Quantum Gravity (Cambridge Monographs on Mathematical Physics)

Abstract

The most difficult unsolved problem in fundamental theoretical physicsis the consistent implementation of the gravitational interaction intoa quantum framework, which would lead to a theory of quantum gravity.Although a final answer is still pending, several promising attemptsdo exist. Despite the general title,this book is about one of them - loop quantum gravity.This approach proceeds from the idea that a direct quantizationof Einstein's theory of general relativity is possible.In contrast to string theory, it presupposes that the unification ofall interactions is not needed as a prerequisite for quantum gravity.Usually onedivides theories of quantum general relativity into covariant andcanonical approaches. Covariant theories employ four-dimensional conceptsin its formulation, one example being the path integral approach.Canonical theories start from a classical Hamiltonian version of thetheory in which spacetime is foliated into spacelike hypersurfaces.Loop quantum gravity is a variant of the canonical approach,the oldest being quantum geometrodynamics where the fundamentalconfiguration variable is the three-metric.Loop quantum gravity has developed from a new choice of canonical variablesintroduced by Abhay Ashtekar in 1986, the new configuration variable beinga connection defined on a three-manifold. Instead of the connection itself,the loop approach employs a non-local version in which the connectionis integrated over closed loops. This is similar to the Wilson loopsused in gauge theories. Carlo Rovelli is one of the pioneers ofloop quantum gravity which he started to develop with Lee Smolinin two papers written in 1988 and 1990. In his book, he presents acomprehensive and competent overview of this approach and provides atthe same time the necessary technical background in order to make thetreatment self-contained. In fact, half of the book is devoted to'preparations' giving a detailed account of Hamiltonian mechanics,quantum mechanics, general relativity and other topics.According to the level of the reader, this part canbe skipped or studied as interesting material on its own.The penetrating theme of the whole book (its leitmotiv)is background independence. In non-gravitational theories,dynamical fields are formulated on a fixed background spacetime that plays therole of an absolute structure in the theory. In general relativity, on theother hand, there is no background structure - all fields are dynamical.This was a confusing point already during the development of general relativityand led Albert Einstein in 1913 erroneously to give up general covariancebefore recognizing his error and presenting his final correct fieldequations that are of course covariant. This story is instructive,circling around the famous 'hole problem', and istold in detail in Rovelli's book. Its solution is that points ona bare manifold do not make sense in physics; everything, includingthe gravitational field, is dragged around by a diffeomorphism - there isjust no background available, only the fields exist. In loop quantum gravity,physical space (called 'quantum geometry') itself is formed byloop-like quantum states: a suitable orthonormal basis is providedby spin-network states (a spin-network is a graph with edges and nodes,where spins are assigned to the edges), and the quantum geometry is asuperposition of such states. Time and space in the usual sense havedisappeared.In the second half of his book, Rovelli discusses at length the majorsuccesses of this approach. First of all, the formalismyields a unique kinematical Hilbert space for the quantum statesobeying the Gauss and diffeomorphism constraints. The situation with theHamiltonian constraint is more subtle. The need for a Hilbert-space structurein quantum gravity is, however, not discussed. After all, the Hilbert-spacestructure in quantum mechanics is tied to the presence of an external timeand the conservation of probability with respect to this external time.But in quantum gravity there is no background structure, in particular noexternal time. Secondly, there exist two important operatorsthat are connected, respectively, with area and volume in the classical limit.These operators have a discrete spectrum and thus provide elementary'quanta' of area and volume. This gives a vague hint of a discrete structureat the Planck scale, about which there were speculations for manydecades.In spite of these promising results, loop quantum gravity is still faraway from a physical theory. This is also reflected in this volumewhere the technical treatment prevails and where physical applicationsare relegated to about 20 pages. These applications deal with quantumcosmology and black holes. The part on loop quantum cosmology summarizesbriefly recent results about a possible singularity avoidance anda new mechanism for inflation. These results are not derived fromloop quantum gravity but from imposing the discrete structure of thefull theory directly on the quantum cosmological models. The part onblack holes discusses the derivation of the Bekenstein-Hawking entropyfrom counting the number of relevant spin-network states. Sincethe theory contains a free parameter (the 'Barbero-Immirzi parameter'),the best one can do is to determine this parameter by demanding thatthe result be the Bekenstein-Hawking entropy. The book does not yetcontain the results of recent papers, published in 2004, that correctthe earlier entropy calculations presented here. From the new value ofthe Barbero-Immirzi parameter, the appealing connection withquasi-normal modes, as discussed in the book, may be lost.The book concludes with a brief discussion of the major open issues.Among these are the following: a well-defined and physically sensiblesemiclassical limit, the precise form of the Hamiltonian, the role ofunification (most of the work in loop quantum gravity deals only withpure gravity) and, last but not least, the issue of quantitative andtestable predictions. Whether loop quantum gravity will become a physicaltheory is not clear. Nor is this clear for string theory or any otherapproach. However, loop quantum gravity provides a fascinating lineof research and has much conceptual appeal. The present volumegives both an introduction and a review of this approach, making itsuitable for advanced students as well as experts. It is certainly ofinterest for the readers of Classical and Quantum Gravity.