Quantum theory of geometry: I. Area operators
TL;DR: In this article, a functional calculus for quantum geometry is developed for a fully nonperturbative treatment of quantum gravity, which is used to begin a systematic construction of a quantum theory of geometry, and Regulated operators corresponding to 2-surfaces are introduced and shown to be self-adjoint on the underlying (kinematical) Hilbert space of states.
Abstract: 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.
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TL;DR: Loop quantum gravity as discussed by the authors is a background-independent, non-perturbative approach to the problem of unification of general relativity and quantum physics, based on a quantum theory of geometry.
Abstract: 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 bird's 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.
1,804 citations
TL;DR: In this article, an improved Hamiltonian constraint operator is introduced in loop quantum cosmology for the isotropic model with a massless scalar field and the big bang is replaced by a quantum bounce.
Abstract: An improved Hamiltonian constraint operator is introduced in loop quantum cosmology. Quantum dynamics of the spatially flat, isotropic model with a massless scalar field is then studied in detail using analytical and numerical methods. The scalar field continues to serve as ''emergent time'', the big bang is again replaced by a quantum bounce, and quantum evolution remains deterministic across the deep Planck regime. However, while with the Hamiltonian constraint used so far in loop quantum cosmology the quantum bounce can occur even at low matter densities, with the new Hamiltonian constraint it occurs only at a Planck-scale density. Thus, the new quantum dynamics retains the attractive features of current evolutions in loop quantum cosmology but, at the same time, cures their main weakness.
1,171 citations
Cites background from "Quantum theory of geometry: I. Area..."
...In quantum geometry, however, one can not continuously shrink the loop to zero area; there is a smallest non-zero area eigenvalue, or an ‘area gap’ ∆ [10, 11, 12]....
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TL;DR: A general overview of ideas, techniques, results and open problems of this candidate theory of quantum gravity is provided, and a guide to the relevant literature is provided.
Abstract: The problem of describing the quantum behavior of gravity, and thus understanding quantum spacetime, is still open. Loop quantum gravity is a well-developed approach to this problem. It is a mathematically well-defined background-independent quantization of general relativity, with its conventional matter couplings. Today research in loop quantum gravity forms a vast area, ranging from mathematical foundations to physical applications. Among the most significant results obtained so far are: (i) The computation of the spectra of geometrical quantities such as area and volume, which yield tentative quantitative predictions for Planck-scale physics. (ii) A physical picture of the microstructure of quantum spacetime, characterized by Planck-scale discreteness. Discreteness emerges as a standard quantum effect from the discrete spectra, and provides a mathematical realization of Wheeler’s “spacetime foam” intuition. (iii) Control of spacetime singularities, such as those in the interior of black holes and the cosmological one. This, in particular, has opened up the possibility of a theoretical investigation into the very early universe and the spacetime regions beyond the Big Bang. (iv) A derivation of the Bekenstein-Hawking black-hole entropy. (v) Low-energy calculations, yielding n-point functions well defined in a background-independent context. The theory is at the roots of, or strictly related to, a number of formalisms that have been developed for describing background-independent quantum field theory, such as spin foams, group field theory, causal spin networks, and others. I give here a general overview of ideas, techniques, results and open problems of this candidate theory of quantum gravity, and a guide to the relevant literature.
851 citations
Cites background from "Quantum theory of geometry: I. Area..."
...Ashtekar, Husain, Loll, Marolf, Rovelli, Samuel, Smolin, Lewandowski, Marolf, Thiemann....
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...The precise construction of this operator requires regularizing the classical expression and then taking the limit of a sequence of operators, in a suitable operator topology [268, 98, 121, 75, 31]....
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...Ashtekar and Lewandowski [174, 31] recover and complete the computation of the spectrum of the area using the connection representation and new regularization techniques....
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TL;DR: In this paper, an application of loop quantum cosmology to homogeneous systems, which removes classical singularities, is presented, where the main effects are introduced into effective classical equations, which allow one to avoid the interpretational problems of quantum theory.
Abstract: Quantum gravity is expected to be necessary in order to understand situations in which classical general relativity breaks down. In particular in cosmology one has to deal with initial singularities, i.e., the fact that the backward evolution of a classical spacetime inevitably comes to an end after a finite amount of proper time. This presents a breakdown of the classical picture and requires an extended theory for a meaningful description. Since small length scales and high curvatures are involved, quantum effects must play a role. Not only the singularity itself but also the surrounding spacetime is then modified. One particular theory is loop quantum cosmology, an application of loop quantum gravity to homogeneous systems, which removes classical singularities. Its implications can be studied at different levels. The main effects are introduced into effective classical equations, which allow one to avoid the interpretational problems of quantum theory. They give rise to new kinds of early-universe phenomenology with applications to inflation and cyclic models. To resolve classical singularities and to understand the structure of geometry around them, the quantum description is necessary. Classical evolution is then replaced by a difference equation for a wave function, which allows an extension of quantum spacetime beyond classical singularities. One main question is how these homogeneous scenarios are related to full loop quantum gravity, which can be dealt with at the level of distributional symmetric states. Finally, the new structure of spacetime arising in loop quantum gravity and its application to cosmology sheds light on more general issues, such as the nature of time.
811 citations
Cites background from "Quantum theory of geometry: I. Area..."
...Quite simple expressions exist for the area and volume operator [260, 21, 22], which are constructed solely from fluxes....
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...Moreover, the contribution from each intersection point can be seen to be analogous to an angular momentum operator in quantum mechanics, which has a discrete spectrum [21]....
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TL;DR: In this article, the precise mathematical structure underlying loop quantum cosmology and the sense in which it implements the full quantization program in a symmetry reduced model has been made explicit, thereby providing a firmer mathematical and conceptual foundation to the subject.
Abstract: Applications of Riemannian quantum geometry to cosmology have had notable successes. In particular, the fundamental discreteness underlying quantum geometry has led to a natural resolution of the big bang singularity. However, the precise mathematical structure underlying loop quantum cosmology and the sense in which it implements the full quantization program in a symmetry reduced model has not been made explicit. The purpose of this paper is to address these issues, thereby providing a firmer mathematical and conceptual foundation to the subject.
794 citations
Cites background from "Quantum theory of geometry: I. Area..."
...In the full theory, the configuration variables are constructed from holonomies he(A) associated with edges e and momentum variables, from E(S, f), momenta E smeared with test fields f on 2-surfaces [13,17,18,15]....
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References
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TL;DR: In this article, the authors studied the spectrum of the volume in nonperturbative quantum gravity, and showed that the spectrum can be computed by diagonalizing finite dimensional matrices, which can be seen as a generalization of the spin networks.
1,212 citations
TL;DR: An important feature of the new form of constraints is the natural embedding of the constraint surface of the Einstein phase space into that of Yang-Mills phase space, which provides new tools to analyze a number of issues in both classical and quantum gravity.
Abstract: The phase space of general relativity is first extended in a standard manner to incorporate spinors. New coordinates are then introduced on this enlarged phase space to simplify the structure of constraint equations. Now, the basic variables, satisfying the canonical Poisson-brackets relations, are the (density-valued) soldering forms \ensuremath{\sigma}\ifmmode \tilde{}\else \~{}\fi{} $^{a}$${\mathrm{}}_{A}$${\mathrm{}}^{B}$ and certain spin-connection one-forms ${A}_{\mathrm{aA}}$${\mathrm{}}^{B}$. Constraints of Einstein's theory simply state that \ensuremath{\sigma}\ifmmode \tilde{}\else \~{}\fi{} $^{a}$ satisfies the Gauss law constraint with respect to ${A}_{a}$ and that the curvature tensor ${F}_{\mathrm{abA}}$${\mathrm{}}^{B}$ and ${A}_{a}$ satisfies certain purely algebraic conditions (involving \ensuremath{\sigma}\ifmmode \tilde{}\else \~{}\fi{} $^{a}$). In particular, the constraints are at worst quadratic in the new variables \ensuremath{\sigma}\ifmmode \tilde{}\else \~{}\fi{} $^{a}$ and ${A}_{a}$. This is in striking contrast with the situation with traditional variables, where constraints contain nonpolynomial functions of the three-metric. Simplification occurs because ${A}_{a}$ has information about both the three-metric and its conjugate momentum. In the four-dimensional space-time picture, ${A}_{a}$ turns out to be a potential for the self-dual part of Weyl curvature. An important feature of the new form of constraints is that it provides a natural embedding of the constraint surface of the Einstein phase space into that of Yang-Mills phase space. This embedding provides new tools to analyze a number of issues in both classical and quantum gravity. Some illustrative applications are discussed. Finally, the (Poisson-bracket) algebra of new constraints is computed. The framework sets the stage for another approach to canonical quantum gravity, discussed in forthcoming papers also by Jacobson, Lee, Renteln, and Smolin.
973 citations
Book•
01 Jan 1991
TL;DR: In this article, the authors present an up-to-date account of a non-perturbative, canonical quantization program for gravity, which was highlighted in virtually every major conference in gravitational physics over the past three years.
Abstract: Notes prepared in Collaboration with Ranjeet S Tate It is now generally recognized that perturbative field theoretical methods that have been highly successful in the quantum description of non-gravitational interactions cannot be used as a means of constructing a quantum theory of gravity. The primary aim of the book is to present an up- to-date account of a non-perturbative, canonical quantization program for gravity. Many of the technical results obtained in the process are of interest also to differential geometry, classical general relativity and QCD. The program as a whole was highlighted in virtually every major conference in gravitational physics over the past three years.
915 citations
TL;DR: In this article, the authors define a new representation for quantum general relativity, in which exact solutions of the quantum constraints may be obtained, by means of a noncanonical graded Poisson algebra of classical observables, defined in terms of Ashtekar's new variables.
759 citations
TL;DR: In this article, a quantization of diffeomorphism invariant theories of connections is studied and the quantum diffeomorphicism constraint is solved and the space of solutions is equipped with an inner product that is shown to satisfy the physical reality conditions.
Abstract: 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.
707 citations