Heller M, Woodin W.H. (eds.) Infinity. New research frontiers (CUP, 2011)(ISBN 1107003873)(O)(327s)_MAml_

infinite divisibility of space: quanta of space 169Loop gravity reaches precisely this conclusion. Indeed, using the equations of thistheory, it is possible to calculate with precision the size and the properties of theindividual quanta of space. For instance, we can compute the number of such “atomsof space” that exist within, say, a centimeter cube. Needless to say, this number is verylarge (of the order of a one followed by a hundred zeros). It is a very large number, butit is not an infinite number.The other tentative theories of quantum gravity arrive at similar conclusions. Forinstance, string theory was first formulated in a “background-dependent” manner,namely, it presupposes, from the very beginning, an arena for physical processes todevelop. However, the recent developments in the theory are all in the direction of thebackground-independent formulation, namely, a formulation in which such an arenais born out of more primitive elements. But even if we start on a fixed background,it turns out that if we attempt to probe arbitrarily small regions of space using stringtheory, we fail necessarily, because any test particle we may think of using for thispurpose would open up into an extended string when we try to confine it too much.In other words, we cannot test arbitrarily small spatial regions in string theory either.Thus, string theory leads to analogous indications that space is not infinitely divisible,although more indirectly.Other approaches such as noncommutative geometry and causal set theory are evenmore radical in this regard. They take the existence of a finite granularity of space asone of their basic assumptions. That is, whereas loop gravity and string theory derivethe granularity of space from our current basic knowledge of the physical world, theseother approaches take it as an input for the definitions of the theory. More precisely,the divisibility problem is in some sense transcended in noncommutative geometry,given that topology can be generalized to such an extent that local concepts becomemeaningless.Let us reflect on the meaning of this finite divisibility of space that appears in theresearch in quantum gravity. It is not difficult to imagine that space is not a continuousquantity. Space can be like a T-shirt, which is smooth and continuous if seen from adistance, but whose fabric reveals a small-scale discrete structure if observed close-up.Imagine billions and billions of extremely tiny chunks that are, themselves, indivisible.Such a replacement of a continuous structure with a discrete structure is typicalof the evolution of science in the last century. The most characteristic example is thediscovery of the atomic structure of matter. Water in a glass, or air in a room, appearsto us as an infinitely divisible continuum. We have learned in the last century that thisapparent continuum is actually an aggregate of discrete atoms, and we are today wellused to this idea. The amount of water in a glass is, ultimately, just the number of atomsin the glass.Similarly, space itself can be thought of as an ensemble of atoms of space, eachhaving finite volume. This should not be interpreted imagining that one individual atomof space has an extension, and this extension has a measure. Rather, the volume of aregion of space, namely, its very “extension,” is nothing else than the number of atomsof space in the region. Extension is our approximate conceptualization of the numberof quanta of space.This picture is not consistent with our common intuition about space. It is alsonot consistent with the Euclidean geometry that we use to formalize it. But this does