Ivancevic_Applied-Diff-Geom

Geometrical Path Integrals and Their Applications 1055Its components u i (x µ , y i , c s ) are given by the expression (6.87) where parameterfunctions ξ r (x) are replaced with the ghosts c r . The componentsv r and v r of the BRST operator ϑ can be derived from the conditionϑ(L ′ ) = −h rs M r q b sα j S j αc q − v r M r q c q + c r ϑ(ϑ(b rα j S j α)) = 0of the BRST invariance of L ′ . This condition falls into the two independentrelationsh rs M r q b sα j S j α + v r M r q = 0,ϑ(c q )(ϑ(c p )(b rα j S j α)) = u(c p )(u(c q )(b rα j S j α)) + u(v r )(b rα j S j α)= u( 1 2 cr pqc p c q + v r )(b rα j S j α) = 0.Hence, we get: v r = −h rs b sα j Sj α, and v r = − 1 2 cr pqc p c q .6.4 Sum over Geometries and TopologiesRecall that the term quantum gravity (or quantum geometrodynamics, orquantum geometry), is usually understood as a consistent fundamentalquantum description of gravitational space–time geometry whose classicallimit is Einstein’s general relativity. Among the possible ramificationsof such a theory are a model for the structure of space–time nearthe Planck scale, a consistent calculational scheme to calculate gravitationaleffects at all energies, a description of quantum geometry nearspace–time singularities and a non–perturbative quantum description of4D black holes. It might also help us in understanding cosmologicalissues about the beginning and end of the universe, i.e., the so–called‘big bang’ and ‘big crunch’ (see e.g., [Penrose (1989); Penrose (1994);Penrose (1997)]).From what we know about the quantum dynamics of other fundamentalinteractions it seems eminently plausible that also the gravitational excitationsshould at very short scales be governed by quantum laws. Now,conventional perturbative path integral expansions of gravity, as well as perturbativeexpansion in the string coupling in the case of unified approaches,both have difficulty in finding any direct or indirect evidence for quantumgravitational effects, be they experimental or observational, which couldgive a feedback for model building. The outstanding problems mentionedabove require a non–perturbative treatment; it is not sufficient to know thefirst few terms of a perturbation series. The real goal is to search for a