Quantum Geometry and Its Applications

In general relativity the gravitational field is encoded in the Riemannian geometry of space-time. Much of the conceptual compactness and mathematical elegance of the theory can be traced back to this central idea. The encoding is also directly responsible for the most dramatic ramifications of the theory: the big-bang, black holes and gravitational waves. However, it also leads one to the conclusion that space-time itself must end and physics must come to a halt at the big-bang and inside black holes, where the gravitational field becomes singular. But this reasoning ignores quantum physics entirely. When the curvature becomes large, of the order of 1/`Pl = c /G~, quantum effects dominate and predictions of general relativity can no longer be trusted. In this ‘Planck regime’, one must use an appropriate synthesis of general relativity and quantum physics, i.e., a quantum gravity theory. The predictions of this theory are likely to be quite different from those of general relativity. In the real, quantum world, evolution may be completely non-singular. Physics may not come to a halt and quantum theory could extend classical spacetime.