Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 303Second, this approach can be used as an approximate method for describingcertain regimes of non–perturbative quantum space–time physics,even if the fundamental dynamics is given by a more complete theory. Inthis spirit, Hawking and collaborators have continued the investigation ofphenomena such as, for instance, pair creation of black holes in a backgroundde Sitter space–time.Effective Perturbative Quantum GravityIf we expand classical GR around, say, the Minkowski metric,g µν (x) = η µν + h µν (x),and construct a conventional QFT for the field h µν (x), we get, as it is wellknow, a non renormalizable theory. A small but intriguing group of papershas recently appeared, based on the proposal of treating this perturbativetheory seriously, as a respectable low energy effective theory by its own.This cannot solve the deep problem of understanding the world in generalrelativistic quantum terms. But it can still be used for studying quantumproperties of space–time in some regimes. This view has been advocatedin a convincing way by John Donoghue, who has developed effective fieldtheory methods for extracting physics from non renormalizable quantumGR [Donoghue (1996)].QFT in Curved Space–TimeQuantum field theory in curved space–time is by now a reasonablyestablished theory (see, e.g., [Wald (1994); Birrel and Davies (1982);Fulling (1989)], predicting physical phenomena of remarkable interest suchas particle creation, vacuum polarization effects and Hawking’s black-holeradiance [Hawking (1975)]. To be sure, there is no direct nor indirect experimentalobservation of any of these phenomena, but the theory is quitecredible as an approximate theory, and many theorists in different fieldswould probably agree that these predicted phenomena are likely to be real.The most natural and general formulation of the theory is within thealgebraic approach [Haag (1992)], in which the primary objects are the localobservables and the states of interest may all be treated on equal footing(as positive linear functionals on the algebra of local observables), even ifthey do not belong to the same Hilbert space.The great merit of QFT on curved space–time is that it has providedus with some very important lessons. The key lesson is that in general