Abhay Ashtekar

Background independent quantum gravity: A Status report

The goal of this review is to present an introduction to loop quantum gravity—a background-independent, non-perturbative approach to the problem of unification of general relativity and quantum physics, based on a quantum theory of geometry. Our presentation is pedagogical. Thus, in addition to providing a birds eye view of the present status of the subject, the review should also serve as a vehicle to enter the field and explore it in detail. To aid non-experts, very little is assumed beyond elements of general relativity, gauge theories and quantum field theory. While the review is essentially self-contained, the emphasis is on communicating the underlying ideas and the significance of results rather than on presenting systematic derivations and detailed proofs. (These can be found in the listed references.) The subject can be approached in different ways. We have chosen one which is deeply rooted in well-established physics and also has sufficient mathematical precision to ensure that there are no hidden infinities. In order to keep the review to a reasonable size, and to avoid overwhelming non-experts, we have had to leave out several interesting topics, results and viewpoints; this is meant to be an introduction to the subject rather than an exhaustive review of it.

Quantum theory of geometry: I. Area operators

A new functional calculus, developed recently for a fully non-perturbative treatment of quantum gravity, is used to begin a systematic construction of a quantum theory of geometry. Regulated operators corresponding to areas of 2-surfaces are introduced and shown to be self-adjoint on the underlying (kinematical) Hilbert space of states. It is shown that their spectra are purely discrete, indicating that the underlying quantum geometry is far from what the continuum picture might suggest. Indeed, the fundamental excitations of quantum geometry are one dimensional, rather like polymers, and the three-dimensional continuum geometry emerges only on coarse graining. The full Hilbert space admits an orthonormal decomposition into finite-dimensional subspaces which can be interpreted as the spaces of states of spin systems. Using this property, the complete spectrum of the area operators is evaluated. The general framework constructed here will be used in a subsequent paper to discuss three-dimensional geometric operators, e.g. the ones corresponding to volumes of regions.

Quantization of diffeomorphism invariant theories of connections with local degrees of freedom

Quantization of diffeomorphism invariant theories of connections is studied and the quantum diffeomorphism constraint is solved. The space of solutions is equipped with an inner product that is shown to satisfy the physical reality conditions. This provides, in particular, a quantization of the Husain–Kuchař model. The main results also pave the way to quantization of other diffeomorphism invariant theories such as general relativity. In the Riemannian case (i.e., signature ++++), the approach appears to contain all the necessary ingredients already. In the Lorentzian case, it will have to be combined in an appropriate fashion with a coherent state transform to incorporate complex connections.

Isolated and Dynamical Horizons and Their Applications

Over the past three decades, black holes have played an important role in quantum gravity, mathematical physics, numerical relativity and gravitational wave phenomenology. However, conceptual settings and mathematical models used to discuss them have varied considerably from one area to another. Over the last five years a new, quasi-local framework was introduced to analyze diverse facets of black holes in a unified manner. In this framework, evolving black holes are modelled by dynamical horizons and black holes in equilibrium by isolated horizons. We review basic properties of these horizons and summarize applications to mathematical physics, numerical relativity, and quantum gravity. This paradigm has led to significant generalizations of several results in black hole physics. Specifically, it has introduced a more physical setting for black hole thermodynamics and for black hole entropy calculations in quantum gravity, suggested a phenomenological model for hairy black holes, provided novel techniques to extract physics from numerical simulations, and led to new laws governing the dynamics of black holes in exact general relativity.

Projective techniques and functional integration for gauge theories

A general framework for integration over certain infinite dimensional spaces is first developed using projective limits of a projective family of compact Hausdorff spaces. The procedure is then applied to gauge theories to carry out integration over the non‐linear, infinite dimensional spaces of connections modulo gauge transformations. This method of evaluating functional integrals can be used either in the Euclidean path integral approach or the Lorentzian canonical approach. A number of measures discussed are diffeomorphism invariant and therefore of interest to (the connection dynamics version of) quantum general relativity. The account is pedagogical; in particular, prior knowledge of projective techniques is not assumed.

Weaving a classical metric with quantum threads

Results that illuminate the physical interpretation of states of nonperturbative quantum gravity are obtained using the recently introduced loop variables. It is shown that (i) while local operators such as the metric at a point may not be well defined, there do exist nonlocal operators, such as the area of a given two-surface, which can be regulated diffeomorphism invariantly and which are finite without renormalization; (ii) there exist quantum states which approximate a given metric at large scales, but such states exhibit a discrete structure at the Planck scale.

A unified treatment of null and spatial infinity in general relativity. I - Universal structure, asymptotic symmetries, and conserved quantities at spatial infinity

A new definition of asymptotic flatness in both null and spacelike directions is introduced. Notions relevant to the null regime are borrowed directly from Penrose’s definition of weak asymptotic simplicity. In the spatial regime, however, a new approach is adopted. The key feature of this approach is that it uses only those notions which refer to space–time as a whole, thereby avoiding the use of a initial value formulation, and, consequently, of a splitting of space–time into space and time. It is shown that the resulting description of asymptotic flatness not only encompasses the essential physical ideas behind the more familiar approaches based on the initial value formulation, but also succeeds in avoiding the global problems that usually arise. A certain 4‐manifold—called Spi (spatial infinity) —is constructed using well‐behaved, asymptotically geodesic, spacelike curves in the physical space–time. The structure of Spi is discussed in detail; in many ways, Spi turns out to be the spatial analog of I. The group of asymptotic symmetries at spatial infinity is examined. In its structure, this group turns out to be very similar to the BMS group. It is further shown that for the class of asymptotically flat space–times satisfying an additional condition on the (asymptotic behavior of the ’’magnetic’’ part of the) Weyl tensor, a Poincare (sub‐) group can be selected from the group of asymptotic symmetries in a canonical way. (This additional condition is rather weak: In essence, it requires only that the angular momentum contribution to the asymptotic curvature be of a higher order than the energy–momentum contribution.) Thus, for this (apparently large) class of space–times, the symmetry group at spatial infinity is just the Poincare group. Scalar, electromagnetic and gravitational fields are then considered, and their limiting behavior at spatial infinity is examined. In each case, the asymptotic field satisfies a simple, linear differential equation. Finally, conserved quantities are constructed using these asymptotic fields. Total charge and 4‐momentum are defined for arbitrary asymptotically flat space–times. These definitions agree with those in the literature, but have a further advantage of being both intrinsic and free of ambiguities which usually arise from global problems. A definition of angular momentum is then proposed for the class of space–times satisfying the additional condition on the (asymptotic behavior of the) Weyl tensor. This definition is intimately intertwined with the fact that, for this class of space–times, the group of asymptotic symmetries at spatial infinity is just the Poincare group; in particular, the definition is free of super‐translation ambiguities. It is shown that this angular momentum has the correct transformation properties. In the next paper, the formalism developed here will be seen to provide a platform for discussing in detail the relationship between the structure of the gravitational field at null infinity and that at spatial infinity.

Differential geometry on the space of connections via graphs and projective limits

Abstract In a quantum mechanical treatment of gauge theories (including general relativity), one is led to consider a certain completion A G of the space A G of guage equivalent connections. This space serves as the quantum configuration space, or, as the space of all Euclidean histories over which one must integrate in the quantum theory A G is a very large is a very large space and serves as a “universal home” for measures in theories in which the Wilson loop observables are well defined. In this paper, A G is considered as the projective limit of a projective family of compact Hausdorff manifolds, labelled by graphs (which can be regarded as “floating lattices” in the physics terminology). Using this characterization, differential geometry is developed through algebraic methods. In particular, we are able to introduce the following notions on A G : differential forms, exterio derivatives, volume forms, vector fields and Lie brackets between them, divergence of a vector field with respect to a volume form, Laplacians and associated heat kernels and heat kernel measures. Thus, although A G is very large, it is small enough to be mathematically interesting and physically useful. A key feature of this approach is that it does not require a background metric. The geometrical framework is therefore well suited for diffeomorphism invariant theories such as quantum general relativity.

Robustness of key features of loop quantum cosmology

A small simplification based on well-motivated approximations is shown to make loop quantum cosmology of the

Isolated horizons: Hamiltonian evolution and the first law

k=0