Table of contents

This is the first in a series of papers outlining an algorithm to explicitly construct finite quantum states of the full theory of gravity in Ashtekar variables. The algorithm is based upon extending some properties of a special state, the Kodama state for pure gravity with cosmological term, to matter-coupled models. We then illustrate a prescription for nonperturbatively constructing the generalized Kodama states, in preparation for subsequent works in this series. We also introduce the concept of the semiclassical-quantum correspondence (SQC). We express the quantum constraints of the full theory as a system of equations to be solved for the constituents of the 'phase' of the wavefunction. Additionally, we provide a variety of representations of the generalized Kodama states including a generalization of the topological instanton term to include matter fields, for which we present arguments for the field-theoretical analogue of cohomology on infinite-dimensional spaces. We demonstrate that the Dirac, reduced phase space and geometric quantization procedures are all equivalent for these generalized Kodama states as a natural consequence of the SQC. We relegate the method of the solution to the constraints and other associated ramifications of the generalized Kodama states to separate works.

In this paper, we construct the generalized Kodama state for the case of a Klein–Gordon scalar field coupled to Ashtekar variables in isotropic minisuperspace by a new method. The criterion for finiteness of the state stems from a minisuperspace reduction of the quantized full theory, rather than the conventional techniques of reduction prior to quantization. We then provide a possible route to the reproduction of a semiclassical limit via these states. This is the result of a new principle of the semiclassical-quantum correspondence (SQC), introduced in the first paper in this series. Lastly, we examine the solution to the minisuperspace case at the semiclassical level for an isotropic CDJ matrix neglecting any quantum corrections and examine some of the implications in relation to results from previous authors on semiclassical orbits of spacetime, including inflation. It is suggested that the application of nonperturbative quantum gravity, by way of the SQC, might potentially lead to some predictions testable below the Planck scale.

As a preparation for its quantization in the loop formalism, the two-dimensional gravitation model of Jackiw and Teitelboim is analysed in the classical canonical formalism. The dynamics is of pure constraints as is well known. A partial gauge fixing of the temporal type being performed, the resulting second class constraints are sorted out and the corresponding Dirac bracket algebra is worked out. Dirac observables of this classical theory are then calculated.

A Killing bubble is a minimal surface that arises as the fixed surface of a spacelike Killing field. We compute the bubble contributions to the Smarr relations and the mass and tension first laws for spacetimes containing both black holes and Killing bubbles. The resulting relations display an interesting interchange symmetry between the properties of black hole horizons and those of KK bubbles. This interchange symmetry reflects the underlying relation between static bubbles and black holes under double analytic continuation of the time and Kaluza–Klein directions. The thermodynamics of bubbles involve a geometrical quantity that we call the bubble surface gravity, which we show has several properties in common with the black hole surface gravity.

The Bousso entropy bound, in its generalized form, is investigated for the case of perfect fluids at local thermodynamic equilibrium and evidence is found that the bound is satisfied if and only if a certain local thermodynamic property holds, emerging when the attempt is made to apply the bound to thin layers of matter. This property consists of the existence of an ultimate lower limit l* to the thickness of the slices for which a statistical-mechanical description is viable, depending l* on the thermodynamical variables which define the state of the system locally. This limiting scale, found to be in general much larger than the Planck scale (so that no Planck scale physics must be necessarily invoked to justify it), appears not related to gravity and this suggests that the generalized entropy bound is likely to be rooted on conventional flat-spacetime statistical mechanics, with the maximum admitted entropy being however actually determined also by gravity. Some examples of ideal fluids are considered in order to identify the mechanisms which can set a lower limit to the statistical-mechanical description and these systems are found to respect the lower limiting scale l*. The photon gas, in particular, appears to seemingly saturate this limiting scale and the consequence is drawn that for systems consisting of a single slice of a photon gas with thickness l*, the generalized Bousso bound is saturated. It is argued that this seems to open the way to a peculiar understanding of black hole entropy: if an entropy can meaningfully (i.e. with a second law) be assigned to a black hole, the value A/4 for it (where A is the area of the black hole) is required simply by (conventional) statistical mechanics coupled to general relativity.

We compute the effects of a compact flat universe on the angular correlation function, the angular power spectrum, the circles-in-the-sky signature, and the covariance matrix of the spherical harmonics coefficients of the cosmic microwave background radiation using the full Boltzmann physics. Our analysis shows that the Wilkinson Microwave Anisotropy Probe (WMAP) three-year data are compatible with the possibility that we live in a flat 3-torus with volume ≃5 × 10³ Gpc³.

The motion of a body endowed with a dipolar as well as a quadrupolar structure is investigated in the Kerr background according to the Dixon model, extending a previous analysis done in the Schwarzschild background. The full set of evolution equations is solved under the simplifying assumptions of constant frame components for both the spin and the quadrupole tensors and that the center of mass moves along an equatorial circular orbit, the total 4-momentum of the body being aligned with it. We find that the motion deviates from the geodesic one due to the internal structure of the body, leading to measurable effects. Corrections to the geodesic value of the orbital period of a close binary system orbiting the galactic center are discussed assuming that the galactic center is a Kerr supermassive black hole.

This paper considers the symmetries of the curvature tensor (curvature collineations) and of the Weyl conformal tensor (Weyl conformal collineations) in general relativity. Some general results are reviewed for later application, some new ones proved and many special cases are investigated. Particular emphasis is laid on the interrelations between these two types of symmetries. A number of instructive examples of such symmetries are given.

We construct the quantum-mechanical evolution operator in the functional Schrödinger picture—the kernel—for a scalar field in spatially homogeneous FLRW spacetimes when the field is (a) free and (b) coupled to a spacetime-dependent source term. The essential element in the construction is the causal propagator, linked to the commutator of two Heisenberg picture scalar fields. We show that the kernels can be expressed solely in terms of the causal propagator and derivatives of the causal propagator. Furthermore, we show that our kernel reveals the standard light cone structure in FLRW spacetimes. We finally apply the result to Minkowski spacetime, to de Sitter spacetime and calculate the forward time evolution of the vacuum in a general FLRW spacetime.

We consider non-rotating geodesic perfect fluid spacetimes which are purely radiative in the sense that the gravitational field satisfies the covariant transverse conditions div H = div E = 0. We show that when the shear tensor σ is degenerate, H, E and σ necessarily commute and hence the resulting spacetimes are hypersurface homogeneous of Bianchi class A (modulo some purely electric exceptions).

The relative positions of the test masses in gravitational-wave detectors will be influenced not only by astrophysical gravitational waves, but also by the fluctuating Newtonian gravitational forces of moving masses in the ground and air around the detector. These effects are often referred to as gravity gradient noise. This paper considers the effects of gravity gradients from density perturbations in the atmosphere, and from massive airborne objects near the detector. These have been discussed previously by Saulson (1984 Phys. Rev. D30 732), who considered the effects of background acoustic pressure waves and of massive objects moving smoothly past the interferometer; the gravity gradients he predicted would be too small to be of serious concern even for advanced interferometric gravitational-wave detectors. In this paper, I revisit these phenomena, considering transient atmospheric shocks, and estimating the effects of sound waves or objects colliding with the ground or buildings around the test masses. I also consider another source of atmospheric density fluctuations: temperature perturbations that are advected past the detector by the wind. I find that background acoustic noise and temperature fluctuations still produce gravity gradient noise that is below the noise floor even of advanced interferometric detectors, although temperature perturbations carried along non-laminar streamlines could produce noise that is within an order of magnitude of the projected noise floor at 10 Hz. A definitive study of this effect may require better models of the wind flow past a given instrument. I also find that transient shockwaves in the atmosphere could potentially produce large spurious signals, with signal-to-noise ratios in the hundreds in an advanced interferometric detector. These signals could be vetoed by means of acoustic sensors outside of the buildings. Massive wind-borne objects such as tumbleweeds could also produce gravity gradient signals with signal-to-noise ratios in the hundreds if they collide with the interferometer buildings, so it may be necessary to build fences preventing such objects from approaching within about 30 m of the test masses.

We discuss results that have been obtained from the implementation of the initial round of testbeds for numerical relativity which was proposed in the first paper of the Apples with Apples Alliance. We present benchmark results for various codes which provide templates for analyzing the testbeds and to draw conclusions about various features of the codes. This allows us to sharpen the initial test specifications, design a new test and add theoretical insight.

We present an explicit cosmological model where inflation and dark energy both could arise from the dynamics of the same scalar field. We present our discussion in the framework where the inflaton field ϕ attains a nearly constant velocity m⁻¹_P|dϕ/dN| ≡ α + βexp(βN) (where N ≡ ln a is the e-folding time) during inflation. We show that the model with |α| < 0.25 and β < 0 can easily satisfy inflationary constraints, including the spectral index of scalar fluctuations (n_s = 0.96 ± 0.013), tensor-to-scalar ratio (r < 0.28) and also the bound imposed on Ω_ϕ during the nucleosynthesis epoch (Ω_ϕ(1 ∼ MeV) < 0.1). In our construction, the scalar field potential always scales proportionally to the square of the Hubble expansion rate. One may thereby account for the two vastly different energy scales associated with the Hubble parameters at early and late epochs. The inflaton energy could also produce an observationally significant effective dark energy at a late epoch without violating local gravity tests.

We analyze the relation between two a priori quite different expansions of the string equations of motion and constraints in a general curved background, namely one based on the covariant Penrose–Fermi expansion of the metric G_μν around a Penrose limit plane wave associated with a null geodesic γ and the other on the Riemann coordinate expansion in the exact metric G_μν of the string embedding variables around the null geodesic γ. Starting with the observation that there is a formal analogy between the exact string equations in a plane wave and the first-order string equations in a general background, we show that this analogy becomes exact provided that one chooses the background string configuration to be the null geodesic γ itself. We then explore the higher-order correspondence between these two expansions and find that for a general curved background they agree to all orders provided that one works in Fermi coordinates and in the lightcone gauge. Requiring moreover the conformal gauge restricts one to the usual class of (Brinkmann) backgrounds admitting simultaneously the lightcone and the conformal gauge, without further restrictions.

The classifications of holonomy groups in Lorentzian and in Euclidean signature are quite different. A group of interest in Lorentzian signature in n dimensions is the maximal proper subgroup of the Lorentz group, Sim(n − 2). Ricci-flat metrics with holonomy were constructed by Kerr and Goldberg, and a single four-dimensional example with a nonzero cosmological constant was exhibited by Ghanam and Thompson. Here we reduce the problem of finding the general n-dimensional Einstein metric of Sim(n − 2) holonomy, with and without a cosmological constant, to solving a set linear generalized Laplace and Poisson equations on an (n − 2)-dimensional Einstein base manifold. Explicit examples may be constructed in terms of generalized harmonic functions. A dimensional reduction of these multi-centre solutions gives new time-dependent Kaluza–Klein black holes and monopoles, including time-dependent black holes in a cosmological background whose spatial sections have non-vanishing curvature.

We investigated Dirac particles' Hawking radiation from the event horizon of the Kerr black hole in terms of the tunneling formalism. Applying the WKB approximation to the general covariant Dirac equation in the Kerr spacetime background, we obtain the tunneling probability for fermions and Hawking temperature of the Kerr black hole. The result obtained by taking the fermion tunneling into account is consistent with the previous literature.

We obtain a non-relativistic diffeomorphism invariant string action as a special limit of the Nambu–Goto action in a FLRW background. We use this action to study non-relativistic string dynamics in an expanding universe and construct an analytic model describing the macroscopic properties of non-relativistic string networks. The non-relativistic constraint equations allow arbitrarily small string velocities and thus a 'frustrated' equation of state for non-interacting strings can be obtained without the need of a velocity damping mechanism. Assuming that colliding string segments reconnect by the exchange of partners, non-relativistic string networks exhibit scaling behaviour, but with enhanced energy densities due to the smaller average string velocity. Non-relativistic string networks can be relevant in several contexts in condensed matter physics and cosmology.

Starting from Plebanski formulation of gravity as a constrained BF theory we propose a new spin foam model for 4D Riemannian quantum gravity that generalizes the well-known Barrett–Crane model and resolves the inherent to it ultra-locality problem. The BF formulation of 4D gravity possesses two sectors: gravitational and topological ones. The model presented here is shown to give a quantization of the gravitational sector, and is dual to the recently proposed spin foam model of Engle et al which, we show, corresponds to the topological sector. Our methods allow us to introduce the Immirzi parameter into the framework of spin foam quantization. We generalize some of our considerations to the Lorentzian setting and obtain a new spin foam model in that context as well.

This volume presents a comprehensive introduction to supersymmetry, concentrating mainly on the Minimal Supersymmetric Standard Model (MSSM) and its possible embedding in a grand unified theory, but also including material on supergravity, non-perturbative aspects of supersymmetry, string theory and cosmology. There is an excellent self-contained appendix on the standard model which could be read first; other appendices provide introductions to spinor representations of the Lorentz group, superfields, and cosmology, and there is a short appendix listing the MSSM renormalisation group beta-functions. The appendices in fact occupy over a quarter of the volume. Substantial knowledge of quantum field theory is required of the reader; and also a working knowledge of group theory as employed in the construction of particle physics models: while there is some useful material on this in the section on grand unification, an appendix on it might perhaps have been a useful addition.

Supersymmetry is introduced via the particle physicist's concern with the hierarchy problem and developed in the component formalism beginning with the Wess–Zumino model and proceeding to supersymmetric gauge theories. The treatment is detailed and authoritative; the author has 25 years of high-level research experience in the area and it shows. The level of presentation is high, and difficult concepts are explained clearly. The examples and associated hints are excellent. One topic I would have liked to see more on is the renormalisation of supersymmetric theories; presentation of the explicit calculation of the anomalous dimension of a chiral superfield (gamma) at one loop for at least the Wess–Zumino model might perhaps have been pedagogically useful. Associated, perhaps, with this omission is an inconsistency in the definition of gamma; the sign of gamma in the treatment in section 8.3.2 clearly differs from its sign in the appendix section E.3.

In the text the formalism of supersymmetry is developed mainly in terms of four-component Majorana spinors, while use of two-component spinors is relegated to the appendices, where the author valiantly presents results in a manner facilitating translation between different conventions, which is very useful. In the main text, however, the reader might even be led to feel that the Wess–Zumino model actions for a Majorana fermion on the one hand and a chiral fermion on the other are different theories, whereas in fact one can of course be rewritten as the other. I feel that the two-component formalism could have been profitably introduced at an earlier stage and used more consistently.

The book concludes with discussions of two 'challenges' to supersymmetry, as presented by the flavour problem and the cosmological constant. These are full of interesting and (indeed) challenging material, at the frontier of present-day research.

This is an excellent and up-to-date book, and very timely vis-à-vis the Large Hadron Collider. It would make an excellent graduate text for graduate students in particle theory. 'Road-maps' for high energy experimentalists and cosmologists are also provided; however for experimentalists a text with more in the way of, for example, explicit calculations of cross-sections would probably be more useful.

To summarise: minor quibbles aside, a fine book with a wealth of exciting material, and to be thoroughly recommended to any theoretically-inclined graduate student who has done a field theory course and wants to really get to grips with any aspect of supersymmetry.

There has been a flurry of books on quantum gravity in the past few years. The first edition of Kiefer's book appeared in 2004, about the same time as Carlo Rovelli's book with the same title. This was soon followed by Thomas Thiemann's 'Modern Canonical Quantum General Relativity'. Although the main focus of each of these books is non-perturbative and non-string approaches to the quantization of general relativity, they are quite orthogonal in temperament, style, subject matter and mathematical detail. Rovelli and Thiemann focus primarily on loop quantum gravity (LQG), whereas Kiefer attempts a broader introduction and review of the subject that includes chapters on string theory and decoherence.

Kiefer's second edition attempts an even wider and somewhat ambitious sweep with 'new sections on asymptotic safety, dynamical triangulation, primordial black holes, the information-loss problem, loop quantum cosmology, and other topics'. The presentation of these current topics is necessarily brief given the size of the book, but effective in encapsulating the main ideas in some cases. For instance the few pages devoted to loop quantum cosmology describe how the mini-superspace reduction of the quantum Hamiltonian constraint of LQG becomes a difference equation, whereas the discussion of 'dynamical triangulations', an approach to defining a discretized Lorentzian path integral for quantum gravity, is less detailed.

The first few chapters of the book provide, in a roughly historical sequence, the covariant and canonical metric variable approach to the subject developed in the 1960s and 70s. The problem(s) of time in quantum gravity are nicely summarized in the chapter on quantum geometrodynamics, followed by a detailed and effective introduction of the WKB approach and the semi-classical approximation. These topics form the traditional core of the subject.

The next three chapters cover LQG, quantization of black holes, and quantum cosmology. Of these the chapter on LQG is the shortest at fourteen pages—a reflection perhaps of the fact that there are two books and a few long reviews of the subject available written by the main protagonists in the field. The chapters on black holes and cosmology provide a more or less standard introduction to black hole thermodynamics, Hawking and Unruh radiation, quantization of the Schwarzschild metric and mini-superspace collapse models, and the DeWitt, Hartle–Hawking and Vilenkin wavefunctions.

The chapter on string theory is an essay-like overview of its quantum gravitational aspects. It provides a nice introduction to selected ideas and a guide to the literature. Here a prescient student may be left wondering why there is no quantum cosmology in string theory, perhaps a deliberate omission to avoid the 'landscape' and its fauna.

In summary, I think this book succeeds in its purpose of providing a broad introduction to quantum gravity, and nicely complements some of the other books on the subject.

The open problem of constructing a consistent and experimentally tested quantum theory of the gravitational field has its place at the heart of fundamental physics. The main approaches can be roughly divided into two classes: either one seeks a unified quantum framework of all interactions or one starts with a direct quantization of general relativity. In the first class, string theory (M-theory) is the only known example. In the second class, one can make an additional methodological distinction: while covariant approaches such as path-integral quantization use the four-dimensional metric as an essential ingredient of their formalism, canonical approaches start with a foliation of spacetime into spacelike hypersurfaces in order to arrive at a Hamiltonian formulation.

The present book is devoted to one of the canonical approaches—loop quantum gravity. It is named modern canonical quantum general relativity by the author because it uses connections and holonomies as central variables, which are analogous to the variables used in Yang–Mills theories. In fact, the canonically conjugate variables are a holonomy of a connection and the flux of a non-Abelian electric field. This has to be contrasted with the older geometrodynamical approach in which the metric of three-dimensional space and the second fundamental form are the fundamental entities, an approach which is still actively being pursued.

It is the author's ambition to present loop quantum gravity in a way in which every step is formulated in a mathematically rigorous form. In his own words: 'loop quantum gravity is an attempt to construct a mathematically rigorous, background-independent, non-perturbative quantum field theory of Lorentzian general relativity and all known matter in four spacetime dimensions, not more and not less'.

The formal Leitmotiv of loop quantum gravity is background independence. Non-gravitational theories are usually quantized on a given non-dynamical background. In contrast, due to the geometrical nature of gravity, no such background exists in quantum gravity. Instead, the notion of a background is supposed to emerge a posteriori as an approximate notion from quantum states of geometry. As a consequence, the standard ultraviolet divergences of quantum field theory do not show up because there is no limit of Δx → 0 to be taken in a given spacetime. On the other hand, it is open whether the theory is free of any type of divergences and anomalies.

A central feature of any canonical approach, independent of the choice of variables, is the existence of constraints. In geometrodynamics, these are the Hamiltonian and diffeomorphism constraints. They also hold in loop quantum gravity, but are supplemented there by the Gauss constraint, which emerges due to the use of triads in the formalism. These constraints capture all the physics of the quantum theory because no spacetime is present anymore (analogous to the absence of trajectories in quantum mechanics), so no additional equations of motion are needed. This book presents a careful and comprehensive discussion of these constraints. In particular, the constraint algebra is calculated in a transparent and explicit way.

The author makes the important assumption that a Hilbert-space structure is still needed on the fundamental level of quantum gravity. In ordinary quantum theory, such a structure is needed for the probability interpretation, in particular for the conservation of probability with respect to external time. It is thus interesting to see how far this concept can be extrapolated into the timeless realm of quantum gravity.

On the kinematical level, that is, before the constraints are imposed, an essentially unique Hilbert space can be constructed in terms of spin-network states. Potentially problematic features are the implementation of the diffeomorphism and Hamiltonian constraints. The Hilbert space H_diff defined on the diffeomorphism subspace can throw states out of the kinematical Hilbert space and is thus not contained in it. Moreover, the Hamiltonian constraint does not seem to preserve H_diff, so its implementation remains open. To avoid some of these problems, the author proposes his 'master constraint programme' in which the infinitely many local Hamiltonian constraints are combined into one master constraint. This is a subject of his current research.

With regard to this situation, it is not surprising that the main results in loop quantum gravity are found on the kinematical level. An especially important feature are the discrete spectra of geometric operators such as the area operator. This quantifies the earlier heuristic ideas about a discreteness at the Planck scale. The hope is that these results survive the consistent implementation of all constraints.

The status of loop quantum gravity is concisely and competently summarized in this volume, whose author is himself one of the pioneers of this approach. What is the relation of this book to the other monograph on loop quantum gravity, written by Carlo Rovelli and published in 2004 under the title Quantum Gravity with the same company? In the words of the present author: 'the two books are complementary in the sense that they can be regarded almost as volume I ('introduction and conceptual framework') and volume II ('mathematical framework and applications') of a general presentation of quantum general relativity in general and loop quantum gravity in particular'. In fact, the present volume gives a complete and self-contained presentation of the required mathematics, especially on the approximately 200 pages of chapters 18–33.

As for the physical applications, the main topic is the microscopic derivation of the black-hole entropy. This is presented in a clear and detailed form. Employing the concept of an isolated horizon (a local generalization of an event horizon), the counting of surface states gives an entropy proportional to the horizon area. It also contains the Barbero–Immirzi parameter β, which is a free parameter of the theory. Demanding, on the other hand, that the entropy be equal to the Bekenstein–Hawking entropy would fix this parameter. Other applications such as loop quantum cosmology are only briefly touched upon.

Since loop quantum gravity is a very active field of research, the author warns that the present book can at best be seen as a snapshot. Part of the overall picture may thus in the future be subject to modifications. For example, recent work by the author using a concept of dust time is not yet covered here. Nevertheless, I expect that this volume will continue to serve as a valuable introduction and reference book. It is essential reading for everyone working on loop quantum gravity.