Canonical Gravity Research Articles

LQG vertex with finite Immirzi parameter

We extend the definition of the “flipped” loop-quantum-gravity vertex to the case of a finite Immirzi parameter γ. We cover both the Euclidean and Lorentzian cases. We show that the resulting dynamics is defined on a Hilbert space isomorphic to the one of loop quantum gravity, and that the area operator has the same discrete spectrum as in loop quantum gravity. This includes the correct dependence on γ, and, remarkably, holds in the Lorentzian case as well. The ad hoc flip of the symplectic structure that was required to derive the flipped vertex is not anymore required for finite γ. These results establish a bridge between canonical loop quantum gravity and the spinfoam formalism in four dimensions.

Gauge formulation of general relativity using conformal and spin symmetries

The gauge symmetry inherent in Maxwell's electromagnetics has a profound impact on modern physics. Following the successful quantization of electromagnetics and other higher order gauge field theories, the gauge principle has been applied in various forms to quantize gravity. A notable development in this direction is loop quantum gravity based on the spin-gauge treatment. This paper considers a further incorporation of the conformal gauge symmetry in canonical general relativity. This is a new conformal decomposition in that it is applied to simplify recently formulated parameter-free construction of spin-gauge variables for gravity. The resulting framework preserves many main features of the existing canonical framework for loop quantum gravity regarding the spin network representation and Thiemann's regularization. However, the Barbero-Immirzi parameter is converted into the conformal factor as a canonical variable. It behaves like a scalar field but is somehow non-dynamical since the Hamiltonian constraint does not depend on its momentum. The essential steps of the mathematical derivation of this parameter-free framework for the spin-gauge variables of gravity are spelled out. The implications for the loop quantum gravity programme are briefly discussed.

Quantum GowdyT3model: Schrödinger representation with unitary dynamics

The linearly polarized Gowdy ${T}^{3}$ model is paradigmatic for studying technical and conceptual issues in the quest for a quantum theory of gravity since, after a suitable and almost complete gauge fixing, it becomes an exactly soluble midisuperspace model. Recently, a new quantization of the model, possessing desired features such as a unitary implementation of the gauge group and of the time evolution, has been put forward and proven to be essentially unique. An appropriate setting for making contact with other approaches to canonical quantum gravity is provided by the Schr\"odinger representation, where states are functionals on the configuration space of the theory. Here we construct this functional description, analyze the time evolution in this context and show that it is also unitary when restricted to physical states, i.e. states which are solutions to the remaining constraint of the theory.

Spacetime Coarse Grainings and the Problem of Time in the Decoherent Histories Approach to Quantum Theory

We investigate the possibility of assigning consistent probabilities to sets of histories characterized by whether they enter a particular subspace of the Hilbert space of a closed system during a given time interval. In particular we investigate the case that this subspace is a region of the configuration space. This corresponds to a particular class of coarse grainings of spacetime regions. We consider the arrival time problem, as a warm up, to deal with the problem of time in reparametrization invariant theories (as for example in canonical quantum gravity) which subsequently we examine. Decoherence conditions and probabilities for those application are derived. The resulting decoherence condition does not depend on the explicit form of the restricted propagator that was problematic for generalizations such as application in quantum cosmology. Closely related to our results, is the problem of tunnelling time as well as the quantum Zeno effect. Some interpretational comments conclude, and we discuss the applicability of this formalism to deal with the arrival time problem and the problem of time in general.

The York map as a Shanmugadhasan canonical transformation in tetrad gravity and the role of non-inertial frames in the geometrical view of the gravitational field

A new parametrization of the 3-metric allows to find explicitly a York map by means of a partial Shanmugadhasan canonical transformation in canonical ADM tetrad gravity. This allows to identify the two pairs of physical tidal degrees of freedom (the Dirac observables of the gravitational field have to be built in term of them) and 14 gauge variables. These gauge quantities, whose role in describing generalized inertial effects is clarified, are all configurational except one, the York time, i.e. the trace \(^{3}K(\tau ,\vec \sigma )\) of the extrinsic curvature of the instantaneous 3-spaces \(\Sigma_{\tau}\) (corresponding to a clock synchronization convention) of a non-inertial frame centered on an arbitrary observer. In \(\Sigma_{\tau}\) the Dirac Hamiltonian is the sum of the weak ADM energy \(E_{\rm ADM} = \int d^3\sigma\, {\mathcal{E}}_{\rm ADM}(\tau ,\vec \sigma )\) (whose density \({\mathcal{E}}_{\rm ADM}(\tau ,\vec \sigma )\) is coordinate-dependent, containing the inertial potentials) and of the first-class constraints. The main results of the paper, deriving from a coherent use of constraint theory, are: (i) The explicit form of the Hamilton equations for the two tidal degrees of freedom of the gravitational field in an arbitrary gauge: a deterministic evolution can be defined only in a completely fixed gauge, i.e. in a non-inertial frame with its pattern of inertial forces. The simplest such gauge is the 3-orthogonal one, but other gauges are discussed and the Hamiltonian interpretation of the harmonic gauges is given. This frame-dependence derives from the geometrical view of the gravitational field and is lost when the theory is reduced to a linear spin 2 field on a background space-time. (ii) A general solution of the super-momentum constraints, which shows the existence of a generalized Gribov ambiguity associated to the 3-diffeomorphism gauge group. It influences: (a) the explicit form of the solution of the super-momentum constraint and then of the Dirac Hamiltonian; (b) the determination of the shift functions and then of the lapse one. (iii) The dependence of the Hamilton equations for the two pairs of dynamical gravitational degrees of freedom (the generalized tidal effects) and for the matter, written in a completely fixed 3-orthogonal Schwinger time gauge, upon the gauge variable \({}^3K(\tau ,\vec \sigma )\) , determining the convention of clock synchronization. The associated relativistic inertial effects, absent in Newtonian gravity and implying inertial forces changing from attractive to repulsive in regions with different sign of \({}^3K(\tau ,\vec \sigma )\) , are completely unexplored and may have astrophysical relevance in the interpretation of the dark side of the universe.

Cosmological vector modes and quantum gravity effects

In contrast to scalar and tensor modes, vector modes of linear perturbations around an expanding Friedmann–Robertson–Walker universe decay. This makes them largely irrelevant for late time cosmology, assuming that all modes started out at a similar magnitude at some early stage. By now, however, bouncing models are frequently considered which exhibit a collapsing phase. Before this phase reaches a minimum size and re-expands, vector modes grow. Such modes are thus relevant for the bounce and may even signal the breakdown of perturbation theory if the growth is too strong. Here, a gauge-invariant formulation of vector mode perturbations in Hamiltonian cosmology is presented. This lays out a framework for studying possible canonical quantum gravity effects, such as those of loop quantum gravity, at an effective level. As an explicit example, typical quantum corrections, namely those coming from inverse densitized triad components and holonomies, are shown to increase the growth rate of vector perturbations in the contracting phase, but only slightly. Effects at the bounce of the background geometry can, however, be much stronger.

Radiation equation of state and loop quantum gravity corrections

The equation of state for radiation is derived in a canonical formulation of the electromagnetic field. This allows one to include correction terms expected from canonical quantum gravity and to infer implications to the universe evolution in radiation dominated epochs. Corrections implied by quantum geometry can be interpreted in physically appealing ways, relating to the conformal invariance of the classical equations.

Algebraic quantum gravity (AQG): I. Conceptual setup

We introduce a new top down approach to canonical quantum gravity, called algebraic quantum gravity (AQG). The quantum kinematics of AQG is determined by an abstract *-algebra generated by a countable set of elementary operators labelled by an algebraic graph. The quantum dynamics of AQG is governed by a single master constraint operator. While AQG is inspired by loop quantum gravity (LQG), it differs drastically from it because in AQG there is fundamentally no topology or differential structure. A natural Hilbert space representation acquires the structure of an infinite tensor product (ITP) whose separable strong equivalence class Hilbert subspaces (sectors) are left invariant by the quantum dynamics. The missing information about the topology and differential structure of the spacetime manifold as well as about the background metric to be approximated is supplied by coherent states. Given such data, the corresponding coherent state defines a sector in the ITP which can be identified with a usual QFT on the given manifold and background. Thus, AQG contains QFT on all curved spacetimes at once, possibly has something to say about topology change and provides the contact with the familiar low energy physics. In particular, in two companion papers we develop semiclassical perturbation theory for AQG and LQG and thereby show that the theory admits a semiclassical limit whose infinitesimal gauge symmetry agrees with that of general relativity. In AQG everything is computable with sufficient precision and no UV divergences arise due to the background independence of the fundamental combinatorial structure. Hence, in contrast to lattice gauge theory on a background metric, no continuum limit has to be taken. There simply is no lattice regulator that must be sent to zero.

Hamiltonian cosmological perturbation theory with loop quantum gravity corrections

Cosmological perturbation equations are derived systematically in a canonical scheme based on Ashtekar variables. A comparison with the covariant derivation and various subtleties in the calculation and choice of gauges are pointed out. Nevertheless, the treatment is more systematic when correction terms of canonical quantum gravity are to be included. This is done throughout the paper for one example of characteristic modifications expected from loop quantum gravity.

Partial and complete observables for canonical general relativity

In this work we will consider the concepts of partial and complete observables for canonical general relativity. These concepts provide a method to calculate Dirac observables. The central result of this work is that one can compute Dirac observables for general relativity by dealing with just one constraint. For this we have to introduce spatial diffeomorphism invariant Hamiltonian constraints. It will turn out that these can be made to be Abelian. Furthermore the methods outlined here provide a connection between observables in the spacetime picture, i.e. quantities invariant under spacetime diffeomorphisms, and Dirac observables in the canonical picture.

Resolving the Degeneracy: Experimental Tests of the New Self Creation Cosmology and a Heterodox Prediction for Gravity Probe B

The new theory of Self Creation Cosmology has been shown to yield a concordant cosmological solution that does not require inflation, exotic non-baryonic Dark matter or Dark Energy to fit observational constraints. In vacuo there is a conformal equivalence between this theory and canonical General Relativity and as a consequence an experimental degeneracy exists as the two theories predict identical results in the standard tests. However, there are three definitive experiments that are able to resolve this degeneracy and distinguish between the two theories. Here these standard tests and definitive experiments are described. One of the definitive predictions, that of the geodetic precession of a gyroscope, has just been measured on the Gravity Probe B satellite, which is at the present time of writing in the data processing stage. This is the first opportunity to falsify Self Creation Cosmology. The theory predicts a 'frame-dragging' result equal to GR but a geodetic precession of only 2/3 the GR value. When applied to the Gravity Probe B satellite, Self Creation Cosmology predicts an E-W gravitomagnetic/frame-dragging precession, equal to that of GR, of 40.9 milliarcsec/yr but a -S gyroscope (geodetic + Thomas) precession of just 4.4096 arcsec/yr.

QUANTUM MECHANICS IN NONINERTIAL FRAMES WITH A MULTITEMPORAL QUANTIZATION SCHEME I: RELATIVISTIC PARTICLES

After a review of the few attempts to define quantum mechanics in noninertial frames, we introduce a family of relativistic nonrigid noninertial frames (equal-time parallel hyper-planes with differentially rotating 3-coordinates) as a gauge fixing of the description of N positive energy particles in the framework of parametrized Minkowski theories. Then we define a multitemporal quantization scheme in which the particles are quantized, but not the gauge variables describing the noninertial frames: they are considered as c-number generalized times. We study the coupled Schrödinger-like equations produced by the first class constraints and we show that there is a physical scalar product independent both from time and generalized times and a unitary evolution. Since a path in the space of the generalized times defines a nonrigid noninertial frame, we can find the associated self-adjoint effective Hamiltonian [Formula: see text] for the noninertial evolution: it differs from the inertial energy operator for the presence of inertial potentials and turns out to be frame-dependent like the energy density in general relativity. After a separation of the relativistic center of mass from the relative variables by means of a recently developed relativistic kinematics, inside [Formula: see text] we can identify the self-adjoint relative energy operator (the invariant mass) [Formula: see text] corresponding to the inertial energy and producing the same levels for the spectra of atoms as in inertial frames. Instead the (in general time-dependent) effective Hamiltonian is responsible for the interferometric effects signaling the noninertiality of the frame. It cannot be interpreted as an energy (there is no relativity principle and no kinematic group in noninertial frames) and generically, like in the case of time-dependent c-number external electromagnetic fields, it has no associated eigenvalue equation defining a noninertial spectrum. This formulation should help to find relativistic Bel inequalities and to define a quantization scheme for canonical gravity after having found a ultraviolet regularization of the Tomonaga–Schwinger formalism in special relativity as required by the Torre–Varadarajan no-go theorem.

Semiclassical states in loop quantum gravity: an introduction

Unifying general relativity and quantum mechanics is a great challenge left to us by Einstein. To face this challenge, considerable progress has been made in non-perturbative canonical (loop) quantum gravity during the past 20 years. The kinematical Hilbert space of the quantum theory is constructed rigorously. However, the semiclassical analysis of the theory is still a crucial and open issue. In this review, we first introduce our work on constructing a semiclassical weave state, using the \(\hat Q\)[ω] operator on the kinematical Hilbert space of loop quantum gravity. Then we give an introduction to the two different approaches currently investigated for constructing coherent states in the kinematical Hilbert space. The current status of semiclassical analysis in loop quantum gravity is then summarized.

Quantum GowdyT3model: A unitary description

The quantization of the family of linearly polarized Gowdy $T^3$ spacetimes is discussed in detail, starting with a canonical analysis in which the true degrees of freedom are described by a scalar field that satisfies a Klein-Gordon type equation in a fiducial time dependent background. A time dependent canonical transformation, which amounts to a change of the basic (scalar) field of the model, brings the system to a description in terms of a Klein-Gordon equation on a background that is now static, although subject to a time dependent potential. The system is quantized by means of a natural choice of annihilation and creation operators. The quantum time evolution is considered and shown to be unitary, allowing both the Schr\"odinger and Heisenberg pictures to be consistently constructed. This has to be contrasted with previous treatments for which time evolution failed to be implementable as a unitary transformation. Possible implications for both canonical quantum gravity and quantum field theory in curved spacetime are commented.

Relational particle models: I. Reconciliation with standard classical and quantum theory

This paper concerns the absolute versus relative motion debate. The Barbour and Bertotti (1982) work may be viewed as an indirectly set up relational formulation of a portion of Newtonian mechanics. I consider further direct formulations of this and argue that the portion in question—universes with zero total angular momentum that are conservative and with kinetic terms that are (homogeneous) quadratic in their velocities—is capable of accommodating a wide range of classical physics phenomena. Furthermore, as I develop in paper II, this relational particle model is a useful toy model for canonical general relativity. I consider what happens if one quantizes relational rather than absolute mechanics, indeed whether the latter is misleading. By exploiting Jacobi coordinates, I show how to access many examples of quantized relational particle models and then interpret these from a relational perspective. By these means, previous suggestions of bad semiclassicality for such models can be eluded. I show how small (particle number) universe relational particle model examples display eigenspectrum truncation, gaps, energy interlocking and counterbalanced total angular momentum. These features mean that these small universe models make interesting toy models for some aspects of closed-universe quantum cosmology. Meanwhile, these features do not compromise the recovery of reality as regards the practicalities of experimentation in a large universe such as our own.

On a class of 2-surface observables in general relativity

The boundary conditions for canonical vacuum general relativity are investigated at the quasi-local level. It is shown that fixing the area element on the 2-surface (rather than the induced 2-metric) is enough to have a well-defined constraint algebra, and a well-defined Poisson algebra of basic Hamiltonians parametrized by shifts that are tangent to and divergence free on . The evolution equations preserve these boundary conditions, and the value of the basic Hamiltonians gives 2+2-covariant, gauge-invariant 2-surface observables. The meaning of these observables is also discussed.

Towards conformal loop quantum gravity

A discussion is given of recent developments in canonical gravity that assimilates the conformal analysis of gravitational degrees of freedom. The work is motivated by the problem of time in quantum gravity and is carried out at the metric and the triad levels. At the metric level, it is shown that by extending the Arnowitt-Deser-Misner (ADM) phase space of general relativity (GR), a conformal form of geometrodynamics can be constructed. In addition to the Hamiltonian and Diffeomorphism constraints, an extra first class constraint is introduced to generate conformal transformations. This phase space consists of York's mean extrinsic curvature time, conformal three-metric and their momenta. At the triad level, the phase space of GR is further enlarged by incorporating spin-gauge as well as conformal symmetries. This leads to a canonical formulation of GR using a new set of real spin connection variables. The resulting gravitational constraints are first class, consisting of the Hamiltonian constraint and the canonical generators for spin-gauge and conformorphism transformations. The formulation has a remarkable feature of being parameter-free. Indeed, it is shown that a conformal parameter of the Barbero-Immirzi type can be absorbed by the conformal symmetry of the extended phase space. This gives rise to an alternative approach to loop quantum gravity that addresses both the conceptual problem of time and the technical problem of functional calculus in quantum gravity.

Clock Synchronization, Dirac Observables and Gauge Variables in Canonical Gravity and the Objectivity of Spacetime.

This is a review of the chrono-geometrical structure of special and general relativity with a special emphasis on the role of non-inertial frames and of the conventions for the synchronization of distant clocks. ADM canonical metric and tetrad gravity are analyzed in a class of space-times suitable to incorporate particle physics by using Dirac theory of constraints, which allows to arrive at a separation of the genuine degrees of freedom of the gravitational field, the Dirac observables describing generalized tidal effects, from its gauge variables, describing generalized inertial effects. A background-independent formulation (the rest-frame instant form of tetrad gravity) emerges, since the chosen boundary conditions at spatial infinity imply the existence of an asymptotic flat metric. By switching off the Newton constant in presence of matter this description deparametrizes to the rest-frame instant form for such matter in the framework of parametrized Minkowski theories. The problem of the objectivity of the spacetime point-events, implied by Einstein's Hole Argument, is analyzed.

Semi-classical quantization of spacetimes with apparent horizons

Coherent or semi-classical states in canonical quantum gravity describe the classical Schwarzschild spacetime. By tracing over the coherent state wavefunction inside the horizon, a density matrix is derived. Bekenstein–Hawking entropy is obtained from the density matrix, modulo the Immirzi parameter. The expectation value of the area and curvature operator is evaluated in these states. The behaviour near the singularity of the curvature operator shows that the singularity is resolved. We then generalize the results to spacetimes with spherically symmetric apparent horizons.

Unambiguous spin-gauge formulation of canonical general relativity with conformorphism invariance

We present a parameter-free gauge formulation of general relativity in terms of a new set of real spin connection variables. The theory is constructed by extending the phase space of the recently formulated conformal geometrodynamics to accommodate a spin-gauge description. This leads to a further enlarged set of first class gravitational constraints consisting of a reduced Hamiltonian constraint and the canonical generators for spin-gauge and conformorphism transformations. Owing to the incorporated conformal symmetry, the new theory is shown to be free from an ambiguity of the Barbero-Immirzi type.