Ivancevic_Applied-Diff-Geom

704 Applied Differential Geometry: A Modern Introductionproperty of being dynamical with the rest of the physical entities, but also(more crucially) that space–time location is relational only. Quantum mechanicshas taught us that any dynamical entity is subject to Heisenberg’suncertainty at small scale. Thus, we need a relational notion of a quantumspace–time, in order to understand Planck scale physics. Thus, fora relativist, the problem of quantum gravity is the problem of bringing avast conceptual revolution, started with quantum mechanics and with generalrelativity, to a conclusion and to a new synthesis (see [Rovelli (1997);Smolin (1997)].) In this synthesis, the notions of space and time need to bedeeply reshaped, in order to keep into account what we have learned withboth our present ‘fundamental’ theories.Unlike perturbative or non–perturbative string theory, loop quantumgravity is formulated without a background space–time, and is thus a genuineattempt to grasp what is quantum space–time at the fundamentallevel. Accordingly, the notion of space–time that emerges from the theoryis profoundly different from the one on which conventional quantum fieldtheory or string theory are based.According to Rovelli, the main merit of string theory is that it provides asuperbly elegant unification of known fundamental physics, and that it hasa well defined perturbation expansion, finite order by order. Its main incompletenessis that its non–perturbative regime is poorly understood, andthat we do not have a background–independent formulation of the stringtheory. In a sense, we do not really know what is the theory we are talkingabout. Because of this poor understanding of the non perturbative regimeof the theory, Planck scale physics and genuine quantum gravitational phenomenaare not easily controlled: except for a few computations, there isnot much Planck scale physics derived from string theory so far. Thereare, however, two sets of remarkable physical results. The first is givenby some very high energy scattering amplitudes that have been computed.An intriguing aspect of these results is that they indirectly suggest thatgeometry below the Planck scale cannot be probed –and thus in a sensedoes not exist– in string theory. The second physical achievement of stringtheory (which followed the D–branes revolution) is the derivation of theBekenstein–Hawking black hole entropy formula for certain kinds of blackholes.On the other hand, the main merit of loop quantum gravity is thatit provides a well–defined and mathematically rigorous formulation of abackground–independent non–perturbative generally covariant quantumfield theory. The theory provides a physical picture and quantitative pre-