There is something almost mystical about the connection between geometry and the laws of nature. An encounter at the seashore with the perfection of a logarithmic spiral in a seashell can be a source of wonder. A wonder which is multiplied when the self-same spirals appear in telescopic images of distant galaxies. That Nature is, at various levels, amenable to a geometrical understanding has been a rich theme in science. Though little more than a belief, it has consistently yielded beautiful insights. But it is perhaps fair to say that at no time in the past has geometry played as profound a role as it currently does in string theory's endeavours to understand quantum gravity. This should not be too surprising. Since the time when Einstein taught us that gravity is geometry, we have learnt to think of space-time not as a passive arena for events, but rather as a dynamical entity. Hence any theory of quantum gravity such as string theory must describe the quantum fluctuations of geometry. Much of the elusiveness of a consistent quantum theory of gravity has to do with this notion of describing quantum fluctuations of geometry. Recall that in the quantum mechanics of particles, instead of classical trajectories from point A to point B, one explores all possible paths from A to B, each weighted by some amplitude. Amplitudes for paths far from the classical one are suppressed if their action is large compared to Planck's constant (h) (note 1). In this regime, a nearly classical ('semi-classical') description can be applied. However, when the action is comparable to Planck's constant, then trajectories far from the classical contribute equally. Such quantum trajectories are by and large far from being smooth - they are more like the paths in Brownian motion (see Figure 1).
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