Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 293‘renaissance’, finding widespread application in astrophysics and cosmology,as well as novel vast experimental support – so that today GR is basicphysics needed for describing a variety of physical systems we have accessto, including advanced technological systems [Ashby (1997)].These two parallel developments have moved fundamental physics to aposition in which it has rarely been in the course of its history: We havetoday a group of fundamental laws, the Standard Model and GR, which–even if it cannot be regarded as a satisfactory global picture of Nature–is perhaps the best confirmed set of fundamental theories after Newton’suniversal gravitation and Maxwell’s electromagnetism. More importantly,there aren’t today experimental facts that openly challenge or escape thisset of fundamental laws. In this unprecedented state of affairs, a largenumber of theoretical physicists from different backgrounds have begun toaddress the piece of the puzzle which is clearly missing: combining thetwo halves of the picture and understanding the quantum properties ofthe gravitational field. Equivalently, understanding the quantum propertiesof space–time. Interest and researches in quantum gravity have thusincreased sharply in recent years. And the problem of understanding whatis a quantum space–time is today at the core of fundamental physics.Today we have some well developed and reasonably well defined tentativetheories of quantum gravity. String theory and loop quantum gravityare the two major examples. Within these theories definite physical resultshave been obtained, such as the explicit computation of the ‘quanta of geometry’and the derivation of the black hole entropy formula. Furthermore,a number of fresh new ideas, like noncommutative geometry, have enteredquantum gravity. For an overview of the problem of quantum gravity, see[Isham (1997)].3.10.4.2 Main Approaches to Quantum GravityString theoryString theory is by far the research direction which is presently mostinvestigated. String theory presently exists at two levels. First, there is awell developed set of techniques that define the string perturbation expansionover a given metric background. Second, the understanding ofthe non–perturbative aspects of the theory has much increased in recentyears [Polchinski (1995)] and in the string community there is a widespreadfaith, supported by numerous indications, in the existence of a yet-to-befoundfull non–perturbative theory, capable of generating the perturbation