operators associated with the volume operator of quantum general relativity. It follows that the infinitesimal quantum dynamics of quantum general relativity is to zeroth order in $\hbar$ indeed given by classical general relativity.
In this article we apply the methods outlined in the previous paper of this series to the particu... more 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, Mour\~ao 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 resolve these issues, in particular, we 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 analysis of canonical quantum gravity and quantum gauge theory coupled to quantum gravity, enable error-controlled approximations and set a new starting point for {\it 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.
Linear cosmological perturbation theory is pivotal to a theoretical understanding of current cosm... more 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 n'th 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 Kuchar as well as the realtional 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 generalisation to higher orders. This will also allow us to settle the stability issue in a future publication.
In our companion paper we identified a complete set of manifestly gauge-invariant observables for... more 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 Einstein's 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.
Radiating black holes pose a number of puzzles for semiclassical and quantum gravity. These inclu... more 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 quantum bound on curvature. We show how this 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.
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Papers by Oliver Winkler